⚠️ Warning: This is a draft ⚠️

This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

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Determining if two triangles in the same plane overlap is an important topic in collision detection.

;Task: Determine which of these pairs of triangles overlap in 2D:

:::* (0,0),(5,0),(0,5) and (0,0),(5,0),(0,6) :::* (0,0),(0,5),(5,0) and (0,0),(0,5),(5,0) :::* (0,0),(5,0),(0,5) and (-10,0),(-5,0),(-1,6) :::* (0,0),(5,0),(2.5,5) and (0,4),(2.5,-1),(5,4) :::* (0,0),(1,1),(0,2) and (2,1),(3,0),(3,2) :::* (0,0),(1,1),(0,2) and (2,1),(3,-2),(3,4)

Optionally, see what the result is when only a single corner is in contact (there is no definitively correct answer): :::* (0,0),(1,0),(0,1) and (1,0),(2,0),(1,1)

PROCEDURE Main IS
TYPE Vertex IS MOD 3;
TYPE Point IS ARRAY (0 .. 1) OF Float;
TYPE Triangle IS ARRAY (Vertex) OF Point;
TYPE Triangle_Vertices IS ARRAY (0 .. 5) OF Float;

FUNCTION Same_Side (A, B, M, N : Point) RETURN Boolean IS
FUNCTION Aff (U : Point) RETURN Float IS
((B (1) - A (1)) * (U (0) - A (0)) + (A (0) - B (0)) * (U (1) - A (1)));
BEGIN
RETURN Aff (M) * Aff (N) >= 0.0;
END Same_Side;

FUNCTION In_Side (T1 , T2 : Triangle) RETURN Boolean IS
(FOR ALL V IN Vertex  =>
(FOR Some P OF T2 => Same_Side (T1 (V + 1), T1 (V + 2), T1 (V), P)));

FUNCTION Overlap (T1, T2 : Triangle) RETURN Boolean IS
(In_Side (T1, T2) AND THEN  In_Side (T2, T1));

FUNCTION "+" (T : Triangle_Vertices) RETURN Triangle IS
((T (0), T (1)), (T (2), T (3)), (T (4), T (5)));

PROCEDURE Put (T1, T2 : Triangle_Vertices) IS
BEGIN
Put_Line (Overlap (+T1, +T2)'Img);
END Put;

BEGIN
Put ((0.0, 0.0, 5.0, 0.0, 0.0, 5.0), (0.0, 0.0, 5.0, 0.0, 0.0, 6.0));
Put ((0.0, 0.0, 0.0, 5.0, 5.0, 0.0), (0.0, 0.0, 0.0, 5.0, 5.0, 0.0));
Put ((0.0, 0.0, 5.0, 0.0, 0.0, 5.0), (-10.0, 0.0, -5.0, 0.0, -1.0, 6.0));
Put ((0.0, 0.0, 5.0, 0.0, 2.5, 5.0), (0.0, 4.0, 2.5, -1.0, 5.0, 4.0));
Put ((0.0, 0.0, 1.0, 1.0, 0.0, 2.0), (2.0, 1.0, 3.0, 0.0, 3.0, 2.0));
Put ((0.0, 0.0, 1.0, 1.0, 0.0, 2.0), (2.0, 1.0, 3.0, -2.0, 3.0, 4.0));
Put ((0.0, 0.0, 1.0, 0.0, 0.0, 1.0), (1.0, 0.0, 2.0, 0.0, 1.0, 1.0));
END Main;

{{out}}

true
true
false
true
false
false
true

C

{{trans|C++}}

#include <errno.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>

typedef struct {
double x, y;
} Point;

double det2D(const Point * const p1, const Point * const p2, const Point * const p3) {
return p1->x * (p2->y - p3->y)
+ p2->x * (p3->y - p1->y)
+ p3->x * (p1->y - p2->y);
}

void checkTriWinding(Point * p1, Point * p2, Point * p3, bool allowReversed) {
double detTri = det2D(p1, p2, p3);
if (detTri < 0.0) {
if (allowReversed) {
double t = p3->x;
p3->x = p2->x;
p2->x = t;

t = p3->y;
p3->y = p2->y;
p2->y = t;
} else {
errno = 1;
}
}
}

bool boundaryCollideChk(const Point *p1, const Point *p2, const Point *p3, double eps) {
return det2D(p1, p2, p3) < eps;
}

bool boundaryDoesntCollideChk(const Point *p1, const Point *p2, const Point *p3, double eps) {
return det2D(p1, p2, p3) <= eps;
}

bool triTri2D(Point t1[], Point t2[], double eps, bool allowReversed, bool onBoundary) {
bool(*chkEdge)(Point*, Point*, Point*, double);
int i;

// Triangles must be expressed anti-clockwise
checkTriWinding(&t1, &t1, &t1, allowReversed);
if (errno != 0) {
return false;
}
checkTriWinding(&t2, &t2, &t2, allowReversed);
if (errno != 0) {
return false;
}

if (onBoundary) {
// Points on the boundary are considered as colliding
chkEdge = boundaryCollideChk;
} else {
// Points on the boundary are not considered as colliding
chkEdge = boundaryDoesntCollideChk;
}

//For edge E of trangle 1,
for (i = 0; i < 3; ++i) {
int j = (i + 1) % 3;

//Check all points of trangle 2 lay on the external side of the edge E. If
//they do, the triangles do not collide.
if (chkEdge(&t1[i], &t1[j], &t2, eps) &&
chkEdge(&t1[i], &t1[j], &t2, eps) &&
chkEdge(&t1[i], &t1[j], &t2, eps)) {
return false;
}
}

//For edge E of trangle 2,
for (i = 0; i < 3; i++) {
int j = (i + 1) % 3;

//Check all points of trangle 1 lay on the external side of the edge E. If
//they do, the triangles do not collide.
if (chkEdge(&t2[i], &t2[j], &t1, eps) &&
chkEdge(&t2[i], &t2[j], &t1, eps) &&
chkEdge(&t2[i], &t2[j], &t1, eps))
return false;
}

//The triangles collide
return true;
}

int main() {
{
Point t1[] = { {0, 0}, {5, 0}, {0, 5} };
Point t2[] = { {0, 0}, {5, 0}, {0, 6} };
printf("%d,true\n", triTri2D(t1, t2, 0.0, false, true));
}

{
Point t1[] = { {0, 0}, {0, 5}, {5, 0} };
Point t2[] = { {0, 0}, {0, 5}, {5, 0} };
printf("%d,true\n", triTri2D(t1, t2, 0.0, true, true));
}

{
Point t1[] = { {0, 0}, {5, 0}, {0, 5} };
Point t2[] = { {-10, 0}, {-5, 0}, {-1, 6} };
printf("%d,false\n", triTri2D(t1, t2, 0.0, false, true));
}

{
Point t1[] = { {0, 0}, {5, 0}, {2.5, 5} };
Point t2[] = { {0, 4}, {2.5, -1}, {5, 4} };
printf("%d,true\n", triTri2D(t1, t2, 0.0, false, true));
}

{
Point t1[] = { {0, 0}, {1, 1}, {0, 2} };
Point t2[] = { {2, 1}, {3, 0}, {3, 2} };
printf("%d,false\n", triTri2D(t1, t2, 0.0, false, true));
}

{
Point t1[] = { {0, 0}, {1, 1}, {0, 2} };
Point t2[] = { {2, 1}, {3, -2}, {3, 4} };
printf("%d,false\n", triTri2D(t1, t2, 0.0, false, true));
}

//Barely touching
{
Point t1[] = { {0, 0}, {1, 0}, {0, 1} };
Point t2[] = { {1, 0}, {2, 0}, {1, 1} };
printf("%d,true\n", triTri2D(t1, t2, 0.0, false, true));
}

//Barely touching
{
Point t1[] = { {0, 0}, {1, 0}, {0, 1} };
Point t2[] = { {1, 0}, {2, 0}, {1, 1} };
printf("%d,false\n", triTri2D(t1, t2, 0.0, false, false));
}

return EXIT_SUCCESS;
}

{{out}}

1,true
1,true
0,false
1,true
0,false
0,false
1,true
0,false

C++

#include <vector>
#include <iostream>
#include <stdexcept>
using namespace std;

typedef std::pair<double, double> TriPoint;

inline double Det2D(TriPoint &p1, TriPoint &p2, TriPoint &p3)
{
return +p1.first*(p2.second-p3.second)
+p2.first*(p3.second-p1.second)
+p3.first*(p1.second-p2.second);
}

void CheckTriWinding(TriPoint &p1, TriPoint &p2, TriPoint &p3, bool allowReversed)
{
double detTri = Det2D(p1, p2, p3);
if(detTri < 0.0)
{
if (allowReversed)
{
TriPoint a = p3;
p3 = p2;
p2 = a;
}
else throw std::runtime_error("triangle has wrong winding direction");
}
}

bool BoundaryCollideChk(TriPoint &p1, TriPoint &p2, TriPoint &p3, double eps)
{
return Det2D(p1, p2, p3) < eps;
}

bool BoundaryDoesntCollideChk(TriPoint &p1, TriPoint &p2, TriPoint &p3, double eps)
{
return Det2D(p1, p2, p3) <= eps;
}

bool TriTri2D(TriPoint *t1,
TriPoint *t2,
double eps = 0.0, bool allowReversed = false, bool onBoundary = true)
{
//Trangles must be expressed anti-clockwise
CheckTriWinding(t1, t1, t1, allowReversed);
CheckTriWinding(t2, t2, t2, allowReversed);

bool (*chkEdge)(TriPoint &, TriPoint &, TriPoint &, double) = NULL;
if(onBoundary) //Points on the boundary are considered as colliding
chkEdge = BoundaryCollideChk;
else //Points on the boundary are not considered as colliding
chkEdge = BoundaryDoesntCollideChk;

//For edge E of trangle 1,
for(int i=0; i<3; i++)
{
int j=(i+1)%3;

//Check all points of trangle 2 lay on the external side of the edge E. If
//they do, the triangles do not collide.
if (chkEdge(t1[i], t1[j], t2, eps) &&
chkEdge(t1[i], t1[j], t2, eps) &&
chkEdge(t1[i], t1[j], t2, eps))
return false;
}

//For edge E of trangle 2,
for(int i=0; i<3; i++)
{
int j=(i+1)%3;

//Check all points of trangle 1 lay on the external side of the edge E. If
//they do, the triangles do not collide.
if (chkEdge(t2[i], t2[j], t1, eps) &&
chkEdge(t2[i], t2[j], t1, eps) &&
chkEdge(t2[i], t2[j], t1, eps))
return false;
}

//The triangles collide
return true;
}

int main()
{
{TriPoint t1[] = {TriPoint(0,0),TriPoint(5,0),TriPoint(0,5)};
TriPoint t2[] = {TriPoint(0,0),TriPoint(5,0),TriPoint(0,6)};
cout << TriTri2D(t1, t2) << "," << true << endl;}

{TriPoint t1[] = {TriPoint(0,0),TriPoint(0,5),TriPoint(5,0)};
TriPoint t2[] = {TriPoint(0,0),TriPoint(0,5),TriPoint(5,0)};
cout << TriTri2D(t1, t2, 0.0, true) << "," << true << endl;}

{TriPoint t1[] = {TriPoint(0,0),TriPoint(5,0),TriPoint(0,5)};
TriPoint t2[] = {TriPoint(-10,0),TriPoint(-5,0),TriPoint(-1,6)};
cout << TriTri2D(t1, t2) << "," << false << endl;}

{TriPoint t1[] = {TriPoint(0,0),TriPoint(5,0),TriPoint(2.5,5)};
TriPoint t2[] = {TriPoint(0,4),TriPoint(2.5,-1),TriPoint(5,4)};
cout << TriTri2D(t1, t2) << "," << true << endl;}

{TriPoint t1[] = {TriPoint(0,0),TriPoint(1,1),TriPoint(0,2)};
TriPoint t2[] = {TriPoint(2,1),TriPoint(3,0),TriPoint(3,2)};
cout << TriTri2D(t1, t2) << "," << false << endl;}

{TriPoint t1[] = {TriPoint(0,0),TriPoint(1,1),TriPoint(0,2)};
TriPoint t2[] = {TriPoint(2,1),TriPoint(3,-2),TriPoint(3,4)};
cout << TriTri2D(t1, t2) << "," << false << endl;}

//Barely touching
{TriPoint t1[] = {TriPoint(0,0),TriPoint(1,0),TriPoint(0,1)};
TriPoint t2[] = {TriPoint(1,0),TriPoint(2,0),TriPoint(1,1)};
cout << TriTri2D(t1, t2, 0.0, false, true) << "," << true << endl;}

//Barely touching
{TriPoint t1[] = {TriPoint(0,0),TriPoint(1,0),TriPoint(0,1)};
TriPoint t2[] = {TriPoint(1,0),TriPoint(2,0),TriPoint(1,1)};
cout << TriTri2D(t1, t2, 0.0, false, false) << "," << false << endl;}

}

{{out}}

1,1
1,1
0,0
1,1
0,0
0,0
1,1
0,0

C#

using System;
using System.Collections.Generic;

namespace TriangleOverlap {
class Triangle {
public Tuple<double, double> P1 { get; set; }
public Tuple<double, double> P2 { get; set; }
public Tuple<double, double> P3 { get; set; }

public Triangle(Tuple<double, double> p1, Tuple<double, double> p2, Tuple<double, double> p3) {
P1 = p1;
P2 = p2;
P3 = p3;
}

public double Det2D() {
return P1.Item1 * (P2.Item2 - P3.Item2)
+ P2.Item1 * (P3.Item2 - P1.Item2)
+ P3.Item1 * (P3.Item1 - P2.Item2);
}

public void CheckTriWinding(bool allowReversed) {
var detTri = Det2D();
if (detTri < 0.0) {
if (allowReversed) {
var a = P3;
P3 = P2;
P2 = a;
} else {
throw new Exception("Triangle has wrong winding direction");
}
}
}

public bool BoundaryCollideChk(double eps) {
return Det2D() < eps;
}

public bool BoundaryDoesntCollideChk(double eps) {
return Det2D() <= eps;
}

public override string ToString() {
return string.Format("Triangle: {0}, {1}, {2}", P1, P2, P3);
}
}

class Program {
static bool BoundaryCollideChk(Triangle t, double eps) {
return t.BoundaryCollideChk(eps);
}

static bool BoundaryDoesntCollideChk(Triangle t, double eps) {
return t.BoundaryDoesntCollideChk(eps);
}

static bool TriTri2D(Triangle t1, Triangle t2, double eps = 0.0, bool allowReversed = false, bool onBoundary = true) {
// Triangles must be expressed anti-clockwise
t1.CheckTriWinding(allowReversed);
t2.CheckTriWinding(allowReversed);
// 'onBoundary' determines whether points on boundary are considered as colliding or not
var chkEdge = onBoundary
? (Func<Triangle, double, bool>)BoundaryCollideChk
: BoundaryDoesntCollideChk;
List<Tuple<double, double>> lp1 = new List<Tuple<double, double>>() { t1.P1, t1.P2, t1.P3 };
List<Tuple<double, double>> lp2 = new List<Tuple<double, double>>() { t2.P1, t2.P2, t2.P3 };

// for each edge E of t1
for (int i = 0; i < 3; i++) {
var j = (i + 1) % 3;
// Check all points of t2 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge(new Triangle(lp1[i], lp1[j], lp2), eps) &&
chkEdge(new Triangle(lp1[i], lp1[j], lp2), eps) &&
chkEdge(new Triangle(lp1[i], lp1[j], lp2), eps)) {
return false;
}
}

// for each edge E of t2
for (int i = 0; i < 3; i++) {
var j = (i + 1) % 3;
// Check all points of t1 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge(new Triangle(lp2[i], lp2[j], lp1), eps) &&
chkEdge(new Triangle(lp2[i], lp2[j], lp1), eps) &&
chkEdge(new Triangle(lp2[i], lp2[j], lp1), eps)) {
return false;
}
}

// The triangles overlap
return true;
}

static void Overlap(Triangle t1, Triangle t2, double eps = 0.0, bool allowReversed = false, bool onBoundary = true) {
if (TriTri2D(t1, t2, eps, allowReversed, onBoundary)) {
Console.WriteLine("overlap");
} else {
Console.WriteLine("do not overlap");
}
}

static void Main(string[] args) {
var t1 = new Triangle(new Tuple<double, double>(0.0, 0.0), new Tuple<double, double>(5.0, 0.0), new Tuple<double, double>(0.0, 5.0));
var t2 = new Triangle(new Tuple<double, double>(0.0, 0.0), new Tuple<double, double>(5.0, 0.0), new Tuple<double, double>(0.0, 6.0));
Console.WriteLine("{0} and\n{1}", t1, t2);
Overlap(t1, t2);
Console.WriteLine();

// need to allow reversed for this pair to avoid exception
t1 = new Triangle(new Tuple<double, double>(0.0, 0.0), new Tuple<double, double>(0.0, 5.0), new Tuple<double, double>(5.0, 0.0));
t2 = t1;
Console.WriteLine("{0} and\n{1}", t1, t2);
Overlap(t1, t2, 0.0, true);
Console.WriteLine();

t1 = new Triangle(new Tuple<double, double>(0.0, 0.0), new Tuple<double, double>(5.0, 0.0), new Tuple<double, double>(0.0, 5.0));
t2 = new Triangle(new Tuple<double, double>(-10.0, 0.0), new Tuple<double, double>(-5.0, 0.0), new Tuple<double, double>(-1.0, 6.0));
Console.WriteLine("{0} and\n{1}", t1, t2);
Overlap(t1, t2);
Console.WriteLine();

t1.P3 = new Tuple<double, double>(2.5, 5.0);
t2 = new Triangle(new Tuple<double, double>(0.0, 4.0), new Tuple<double, double>(2.5, -1.0), new Tuple<double, double>(5.0, 4.0));
Console.WriteLine("{0} and\n{1}", t1, t2);
Overlap(t1, t2);
Console.WriteLine();

t1 = new Triangle(new Tuple<double, double>(0.0, 0.0), new Tuple<double, double>(1.0, 1.0), new Tuple<double, double>(0.0, 2.0));
t2 = new Triangle(new Tuple<double, double>(2.0, 1.0), new Tuple<double, double>(3.0, 0.0), new Tuple<double, double>(3.0, 2.0));
Console.WriteLine("{0} and\n{1}", t1, t2);
Overlap(t1, t2);
Console.WriteLine();

t2 = new Triangle(new Tuple<double, double>(2.0, 1.0), new Tuple<double, double>(3.0, -2.0), new Tuple<double, double>(3.0, 4.0));
Console.WriteLine("{0} and\n{1}", t1, t2);
Overlap(t1, t2);
Console.WriteLine();

t1 = new Triangle(new Tuple<double, double>(0.0, 0.0), new Tuple<double, double>(1.0, 0.0), new Tuple<double, double>(0.0, 1.0));
t2 = new Triangle(new Tuple<double, double>(1.0, 0.0), new Tuple<double, double>(2.0, 0.0), new Tuple<double, double>(1.0, 1.1));
Console.WriteLine("{0} and\n{1}", t1, t2);
Console.WriteLine("which have only a single corner in contact, if boundary points collide");
Overlap(t1, t2);
Console.WriteLine();

Console.WriteLine("{0} and\n{1}", t1, t2);
Console.WriteLine("which have only a single corner in contact, if boundary points do not collide");
Overlap(t1, t2, 0.0, false, false);
}
}
}

{{out}}

Triangle: (0, 0), (5, 0), (0, 5) and
Triangle: (0, 0), (5, 0), (0, 6)
overlap

Triangle: (0, 0), (0, 5), (5, 0) and
Triangle: (0, 0), (0, 5), (5, 0)
overlap

Triangle: (0, 0), (5, 0), (0, 5) and
Triangle: (-10, 0), (-5, 0), (-1, 6)
do not overlap

Triangle: (0, 0), (5, 0), (2.5, 5) and
Triangle: (0, 4), (2.5, -1), (5, 4)
overlap

Triangle: (0, 0), (1, 1), (0, 2) and
Triangle: (2, 1), (3, 0), (3, 2)
do not overlap

Triangle: (0, 0), (1, 1), (0, 2) and
Triangle: (2, 1), (3, -2), (3, 4)
do not overlap

Triangle: (0, 0), (1, 0), (0, 1) and
Triangle: (1, 0), (2, 0), (1, 1.1)
which have only a single corner in contact, if boundary points collide
do not overlap

Triangle: (0, 0), (1, 0), (0, 1) and
Triangle: (1, 0), (2, 0), (1, 1.1)
which have only a single corner in contact, if boundary points do not collide
do not overlap

D

{{trans|Kotlin}}

import std.stdio;
import std.typecons;

alias Pair = Tuple!(real, real);

struct Triangle {
Pair p1;
Pair p2;
Pair p3;

void toString(scope void delegate(const(char)[]) sink) const {
import std.format;
sink("Triangle: ");
formattedWrite!"%s"(sink, p1);
sink(", ");
formattedWrite!"%s"(sink, p2);
sink(", ");
formattedWrite!"%s"(sink, p3);
}
}

auto det2D(Triangle t) {
return t.p1 *(t.p2 - t.p3)
+ t.p2 *(t.p3 - t.p1)
+ t.p3 *(t.p1 - t.p2);
}

void checkTriWinding(Triangle t, bool allowReversed) {
auto detTri = t.det2D();
if (detTri < 0.0) {
if (allowReversed) {
auto a = t.p3;
t.p3 = t.p2;
t.p2 = a;
} else {
throw new Exception("Triangle has wrong winding direction");
}
}
}

auto boundaryCollideChk(Triangle t, real eps) {
return t.det2D() < eps;
}

auto boundaryDoesntCollideChk(Triangle t, real eps) {
return t.det2D() <= eps;
}

bool triTri2D(Triangle t1, Triangle t2, real eps = 0.0, bool allowReversed = false, bool onBoundary = true) {
// Triangles must be expressed anti-clockwise
checkTriWinding(t1, allowReversed);
checkTriWinding(t2, allowReversed);
// 'onBoundary' determines whether points on boundary are considered as colliding or not
auto chkEdge = onBoundary ? &boundaryCollideChk : &boundaryDoesntCollideChk;
auto lp1 = [t1.p1, t1.p2, t1.p3];
auto lp2 = [t2.p1, t2.p2, t2.p3];

// for each edge E of t1
foreach (i; 0..3) {
auto j = (i + 1) % 3;
// Check all points of t2 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge(Triangle(lp1[i], lp1[j], lp2), eps) &&
chkEdge(Triangle(lp1[i], lp1[j], lp2), eps) &&
chkEdge(Triangle(lp1[i], lp1[j], lp2), eps)) {
return false;
}
}

// for each edge E of t2
foreach (i; 0..3) {
auto j = (i + 1) % 3;
// Check all points of t1 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge(Triangle(lp2[i], lp2[j], lp1), eps) &&
chkEdge(Triangle(lp2[i], lp2[j], lp1), eps) &&
chkEdge(Triangle(lp2[i], lp2[j], lp1), eps)) {
return false;
}
}

// The triangles overlap
return true;
}

void overlap(Triangle t1, Triangle t2, real eps = 0.0, bool allowReversed = false, bool onBoundary = true) {
if (triTri2D(t1, t2, eps, allowReversed, onBoundary)) {
writeln("overlap");
} else {
writeln("do not overlap");
}
}

void main() {
auto t1 = Triangle(Pair(0.0, 0.0), Pair(5.0, 0.0), Pair(0.0, 5.0));
auto t2 = Triangle(Pair(0.0, 0.0), Pair(5.0, 0.0), Pair(0.0, 6.0));
writeln(t1, " and\n", t2);
overlap(t1, t2);
writeln;

// need to allow reversed for this pair to avoid exception
t1 = Triangle(Pair(0.0, 0.0), Pair(0.0, 5.0), Pair(5.0, 0.0));
t2 = t1;
writeln(t1, " and\n", t2);
overlap(t1, t2, 0.0, true);
writeln;

t1 = Triangle(Pair(0.0, 0.0), Pair(5.0, 0.0), Pair(0.0, 5.0));
t2 = Triangle(Pair(-10.0, 0.0), Pair(-5.0, 0.0), Pair(-1.0, 6.0));
writeln(t1, " and\n", t2);
overlap(t1, t2);
writeln;

t1.p3 = Pair(2.5, 5.0);
t2 = Triangle(Pair(0.0, 4.0), Pair(2.5, -1.0), Pair(5.0, 4.0));
writeln(t1, " and\n", t2);
overlap(t1, t2);
writeln;

t1 = Triangle(Pair(0.0, 0.0), Pair(1.0, 1.0), Pair(0.0, 2.0));
t2 = Triangle(Pair(2.0, 1.0), Pair(3.0, 0.0), Pair(3.0, 2.0));
writeln(t1, " and\n", t2);
overlap(t1, t2);
writeln;

t2 = Triangle(Pair(2.0, 1.0), Pair(3.0, -2.0), Pair(3.0, 4.0));
writeln(t1, " and\n", t2);
overlap(t1, t2);
writeln;

t1 = Triangle(Pair(0.0, 0.0), Pair(1.0, 0.0), Pair(0.0, 1.0));
t2 = Triangle(Pair(1.0, 0.0), Pair(2.0, 0.0), Pair(1.0, 1.1));
writeln(t1, " and\n", t2);
writeln("which have only a single corner in contact, if boundary points collide");
overlap(t1, t2);
writeln;

writeln(t1, " and\n", t2);
writeln("which have only a single corner in contact, if boundary points do not collide");
overlap(t1, t2, 0.0, false, false);
}

{{out}}

Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(5, 0), Tuple!(real, real)(0, 5) and
Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(5, 0), Tuple!(real, real)(0, 6)
overlap

Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(0, 5), Tuple!(real, real)(5, 0) and
Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(0, 5), Tuple!(real, real)(5, 0)
overlap

Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(5, 0), Tuple!(real, real)(0, 5) and
Triangle: Tuple!(real, real)(-10, 0), Tuple!(real, real)(-5, 0), Tuple!(real, real)(-1, 6)
do not overlap

Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(5, 0), Tuple!(real, real)(2.5, 5) and
Triangle: Tuple!(real, real)(0, 4), Tuple!(real, real)(2.5, -1), Tuple!(real, real)(5, 4)
overlap

Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(1, 1), Tuple!(real, real)(0, 2) and
Triangle: Tuple!(real, real)(2, 1), Tuple!(real, real)(3, 0), Tuple!(real, real)(3, 2)
do not overlap

Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(1, 1), Tuple!(real, real)(0, 2) and
Triangle: Tuple!(real, real)(2, 1), Tuple!(real, real)(3, -2), Tuple!(real, real)(3, 4)
do not overlap

Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(1, 0), Tuple!(real, real)(0, 1) and
Triangle: Tuple!(real, real)(1, 0), Tuple!(real, real)(2, 0), Tuple!(real, real)(1, 1.1)
which have only a single corner in contact, if boundary points collide
overlap

Triangle: Tuple!(real, real)(0, 0), Tuple!(real, real)(1, 0), Tuple!(real, real)(0, 1) and
Triangle: Tuple!(real, real)(1, 0), Tuple!(real, real)(2, 0), Tuple!(real, real)(1, 1.1)
which have only a single corner in contact, if boundary points do not collide
do not overlap

open System

type Point = double * double
type Triangle = Point * Point * Point

let Det2D (t:Triangle) =
let (p1, p2, p3) = t
let (p1x, p1y) = p1
let (p2x, p2y) = p2
let (p3x, p3y) = p3

p1x * (p2y - p3y) +
p2x * (p3y - p1y) +
p3x * (p1y - p2y)

let CheckTriWinding allowReversed t =
let detTri = Det2D t
if detTri < 0.0 then
if allowReversed then
let (p1, p2, p3) = t
(p1, p3, p2)
else
raise (Exception "Triangle has wrong winding direction")
else
t

let boundaryCollideChk eps t =
(Det2D t) < eps

let boundaryDoesntCollideChk eps t =
(Det2D t) <= eps

let TriTri2D eps allowReversed onBoundary t1 t2 =
// Triangles must be expressed anti-clockwise
let t3 = CheckTriWinding allowReversed t1
let t4 = CheckTriWinding allowReversed t2

// 'onBoundary' determines whether points on boundary are considered as colliding or not
let chkEdge = if onBoundary then boundaryCollideChk else boundaryDoesntCollideChk
let (t1p1, t1p2, t1p3) = t3
let (t2p1, t2p2, t2p3) = t4

// Check all points of t2 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge eps (t1p1, t1p2, t2p1)) && (chkEdge eps (t1p1, t1p2, t2p2)) && (chkEdge eps (t1p1, t1p2, t2p3)) then
false
else if (chkEdge eps (t1p2, t1p3, t2p1)) && (chkEdge eps (t1p2, t1p3, t2p2)) && (chkEdge eps (t1p2, t1p3, t2p3)) then
false
else if (chkEdge eps (t1p3, t1p1, t2p1)) && (chkEdge eps (t1p3, t1p1, t2p2)) && (chkEdge eps (t1p3, t1p1, t2p3)) then
false

// Check all points of t1 lay on the external side of edge E.
// If they do, the triangles do not overlap.
else if (chkEdge eps (t2p1, t2p2, t1p1)) && (chkEdge eps (t2p1, t2p2, t1p2)) && (chkEdge eps (t2p1, t2p2, t1p3)) then
false
else if (chkEdge eps (t2p2, t2p3, t1p1)) && (chkEdge eps (t2p2, t2p3, t1p2)) && (chkEdge eps (t2p2, t2p3, t1p3)) then
false
else if (chkEdge eps (t2p3, t2p1, t1p1)) && (chkEdge eps (t2p3, t2p1, t1p2)) && (chkEdge eps (t2p3, t2p1, t1p3)) then
false

else
// The triangles overlap
true

let Print t1 t2 =
Console.WriteLine("{0} and\n{1}\n{2}\n", t1, t2, if TriTri2D 0.0 false true t1 t2 then "overlap" else "do not overlap")

[<EntryPoint>]
let main _ =
let t1 = ((0.0, 0.0), (5.0, 0.0), (0.0, 5.0))
let t2 = ((0.0, 0.0), (5.0, 0.0), (0.0, 6.0))
Print t1 t2

let t3 = ((0.0, 0.0), (0.0, 5.0), (5.0, 0.0))
Console.WriteLine("{0} and\n{1}\n{2}\n", t3, t3, if TriTri2D 0.0 true true t3 t3 then "overlap (reversed)" else "do not overlap")

let t4 = ((0.0, 0.0), (5.0, 0.0), (0.0, 5.0))
let t5 = ((-10.0, 0.0), (-5.0, 0.0), (-1.0, 6.0))
Print t4 t5

let t6 = ((0.0, 0.0), (5.0, 0.0), (2.5, 5.0))
let t7 = ((0.0, 4.0), (2.5, -1.0), (5.0, 4.0))
Print t6 t7

let t8 = ((0.0, 0.0), (1.0, 1.0), (0.0, 2.0))
let t9 = ((2.0, 1.0), (3.0, 0.0), (3.0, 2.0))
Print t8 t9

let t10 = ((2.0, 1.0), (3.0, -2.0), (3.0, 4.0))
Print t8 t10

let t11 = ((0.0, 0.0), (1.0, 0.0), (0.0, 1.0))
let t12 = ((1.0, 0.0), (2.0, 0.0), (1.0, 1.1))
printfn "The following triangles which have only a single corner in contact, if boundary points collide"
Print t11 t12

Console.WriteLine("{0} and\n{1}\nwhich have only a single corner in contact, if boundary points do not collide\n{2}", t11, t12, if TriTri2D 0.0 false false t11 t12 then "overlap" else "do not overlap")

0 // return an integer exit code

{{out}}

((0, 0), (5, 0), (0, 5)) and
((0, 0), (5, 0), (0, 6))
overlap

((0, 0), (0, 5), (5, 0)) and
((0, 0), (0, 5), (5, 0))
overlap (reversed)

((0, 0), (5, 0), (0, 5)) and
((-10, 0), (-5, 0), (-1, 6))
do not overlap

((0, 0), (5, 0), (2.5, 5)) and
((0, 4), (2.5, -1), (5, 4))
overlap

((0, 0), (1, 1), (0, 2)) and
((2, 1), (3, 0), (3, 2))
do not overlap

((0, 0), (1, 1), (0, 2)) and
((2, 1), (3, -2), (3, 4))
do not overlap

The following triangles which have only a single corner in contact, if boundary points collide
((0, 0), (1, 0), (0, 1)) and
((1, 0), (2, 0), (1, 1.1))
overlap

((0, 0), (1, 0), (0, 1)) and
((1, 0), (2, 0), (1, 1.1))
which have only a single corner in contact, if boundary points do not collide
do not overlap

Go

{{trans|Kotlin}}

package main

import "fmt"

type point struct {
x, y float64
}

func (p point) String() string {
return fmt.Sprintf("(%.1f, %.1f)", p.x, p.y)
}

type triangle struct {
p1, p2, p3 point
}

func (t *triangle) String() string {
return fmt.Sprintf("Triangle %s, %s, %s", t.p1, t.p2, t.p3)
}

func (t *triangle) det2D() float64 {
return t.p1.x * (t.p2.y - t.p3.y) +
t.p2.x * (t.p3.y - t.p1.y) +
t.p3.x * (t.p1.y - t.p2.y)
}

func (t *triangle) checkTriWinding(allowReversed bool) {
detTri := t.det2D()
if detTri < 0.0 {
if allowReversed {
a := t.p3
t.p3 = t.p2
t.p2 = a
} else {
panic("Triangle has wrong winding direction.")
}
}
}

func boundaryCollideChk(t *triangle, eps float64) bool {
return t.det2D() < eps
}

func boundaryDoesntCollideChk(t *triangle, eps float64) bool {
return t.det2D() <= eps
}

func triTri2D(t1, t2 *triangle, eps float64, allowReversed, onBoundary bool) bool {
// Triangles must be expressed anti-clockwise.
t1.checkTriWinding(allowReversed)
t2.checkTriWinding(allowReversed)

// 'onBoundary' determines whether points on boundary are considered as colliding or not.
var chkEdge func (*triangle, float64) bool
if onBoundary {
chkEdge = boundaryCollideChk
} else {
chkEdge = boundaryDoesntCollideChk
}
lp1 := point{t1.p1, t1.p2, t1.p3}
lp2 := point{t2.p1, t2.p2, t2.p3}

// for each edge E of t1
for i := 0; i < 3; i++ {
j := (i + 1) % 3
// Check all points of t2 lay on the external side of edge E.
// If they do, the triangles do not overlap.
tri1 := &triangle{lp1[i], lp1[j], lp2}
tri2 := &triangle{lp1[i], lp1[j], lp2}
tri3 := &triangle{lp1[i], lp1[j], lp2}
if chkEdge(tri1, eps) && chkEdge(tri2, eps) && chkEdge(tri3, eps) {
return false
}
}

// for each edge E of t2
for i := 0; i < 3; i++ {
j := (i + 1) % 3
// Check all points of t1 lay on the external side of edge E.
// If they do, the triangles do not overlap.
tri1 := &triangle{lp2[i], lp2[j], lp1}
tri2 := &triangle{lp2[i], lp2[j], lp1}
tri3 := &triangle{lp2[i], lp2[j], lp1}
if chkEdge(tri1, eps) && chkEdge(tri2, eps) && chkEdge(tri3, eps) {
return false
}
}

// The triangles overlap.
return true
}

func iff(cond bool, s1, s2 string) string {
if cond {
return s1
}
return s2
}

func main() {
t1 := &triangle{point{0.0, 0.0}, point{5.0, 0.0}, point{0.0, 5.0}}
t2 := &triangle{point{0.0, 0.0}, point{5.0, 0.0}, point{0.0, 6.0}}
fmt.Printf("%s and\n%s\n", t1, t2)
overlapping := triTri2D(t1, t2, 0.0, false, true)
fmt.Println(iff(overlapping, "overlap", "do not overlap"))

// Need to allow reversed for this pair to avoid panic.
t1 = &triangle{point{0.0, 0.0}, point{0.0, 5.0}, point{5.0, 0.0}}
t2 = t1
fmt.Printf("\n%s and\n%s\n", t1, t2)
overlapping = triTri2D(t1, t2, 0.0, true, true)
fmt.Println(iff(overlapping, "overlap (reversed)", "do not overlap"))

t1 = &triangle{point{0.0, 0.0}, point{5.0, 0.0}, point{0.0, 5.0}}
t2 = &triangle{point{-10.0, 0.0}, point{-5.0, 0.0}, point{-1.0, 6.0}}
fmt.Printf("\n%s and\n%s\n", t1, t2)
overlapping = triTri2D(t1, t2, 0.0, false, true)
fmt.Println(iff(overlapping, "overlap", "do not overlap"))

t1.p3 = point{2.5, 5.0}
t2 = &triangle{point{0.0, 4.0}, point{2.5, -1.0}, point{5.0, 4.0}}
fmt.Printf("\n%s and\n%s\n", t1, t2)
overlapping = triTri2D(t1, t2, 0.0, false, true)
fmt.Println(iff(overlapping, "overlap", "do not overlap"))

t1 = &triangle{point{0.0, 0.0}, point{1.0, 1.0}, point{0.0, 2.0}}
t2 = &triangle{point{2.0, 1.0}, point{3.0, 0.0}, point{3.0, 2.0}}
fmt.Printf("\n%s and\n%s\n", t1, t2)
overlapping = triTri2D(t1, t2, 0.0, false, true)
fmt.Println(iff(overlapping, "overlap", "do not overlap"))

t2 = &triangle{point{2.0, 1.0}, point{3.0, -2.0}, point{3.0, 4.0}}
fmt.Printf("\n%s and\n%s\n", t1, t2)
overlapping = triTri2D(t1, t2, 0.0, false, true)
fmt.Println(iff(overlapping, "overlap", "do not overlap"))

t1 = &triangle{point{0.0, 0.0}, point{1.0, 0.0}, point{0.0, 1.0}}
t2 = &triangle{point{1.0, 0.0}, point{2.0, 0.0}, point{1.0, 1.1}}
fmt.Printf("\n%s and\n%s\n", t1, t2)
println("which have only a single corner in contact, if boundary points collide")
overlapping = triTri2D(t1, t2, 0.0, false, true)
fmt.Println(iff(overlapping, "overlap", "do not overlap"))

fmt.Printf("\n%s and\n%s\n", t1, t2)
fmt.Println("which have only a single corner in contact, if boundary points do not collide")
overlapping = triTri2D(t1, t2, 0.0, false, false)
fmt.Println(iff(overlapping, "overlap", "do not overlap"))
}

{{out}}

Same as Kotlin entry.

Java

{{trans|Kotlin}} {{works with|Java|8}}

import java.util.function.BiFunction;

public class TriangleOverlap {
private static class Pair {
double first;
double second;

Pair(double first, double second) {
this.first = first;
this.second = second;
}

@Override
public String toString() {
return String.format("(%s, %s)", first, second);
}
}

private static class Triangle {
Pair p1, p2, p3;

Triangle(Pair p1, Pair p2, Pair p3) {
this.p1 = p1;
this.p2 = p2;
this.p3 = p3;
}

@Override
public String toString() {
return String.format("Triangle: %s, %s, %s", p1, p2, p3);
}
}

private static double det2D(Triangle t) {
Pair p1 = t.p1;
Pair p2 = t.p2;
Pair p3 = t.p3;
return p1.first * (p2.second - p3.second)
+ p2.first * (p3.second - p1.second)
+ p3.first * (p1.second - p2.second);
}

private static void checkTriWinding(Triangle t, boolean allowReversed) {
double detTri = det2D(t);
if (detTri < 0.0) {
if (allowReversed) {
Pair a = t.p3;
t.p3 = t.p2;
t.p2 = a;
} else throw new RuntimeException("Triangle has wrong winding direction");
}
}

private static boolean boundaryCollideChk(Triangle t, double eps) {
return det2D(t) < eps;
}

private static boolean boundaryDoesntCollideChk(Triangle t, double eps) {
return det2D(t) <= eps;
}

private static boolean triTri2D(Triangle t1, Triangle t2) {
return triTri2D(t1, t2, 0.0, false, true);
}

private static boolean triTri2D(Triangle t1, Triangle t2, double eps, boolean allowedReversed) {
return triTri2D(t1, t2, eps, allowedReversed, true);
}

private static boolean triTri2D(Triangle t1, Triangle t2, double eps, boolean allowedReversed, boolean onBoundary) {
// Triangles must be expressed anti-clockwise
checkTriWinding(t1, allowedReversed);
checkTriWinding(t2, allowedReversed);
// 'onBoundary' determines whether points on boundary are considered as colliding or not
BiFunction<Triangle, Double, Boolean> chkEdge = onBoundary ? TriangleOverlap::boundaryCollideChk : TriangleOverlap::boundaryDoesntCollideChk;
Pair[] lp1 = new Pair[]{t1.p1, t1.p2, t1.p3};
Pair[] lp2 = new Pair[]{t2.p1, t2.p2, t2.p3};

// for each edge E of t1
for (int i = 0; i < 3; ++i) {
int j = (i + 1) % 3;
// Check all points of t2 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge.apply(new Triangle(lp1[i], lp1[j], lp2), eps) &&
chkEdge.apply(new Triangle(lp1[i], lp1[j], lp2), eps) &&
chkEdge.apply(new Triangle(lp1[i], lp1[j], lp2), eps)) return false;
}

// for each edge E of t2
for (int i = 0; i < 3; ++i) {
int j = (i + 1) % 3;
// Check all points of t1 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge.apply(new Triangle(lp2[i], lp2[j], lp1), eps) &&
chkEdge.apply(new Triangle(lp2[i], lp2[j], lp1), eps) &&
chkEdge.apply(new Triangle(lp2[i], lp2[j], lp1), eps)) return false;
}

// The triangles overlap
return true;
}

public static void main(String[] args) {
Triangle t1 = new Triangle(new Pair(0.0, 0.0), new Pair(5.0, 0.0), new Pair(0.0, 5.0));
Triangle t2 = new Triangle(new Pair(0.0, 0.0), new Pair(5.0, 0.0), new Pair(0.0, 6.0));
System.out.printf("%s and\n%s\n", t1, t2);
if (triTri2D(t1, t2)) {
System.out.println("overlap");
} else {
System.out.println("do not overlap");
}

// need to allow reversed for this pair to avoid exception
t1 = new Triangle(new Pair(0.0, 0.0), new Pair(0.0, 5.0), new Pair(5.0, 0.0));
t2 = t1;
System.out.printf("\n%s and\n%s\n", t1, t2);
if (triTri2D(t1, t2, 0.0, true)) {
System.out.println("overlap (reversed)");
} else {
System.out.println("do not overlap");
}

t1 = new Triangle(new Pair(0.0, 0.0), new Pair(5.0, 0.0), new Pair(0.0, 5.0));
t2 = new Triangle(new Pair(-10.0, 0.0), new Pair(-5.0, 0.0), new Pair(-1.0, 6.0));
System.out.printf("\n%s and\n%s\n", t1, t2);
if (triTri2D(t1, t2)) {
System.out.println("overlap");
} else {
System.out.println("do not overlap");
}

t1.p3 = new Pair(2.5, 5.0);
t2 = new Triangle(new Pair(0.0, 4.0), new Pair(2.5, -1.0), new Pair(5.0, 4.0));
System.out.printf("\n%s and\n%s\n", t1, t2);
if (triTri2D(t1, t2)) {
System.out.println("overlap");
} else {
System.out.println("do not overlap");
}

t1 = new Triangle(new Pair(0.0, 0.0), new Pair(1.0, 1.0), new Pair(0.0, 2.0));
t2 = new Triangle(new Pair(2.0, 1.0), new Pair(3.0, 0.0), new Pair(3.0, 2.0));
System.out.printf("\n%s and\n%s\n", t1, t2);
if (triTri2D(t1, t2)) {
System.out.println("overlap");
} else {
System.out.println("do not overlap");
}

t2 = new Triangle(new Pair(2.0, 1.0), new Pair(3.0, -2.0), new Pair(3.0, 4.0));
System.out.printf("\n%s and\n%s\n", t1, t2);
if (triTri2D(t1, t2)) {
System.out.println("overlap");
} else {
System.out.println("do not overlap");
}

t1 = new Triangle(new Pair(0.0, 0.0), new Pair(1.0, 0.0), new Pair(0.0, 1.0));
t2 = new Triangle(new Pair(1.0, 0.0), new Pair(2.0, 0.0), new Pair(1.0, 1.1));
System.out.printf("\n%s and\n%s\n", t1, t2);
System.out.println("which have only a single corner in contact, if boundary points collide");
if (triTri2D(t1, t2)) {
System.out.println("overlap");
} else {
System.out.println("do not overlap");
}

System.out.printf("\n%s and\n%s\n", t1, t2);
System.out.println("which have only a single corner in contact, if boundary points do not collide");
if (triTri2D(t1, t2, 0.0, false, false)) {
System.out.println("overlap");
} else {
System.out.println("do not overlap");
}
}
}

{{out}}

Triangle: (0.0, 0.0), (5.0, 0.0), (0.0, 5.0) and
Triangle: (0.0, 0.0), (5.0, 0.0), (0.0, 6.0)
overlap

Triangle: (0.0, 0.0), (0.0, 5.0), (5.0, 0.0) and
Triangle: (0.0, 0.0), (0.0, 5.0), (5.0, 0.0)
overlap (reversed)

Triangle: (0.0, 0.0), (5.0, 0.0), (0.0, 5.0) and
Triangle: (-10.0, 0.0), (-5.0, 0.0), (-1.0, 6.0)
do not overlap

Triangle: (0.0, 0.0), (5.0, 0.0), (2.5, 5.0) and
Triangle: (0.0, 4.0), (2.5, -1.0), (5.0, 4.0)
overlap

Triangle: (0.0, 0.0), (1.0, 1.0), (0.0, 2.0) and
Triangle: (2.0, 1.0), (3.0, 0.0), (3.0, 2.0)
do not overlap

Triangle: (0.0, 0.0), (1.0, 1.0), (0.0, 2.0) and
Triangle: (2.0, 1.0), (3.0, -2.0), (3.0, 4.0)
do not overlap

Triangle: (0.0, 0.0), (1.0, 0.0), (0.0, 1.0) and
Triangle: (1.0, 0.0), (2.0, 0.0), (1.0, 1.1)
which have only a single corner in contact, if boundary points collide
overlap

Triangle: (0.0, 0.0), (1.0, 0.0), (0.0, 1.0) and
Triangle: (1.0, 0.0), (2.0, 0.0), (1.0, 1.1)
which have only a single corner in contact, if boundary points do not collide
do not overlap

Julia

{{trans|Python}} '''Module''':

module Triangles

using LinearAlgebra

export AntiClockwise, Both, StrictCheck, MildCheck

abstract type Widing end
struct AntiClockwise <: Widing end
struct Both          <: Widing end

function _check_triangle_winding(t, widing::AntiClockwise)
trisq = fill!(Matrix{eltype(t)}(undef, 3, 3), 1)
trisq[:, 1:2] .= t
det(trisq) < 0 && throw(ArgumentError("triangle has wrong winding direction"))
return trisq
end

function _check_triangle_winding(t, widing::Both)
trisq = fill!(Matrix{eltype(t)}(undef, 3, 3), 1)
trisq[:, 1:2] .= t
if det(trisq) < 0
tmp = trisq[2, :]
trisq[2, :] .= trisq[1, :]
trisq[1, :] .= tmp
end
return trisq
end

abstract type OnBoundaryCheck end
struct StrictCheck <: OnBoundaryCheck end
struct MildCheck   <: OnBoundaryCheck end

_checkedge(::StrictCheck, x, ϵ) = det(x) < ϵ
_checkedge(::MildCheck, x, ϵ)   = det(x) ϵ

function overlap(T₁, T₂, onboundary::OnBoundaryCheck=MildCheck(),; ϵ=0.0, widing::Widing=AntiClockwise())
# Trangles must be expressed anti-clockwise
T₁ = _check_triangle_winding(T₁, widing)
T₂ = _check_triangle_winding(T₂, widing)

edge = similar(T₁)
for (A, B) in ((T₁, T₂), (T₂, T₁)), i in 1:3
circshift!(edge, A, (i, 0))
@views if all(_checkedge(onboundary, vcat(edge[1:2, :], B[r, :]'), ϵ) for r in 1:3)
return false
end
end

return true
end

end  # module Triangles

'''Main''':

using .Triangles

t1 = [0 0; 5 0; 0 5]
t2 = [0 0; 5 0; 0 6]
@show Triangles.overlap(t1, t2)

t1 = [0 0; 0 5; 5 0]
t2 = [0 0; 0 6; 5 0]
@show Triangles.overlap(t1, t2, widing=Both())

t1 = [0 0; 5 0; 0 5]
t2 = [-10 0; -5 0; -1 6]
@show Triangles.overlap(t1, t2)

t1 = [0 0; 5 0; 2.5 5]
t2 = [0 4; 2.5 -1; 5 4]
@show Triangles.overlap(t1, t2)

t1 = [0 0; 1 1; 0 2]
t2 = [2 1; 3 0; 3 2]
@show Triangles.overlap(t1, t2)

t1 = [0 0; 1 1; 0 2]
t2 = [2 1; 3 -2; 3 4]
@show Triangles.overlap(t1, t2)

# Barely touching
t1 = [0 0; 1 0; 0 1]
t2 = [1 0; 2 0; 1 1]
@show Triangles.overlap(t1, t2, StrictCheck())

# Barely touching
t1 = [0 0; 1 0; 0 1]
t2 = [1 0; 2 0; 1 1]
@show Triangles.overlap(t1, t2, MildCheck())

{{out}}

Triangles.overlap(t1, t2) = true
Triangles.overlap(t1, t2, widing=Both()) = true
Triangles.overlap(t1, t2) = false
Triangles.overlap(t1, t2) = true
Triangles.overlap(t1, t2) = false
Triangles.overlap(t1, t2) = false
Triangles.overlap(t1, t2, StrictCheck()) = true
Triangles.overlap(t1, t2, MildCheck()) = false

Kotlin

{{trans|C++}}

// version 1.1.0

typealias Point = Pair<Double, Double>

data class Triangle(var p1: Point, var p2: Point, var p3: Point) {
override fun toString() = "Triangle: \$p1, \$p2, \$p3"
}

fun det2D(t: Triangle): Double {
val (p1, p2, p3) = t
return  p1.first * (p2.second - p3.second) +
p2.first * (p3.second - p1.second) +
p3.first * (p1.second - p2.second)
}

fun checkTriWinding(t: Triangle, allowReversed: Boolean) {
val detTri = det2D(t)
if (detTri < 0.0) {
if (allowReversed) {
val a = t.p3
t.p3  = t.p2
t.p2 =  a
}
else throw RuntimeException("Triangle has wrong winding direction")
}
}

fun boundaryCollideChk(t: Triangle, eps: Double) = det2D(t) < eps

fun boundaryDoesntCollideChk(t: Triangle, eps: Double) = det2D(t) <= eps

fun triTri2D(t1: Triangle, t2: Triangle, eps: Double = 0.0,
allowReversed: Boolean = false, onBoundary: Boolean = true): Boolean {
// Triangles must be expressed anti-clockwise
checkTriWinding(t1, allowReversed)
checkTriWinding(t2, allowReversed)
// 'onBoundary' determines whether points on boundary are considered as colliding or not
val chkEdge = if (onBoundary) ::boundaryCollideChk else ::boundaryDoesntCollideChk
val lp1 = listOf(t1.p1, t1.p2, t1.p3)
val lp2 = listOf(t2.p1, t2.p2, t2.p3)

// for each edge E of t1
for (i in 0 until 3) {
val j = (i + 1) % 3
// Check all points of t2 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge(Triangle(lp1[i], lp1[j], lp2), eps) &&
chkEdge(Triangle(lp1[i], lp1[j], lp2), eps) &&
chkEdge(Triangle(lp1[i], lp1[j], lp2), eps)) return false
}

// for each edge E of t2
for (i in 0 until 3) {
val j = (i + 1) % 3
// Check all points of t1 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge(Triangle(lp2[i], lp2[j], lp1), eps) &&
chkEdge(Triangle(lp2[i], lp2[j], lp1), eps) &&
chkEdge(Triangle(lp2[i], lp2[j], lp1), eps)) return false
}

// The triangles overlap
return true
}

fun main(args: Array<String>) {
var t1 = Triangle(0.0 to 0.0, 5.0 to 0.0, 0.0 to 5.0)
var t2 = Triangle(0.0 to 0.0, 5.0 to 0.0, 0.0 to 6.0)
println("\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

// need to allow reversed for this pair to avoid exception
t1 = Triangle(0.0 to 0.0, 0.0 to 5.0, 5.0 to 0.0)
t2 = t1
println("\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2, 0.0, true)) "overlap (reversed)" else "do not overlap")

t1 = Triangle(0.0 to 0.0, 5.0 to 0.0, 0.0 to 5.0)
t2 = Triangle(-10.0 to 0.0, -5.0 to 0.0, -1.0 to 6.0)
println("\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

t1.p3 = 2.5 to 5.0
t2 = Triangle(0.0 to 4.0, 2.5 to -1.0, 5.0 to 4.0)
println("\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

t1 = Triangle(0.0 to 0.0, 1.0 to 1.0, 0.0 to 2.0)
t2 = Triangle(2.0 to 1.0, 3.0 to 0.0, 3.0 to 2.0)
println("\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

t2 = Triangle(2.0 to 1.0, 3.0 to -2.0, 3.0 to 4.0)
println("\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

t1 = Triangle(0.0 to 0.0, 1.0 to 0.0, 0.0 to 1.0)
t2 = Triangle(1.0 to 0.0, 2.0 to 0.0, 1.0 to 1.1)
println("\n\$t1 and\n\$t2")
println("which have only a single corner in contact, if boundary points collide")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

println("\n\$t1 and\n\$t2")
println("which have only a single corner in contact, if boundary points do not collide")
println(if (triTri2D(t1, t2, 0.0, false, false)) "overlap" else "do not overlap")
}

{{out}}

Triangle: (0.0, 0.0), (5.0, 0.0), (0.0, 5.0) and
Triangle: (0.0, 0.0), (5.0, 0.0), (0.0, 6.0)
overlap

Triangle: (0.0, 0.0), (0.0, 5.0), (5.0, 0.0) and
Triangle: (0.0, 0.0), (0.0, 5.0), (5.0, 0.0)
overlap (reversed)

Triangle: (0.0, 0.0), (5.0, 0.0), (0.0, 5.0) and
Triangle: (-10.0, 0.0), (-5.0, 0.0), (-1.0, 6.0)
do not overlap

Triangle: (0.0, 0.0), (5.0, 0.0), (2.5, 5.0) and
Triangle: (0.0, 4.0), (2.5, -1.0), (5.0, 4.0)
overlap

Triangle: (0.0, 0.0), (1.0, 1.0), (0.0, 2.0) and
Triangle: (2.0, 1.0), (3.0, 0.0), (3.0, 2.0)
do not overlap

Triangle: (0.0, 0.0), (1.0, 1.0), (0.0, 2.0) and
Triangle: (2.0, 1.0), (3.0, -2.0), (3.0, 4.0)
do not overlap

Triangle: (0.0, 0.0), (1.0, 0.0), (0.0, 1.0) and
Triangle: (1.0, 0.0), (2.0, 0.0), (1.0, 1.1)
which have only a single corner in contact, if boundary points collide
overlap

Triangle: (0.0, 0.0), (1.0, 0.0), (0.0, 1.0) and
Triangle: (1.0, 0.0), (2.0, 0.0), (1.0, 1.1)
which have only a single corner in contact, if boundary points do not collide
do not overlap

Lua

{{trans|C++}}

function det2D(p1,p2,p3)
return p1.x * (p2.y - p3.y)
+ p2.x * (p3.y - p1.y)
+ p3.x * (p1.y - p2.y)
end

function checkTriWinding(p1,p2,p3,allowReversed)
local detTri = det2D(p1,p2,p3)
if detTri < 0.0 then
if allowReversed then
local t = p3
p3 = p2
p2 = t
else
error("triangle has wrong winding direction")
end
end
return nil
end

function boundaryCollideChk(p1,p2,p3,eps)
return det2D(p1,p2,p3) < eps
end

function boundaryDoesntCollideChk(p1,p2,p3,eps)
return det2D(p1,p2,p3) <= eps
end

function triTri2D(t1,t2,eps,allowReversed,onBoundary)
eps = eps or 0.0
allowReversed = allowReversed or false
onBoundary = onBoundary or true

-- triangles must be expressed anti-clockwise
checkTriWinding(t1, t1, t1, allowReversed)
checkTriWinding(t2, t2, t2, allowReversed)

local chkEdge
if onBoundary then
-- points on the boundary are considered as colliding
chkEdge = boundaryCollideChk
else
-- points on the boundary are not considered as colliding
chkEdge = boundaryDoesntCollideChk
end

-- for edge E of triangle 1
for i=0,2 do
local j = (i+1)%3

-- check all points of triangle 2 lay on the external side of the edge E.
-- If they do, the triangles do not collide
if chkEdge(t1[i+1], t1[j+1], t2, eps) and
chkEdge(t1[i+1], t1[j+1], t2, eps) and
chkEdge(t1[i+1], t1[j+1], t2, eps) then
return false
end
end

-- for edge E of triangle 2
for i=0,2 do
local j = (i+1)%3

-- check all points of triangle 1 lay on the external side of the edge E.
-- If they do, the triangles do not collide
if chkEdge(t2[i+1], t2[j+1], t1, eps) and
chkEdge(t2[i+1], t2[j+1], t1, eps) and
chkEdge(t2[i+1], t2[j+1], t1, eps) then
return false
end
end

-- the triangles collide
return true
end

function formatTri(t)
return "Triangle: ("..t.x..", "..t.y
.."), ("..t.x..", "..t.y
.."), ("..t.x..", "..t.y..")"
end

function overlap(t1,t2,eps,allowReversed,onBoundary)
if triTri2D(t1,t2,eps,allowReversed,onBoundary) then
return "overlap\n"
else
return "do not overlap\n"
end
end

-- Main
local t1 = {{x=0,y=0},{x=5,y=0},{x=0,y=5}}
local t2 = {{x=0,y=0},{x=5,y=0},{x=0,y=6}}
print(formatTri(t1).." and")
print(formatTri(t2))
print(overlap(t1,t2))

t1 = {{x=0,y=0},{x=0,y=5},{x=5,y=0}}
t2 = {{x=0,y=0},{x=0,y=5},{x=5,y=0}}
print(formatTri(t1).." and")
print(formatTri(t2))
print(overlap(t1,t2,0.0,true))

t1 = {{x=0,y=0},{x=5,y=0},{x=0,y=5}}
t2 = {{x=-10,y=0},{x=-5,y=0},{x=-1,y=6}}
print(formatTri(t1).." and")
print(formatTri(t2))
print(overlap(t1,t2))

t1 = {{x=0,y=0},{x=5,y=0},{x=2.5,y=5}}
t2 = {{x=0,y=4},{x=2.5,y=-1},{x=5,y=4}}
print(formatTri(t1).." and")
print(formatTri(t2))
print(overlap(t1,t2))

t1 = {{x=0,y=0},{x=1,y=1},{x=0,y=2}}
t2 = {{x=2,y=1},{x=3,y=0},{x=3,y=2}}
print(formatTri(t1).." and")
print(formatTri(t2))
print(overlap(t1,t2))

t1 = {{x=0,y=0},{x=1,y=1},{x=0,y=2}}
t2 = {{x=2,y=1},{x=3,y=-2},{x=3,y=4}}
print(formatTri(t1).." and")
print(formatTri(t2))
print(overlap(t1,t2))

-- Barely touching
t1 = {{x=0,y=0},{x=1,y=0},{x=0,y=1}}
t2 = {{x=1,y=0},{x=2,y=0},{x=1,y=1}}
print(formatTri(t1).." and")
print(formatTri(t2))
print(overlap(t1,t2,0.0,false,true))

-- Barely touching
local t1 = {{x=0,y=0},{x=1,y=0},{x=0,y=1}}
local t2 = {{x=1,y=0},{x=2,y=0},{x=1,y=1}}
print(formatTri(t1).." and")
print(formatTri(t2))
print(overlap(t1,t2,0.0,false,false))

{{out}}

Triangle: (0, 0), (5, 0), (0, 5) and
Triangle: (0, 0), (5, 0), (0, 6)
overlap

Triangle: (0, 0), (0, 5), (5, 0) and
Triangle: (0, 0), (0, 5), (5, 0)
overlap

Triangle: (0, 0), (5, 0), (0, 5) and
Triangle: (-10, 0), (-5, 0), (-1, 6)
do not overlap

Triangle: (0, 0), (5, 0), (2.5, 5) and
Triangle: (0, 4), (2.5, -1), (5, 4)
overlap

Triangle: (0, 0), (1, 1), (0, 2) and
Triangle: (2, 1), (3, 0), (3, 2)
do not overlap

Triangle: (0, 0), (1, 1), (0, 2) and
Triangle: (2, 1), (3, -2), (3, 4)
do not overlap

Triangle: (0, 0), (1, 0), (0, 1) and
Triangle: (1, 0), (2, 0), (1, 1)
overlap

Triangle: (0, 0), (1, 0), (0, 1) and
Triangle: (1, 0), (2, 0), (1, 1)
overlap

MODULE Overlap;
FROM EXCEPTIONS IMPORT AllocateSource,ExceptionSource,GetMessage,RAISE;
FROM LongStr IMPORT RealToFixed;

TYPE
Point = RECORD
x,y : LONGREAL;
END;
Triangle = RECORD
p1,p2,p3 : Point;
END;

VAR
TextWinExSrc : ExceptionSource;

PROCEDURE WritePoint(p : Point);
VAR buf : ARRAY[0..31] OF CHAR;
BEGIN
WriteString("(");
RealToFixed(p.x, 2, buf);
WriteString(buf);
WriteString(", ");
RealToFixed(p.y, 2, buf);
WriteString(buf);
WriteString(")")
END WritePoint;

PROCEDURE WriteTriangle(t : Triangle);
BEGIN
WriteString("Triangle: ");
WritePoint(t.p1);
WriteString(", ");
WritePoint(t.p2);
WriteString(", ");
WritePoint(t.p3)
END WriteTriangle;

PROCEDURE Det2D(p1,p2,p3 : Point) : LONGREAL;
BEGIN
RETURN p1.x * (p2.y - p3.y)
+ p2.x * (p3.y - p1.y)
+ p3.x * (p1.y - p2.y)
END Det2D;

PROCEDURE CheckTriWinding(VAR p1,p2,p3 : Point; allowReversed : BOOLEAN);
VAR
detTri : LONGREAL;
t : Point;
BEGIN
detTri := Det2D(p1, p2, p3);
IF detTri < 0.0 THEN
IF allowReversed THEN
t := p3;
p3 := p2;
p2 := t
ELSE
RAISE(TextWinExSrc, 0, "triangle has wrong winding direction")
END
END
END CheckTriWinding;

PROCEDURE BoundaryCollideChk(p1,p2,p3 : Point; eps : LONGREAL) : BOOLEAN;
BEGIN
RETURN Det2D(p1, p2, p3) < eps
END BoundaryCollideChk;

PROCEDURE BoundaryDoesntCollideChk(p1,p2,p3 : Point; eps : LONGREAL) : BOOLEAN;
BEGIN
RETURN Det2D(p1, p2, p3) <= eps
END BoundaryDoesntCollideChk;

PROCEDURE TriTri2D(t1,t2 : Triangle; eps : LONGREAL; allowReversed,onBoundary : BOOLEAN) : BOOLEAN;
TYPE
Points = ARRAY[0..2] OF Point;
VAR
chkEdge : PROCEDURE(Point, Point, Point, LONGREAL) : BOOLEAN;
lp1,lp2 : Points;
i,j : CARDINAL;
BEGIN
(* Triangles must be expressed anti-clockwise *)
CheckTriWinding(t1.p1, t1.p2, t1.p3, allowReversed);
CheckTriWinding(t2.p1, t2.p2, t2.p3, allowReversed);

(* 'onBoundary' determines whether points on boundary are considered as colliding or not *)
IF onBoundary THEN
chkEdge := BoundaryCollideChk
ELSE
chkEdge := BoundaryDoesntCollideChk
END;

lp1 := Points{t1.p1, t1.p2, t1.p3};
lp2 := Points{t2.p1, t2.p2, t2.p3};

(* for each edge E of t1 *)
FOR i:=0 TO 2 DO
j := (i + 1) MOD 3;
(* Check all points of t2 lay on the external side of edge E.
If they do, the triangles do not overlap. *)
IF     chkEdge(lp1[i], lp1[j], lp2, eps)
AND chkEdge(lp1[i], lp1[j], lp2, eps)
AND chkEdge(lp1[i], lp1[j], lp2, eps)
THEN
RETURN FALSE
END
END;

(* for each edge E of t2 *)
FOR i:=0 TO 2 DO
j := (i + 1) MOD 3;
(* Check all points of t1 lay on the external side of edge E.
If they do, the triangles do not overlap. *)
IF     chkEdge(lp2[i], lp2[j], lp1, eps)
AND chkEdge(lp2[i], lp2[j], lp1, eps)
AND chkEdge(lp2[i], lp2[j], lp1, eps)
THEN
RETURN FALSE
END
END;

(* The triangles overlap *)
RETURN TRUE
END TriTri2D;

PROCEDURE CheckOverlap(t1,t2 : Triangle; eps : LONGREAL; allowReversed,onBoundary : BOOLEAN);
BEGIN
WriteTriangle(t1);
WriteString(" and");
WriteLn;
WriteTriangle(t2);
WriteLn;

IF TriTri2D(t1, t2, eps, allowReversed, onBoundary) THEN
WriteString("overlap")
ELSE
WriteString("do not overlap")
END;
WriteLn
END CheckOverlap;

(* main *)
VAR
t1,t2 : Triangle;
BEGIN
t1 := Triangle{{0.0,0.0},{5.0,0.0},{0.0,5.0}};
t2 := Triangle{{0.0,0.0},{5.0,0.0},{0.0,6.0}};
CheckOverlap(t1, t2, 0.0, FALSE, TRUE);
WriteLn;

t1 := Triangle{{0.0,0.0},{0.0,5.0},{5.0,0.0}};
t2 := Triangle{{0.0,0.0},{0.0,5.0},{5.0,0.0}};
CheckOverlap(t1, t2, 0.0, TRUE, TRUE);
WriteLn;

t1 := Triangle{{0.0,0.0},{5.0,0.0},{0.0,5.0}};
t2 := Triangle{{-10.0,0.0},{-5.0,0.0},{-1.0,6.0}};
CheckOverlap(t1, t2, 0.0, FALSE, TRUE);
WriteLn;

t1 := Triangle{{0.0,0.0},{5.0,0.0},{2.5,5.0}};
t2 := Triangle{{0.0,4.0},{2.5,-1.0},{5.0,4.0}};
CheckOverlap(t1, t2, 0.0, FALSE, TRUE);
WriteLn;

t1 := Triangle{{0.0,0.0},{1.0,1.0},{0.0,2.0}};
t2 := Triangle{{2.0,1.0},{3.0,0.0},{3.0,2.0}};
CheckOverlap(t1, t2, 0.0, FALSE, TRUE);
WriteLn;

t1 := Triangle{{0.0,0.0},{1.0,1.0},{0.0,2.0}};
t2 := Triangle{{2.0,1.0},{3.0,-2.0},{3.0,4.0}};
CheckOverlap(t1, t2, 0.0, FALSE, TRUE);
WriteLn;

t1 := Triangle{{0.0,0.0},{1.0,0.0},{0.0,1.0}};
t2 := Triangle{{1.0,0.0},{2.0,0.0},{1.0,1.1}};
CheckOverlap(t1, t2, 0.0, FALSE, TRUE);
WriteLn;

t1 := Triangle{{0.0,0.0},{1.0,0.0},{0.0,1.0}};
t2 := Triangle{{1.0,0.0},{2.0,0.0},{1.0,1.1}};
CheckOverlap(t1, t2, 0.0, FALSE, FALSE);
WriteLn;

END Overlap.

ooRexx

/*--------------------------------------------------------------------
* Determine if two triangles overlap
* Fully (?) tested with integer coordinates of the 6 corners
* This was/is an exercise with ooRexx
* Removed the fraction arithmetic
*-------------------------------------------------------------------*/
Parse Version v

oid='trioo.txt'; 'erase' oid
Call o v
case=0
cc=0
Call trio_test '0 0   4 0   0 4   1 1   2  1   1 2'
Call trio_test '0 0 0 6 8 3 8 0 8 8 0 3'
Call trio_test '0 0 0 2 2 0 0 0 4 0 0 6'
/* The task's specified input */
Call trio_test '0 0 5 0 0 5 0 0 5 0 0 6'
Call trio_test '0 0 0 5 5 0 0 0 0 5 5 0'
Call trio_test '0 0 5 0 0 5 -10 0 -5 0 -1 6'
Call trio_test '0 0 5 0 2.5 5 0 4 2.5 -1 5 4'
Call trio_test '0 0 1 1 0 2 2 1 3 0 3 2'
Call trio_test '0 0 1 1 0 2 2 1 3 -2 3 4'
Call trio_test '0 0 1 0 0 1 1 0 2 0 1 1'
Exit
/* Other test cases */
Call trio_test '0 0   0 4   4 0   0 2   2  2   2 0'
Call trio_test '0 0   0 5   5 0   0 0   0  5   5 0'
Call trio_test '0 0   0 5   5 0   0 0   0  5   7 0'
Call trio_test '0 0   1 0   0 1   1 0   2  0   1 1'
Call trio_test '0 0   1 1   0 2   2 1   3  0   3 2'
Call trio_test '0 0   1 1   0 2   2 1   3 -2   3 4'
Call trio_test '0 0   2 0   2 2   3 3   5  3   5 5'
Call trio_test '0 0   2 0   2 3   0 0   2  0   2 3'
Call trio_test '0 0   4 0   0 4   0 2   2  0   2 2'
Call trio_test '0 0   4 0   0 4   1 1   2  1   1 2'
Call trio_test '0 0   5 0   0 2   5 0   8  0   4 8'
Call trio_test '0 0   5 0   0 5   0 0   5  0   0 6'
Call trio_test '0 0   5 0   0 5 -10 0  -5  0  -1 6'
Call trio_test '0 0   5 0   0 5  -5 0  -1  6  -3 0'
Call trio_test '0 0   5 0   3 5   0 4   3 -1   5 4'
Call trio_test '0 0   6 0   4 6   1 1   4  2   7 1'
Call trio_test '0 1   0 4   2 2   3 1   3  4   5 2'
Call trio_test '1 0   3 0   2 2   1 3   3  3   2 2'
Call trio_test '1 0   3 0   2 2   1 3   3  3   2 5'
Call trio_test '1 1   4 2   7 1   0 0   8  0   4 8'
Call trio_test '2 0   2 6   1 8   0 1   0  5   8 3'
Call trio_test '0 0   4 0   0 4   1 1   2  1   1 2'
Say case 'cases tested'
Say cc
Exit

trio_test:
Parse Arg tlist
cc+=1
tlist=space(tlist)
tl1=tlist                              ; Call trio_t tl1
tl2=reversex(tlist)                    ; Call trio_t tl2
tl3=''
tl=tlist
Do While tl<>''
Parse Var tl x y tl
tl3=tl3 y x
End
Call trio_t tl3
tl4=reversex(tl3)                      ; Call trio_t tl4
tl5=subword(tl4,7) subword(tl4,1,6)    ; Call trio_t tl5
tl6=subword(tl5,7) subword(tl5,1,6)    ; Call trio_t tl6
Return

trio_t:
Parse Arg tlist
tlist=space(tlist)
Say tlist
case+=1
Parse Arg ax ay bx by cx cy dx dy ex ey fx fy
/*---------------------------------------------------------------------
* First build the objects needed
*--------------------------------------------------------------------*/
a=.point~new(ax,ay); b=.point~new(bx,by); c=.point~new(cx,cy)
d=.point~new(dx,dy); e=.point~new(ex,ey); f=.point~new(fx,fy)
abc=.triangle~new(a,b,c)
def=.triangle~new(d,e,f)
Call o 'Triangle: ABC:' abc ,1
Call o 'Edges of ABC:'; Do i=1 To 3; Call o ' 'abc~edge(i); End
Call o 'Triangle: DEF:' def ,1
Call o 'Edges of DEF:'; Do i=1 To 3; Call o ' 'def~edge(i); End
pixl=' '
Do i=1 To 3
pixl=pixl abc~draw(i,'O')
pixl=pixl def~draw(i,'*')
End
res=0
fc=0
touch=0
bordl=''
Do i=1 To 3
p1=abc~point(i)
p2=def~point(i)
Do j=1 To 3
e1=abc~edge(j)
e2=def~edge(j)
If e1~contains(p2) Then Do
Call o e1 'contains' p2
ps=p2~string
If wordpos(ps,bordl)=0 Then Do
bordl=bordl ps
touch+=1
End
End
Else
Call o e1 'does not contain' p2 i j
If e2~contains(p1) Then Do
Call o e2 'contains' p1
ps=p1~string
If wordpos(ps,bordl)=0 Then Do
bordl=bordl ps
touch+=1
End
End
Else
Call o e2 'does not contain' p1
End
End

wb=words(bordl)                     /* how many of them?             */
If wb>0 Then
Call o 'Corner(s) that touch the other triangle:' bordl,1

/*---------------------------------------------------------------------
* How many of them are corners of both triangles
*--------------------------------------------------------------------*/
m=0
cmatch=''
do i=1 To 3
If wordpos(abc~point(i),bordl)>0 &,
wordpos(abc~point(i),def)>0 Then Do
cmatch=cmatch abc~point(i)
m+=1
End
End

/*---------------------------------------------------------------------
* With two or three touching corners we show the result and return
*--------------------------------------------------------------------*/
Select
When wb=3 Then Do                 /* all three touch               */
Call draw(pixl)
Select
When m=3 Then
Call o 'Triangles are identical',1
When m=2 Then
Call o 'Triangles have an edge in common:' cmatch,1
Otherwise
Call o 'Triangles overlap and touch on' bordl,1
End
Call o '',1
--   Pull .
Return
End
When wb=2 Then Do                 /* two of them match             */
Call draw(pixl)
If m=2 Then
Call o 'Triangles have an edge in common:' cmatch,1
Else
Call o 'Triangles overlap and touch on' bordl,1
Call o ''
--   Pull .
Return
End
When wb=1 Then Do                 /* one of them matches          */
Call o 'Triangles touch on' bordl,1 /* other parts may overlap  */
Call o '  we analyze further',1
End
Otherwise                         /* we know nothing yet           */
Nop
End

/*---------------------------------------------------------------------
* Now we look for corners of abc that are within the triangle def
*--------------------------------------------------------------------*/
in_def=0
Do i=1 To 3
p=abc~point(i)
Call o 'p  ='p
Call o 'def='def
If def~contains(p) &,
wordpos(p,bordl)=0 Then Do
Call o def 'contains' p
in_def+=1
End
End

If in_def=3 Then Do
Call o abc 'is fully contained in' def,1
Call o '',1
Call draw(pixl)
fc=1
End
res=(in_def>0)
/*---------------------------------------------------------------------
* Now we look for corners of def that are within the triangle abc
*--------------------------------------------------------------------*/
If res=0 Then Do
in_abc=0
If res=0 Then Do
Do i=1 To 3
p=def~point(i)
Call o 'p  ='p
Call o 'def='def
If abc~contains(p) &,
wordpos(p,bordl)=0 Then Do
Call o abc 'contains' p
in_abc+=1
End
End
End
If in_abc=3 Then Do
Call o def 'is fully contained in' abc,1
Call o '',1
Call draw(pixl)
fc=1
End
res=(in_abc>0)

End

/*---------------------------------------------------------------------
* Now we check if some edge of abc crosses any edge of def
*--------------------------------------------------------------------*/
If res=0 Then Do
Do i=1 To 3
Do j=1 To 3
e1=abc~edge(i); Call o 'e1='e1
e2=def~edge(j); Call o 'e2='e2
Call o 'crossing???'
res=e1~crosses(e2)
If res Then Do
End
If res Then
Call o 'edges cross'
Else
Call o 'edges don''t cross'
End
End
End

If fc=0 Then Do                     /* no fully contained            */
Call draw(pixl)
If res=0 Then                     /* no overlap                    */
If wb=1 Then                    /* but one touching corner       */
call o abc 'and' def 'don''t overlap but touch on' bordl,1
Else
call o abc 'and' def 'don''t overlap',1
Else                              /* overlap                       */
If wb>0 Then                    /* one touching corner           */
call o abc 'and' def 'overlap and touch on' bordl,1
Else
call o abc 'and' def 'overlap',1
Call o '',1
--  Pull .
End
Return

/*---------------------------------------------------------------------
* And here are all the classes and methods needed:
* point         init, x, y, string
* triangle      init, point, edge, contains, string
* edge          init, p1, p2, kdx, contains, crosses, string
*--------------------------------------------------------------------*/

::class point public
::attribute x
::attribute y
::method init
expose x y
use arg x,y
::method string
expose x y
return "("||x","y")"

::class triangle public
::method init
expose point edge
use arg p1,p2,p3
point=.array~new
point=p1
point=p2
point=p3
edge=.array~new
Do i=1 To 3
ia=i+1; If ia=4 Then ia=1
edge[i]=.edge~new(point[i],point[ia])
End
::method point
expose point
use arg n
Return point[n]
::method edge
expose edge
use arg n
Return edge[n]
::method contains
expose point edge
use arg pp
Call o self
Call o 'pp='pp
xmin=1.e9
ymin=1.e9
xmax=-1.e9
ymax=-1.e9
Do i=1 To 3
e=edge[i]
Parse Value e~kdx With ka.i da.i xa.i
Call o show_g(ka.i,da.i,xa.i)
p1=e~p1
p2=e~p2
xmin=min(xmin,p1~x,p2~x)
xmax=max(xmax,p1~x,p2~x)
ymin=min(ymin,p1~y,p2~y)
ymax=max(ymax,p1~y,p2~y)
End
If pp~x<xmin|pp~x>xmax|pp~y<ymin|pp~y>ymax Then
res=0
Else Do
e=edge
e2=edge
p1=e2~p1
p2=e2~p2
Call o 'e:' e
Select
When ka.1='*' Then Do
y2=ka.2*pp~x+da.2
y3=ka.3*pp~x+da.3
res=between(y2,pp~y,y3)
End
When ka.2='*' Then Do
y2=ka.1*pp~x+da.1
res=between(p1~y,y2,p2~y)
End
Otherwise Do
dap=pp~y-ka.1*pp~x
If ka.3='*' Then
x3=xa.3
Else
x3=(da.3-dap)/(ka.1-ka.3)
x2=(da.2-dap)/(ka.1-ka.2)
res=between(x2,pp~x,x3)
End
End
End
Return res
::method string
expose point
ol=''
Do p over point
ol=ol p~string
End
return ol
::method draw
expose point
Use Arg i,c
p=self~point(i)
Return p~x p~y c
::class edge public
::method init
expose edge p1 p2
use arg p1,p2
edge=.array~new
edge=p1
edge=p2
::method p1
expose edge p1 p2
return p1
::method p2
expose edge p1 p2
return p2
::method kdx
expose edge p1 p2
x1=p1~x
y1=p1~y
x2=p2~x
y2=p2~y
If x1=x2 Then Do
Parse Value '*' '-' x1 With ka da xa
Call o show_g(ka,da,xa)
End
Else Do
ka=(y2-y1)/(x2-x1)
da=y2-ka*x2
xa='*'
End
Return ka da xa
::method contains
Use Arg p
p1=self~p1
p2=self~p2
parse Value self~kdx With k d x
If k='*' Then Do
res=(p~x=p1~x)&between(p1~y,p~y,p2~y,'I')
End
Else Do
ey=k*p~x+d
res=(ey=p~y)&between(p1~x,p~x,p2~x,'I')
End
If res Then Call o self 'contains' p
Else Call o self 'does not contain' p
Return res
::method crosses
expose p1 p2
Use Arg e
q1=e~p1
q2=e~p2
Call o 'Test if' e 'crosses' self
Call o self~kdx
Call o e~kdx
Parse Value self~kdx With ka da xa; Call o ka da xa
Call o show_g(ka,da,xa)
Parse Value    e~kdx With kb db xb; Call o kb db xb
Call o show_g(kb,db,xb)
Call o 'ka='ka
Call o 'kb='kb
Select
When ka='*' Then Do
If kb='*' Then Do
res=(xa=xb)
End
Else Do
Call o 'kb='kb 'xa='||xa  'db='db
yy=kb*xa+db
res=between(q1~y,yy,q2~y)
End
End
When kb='*' Then Do
yy=ka*xb+da
res=between(p1~y,yy,p2~y)
End
When ka=kb Then Do
If da=db Then Do
If min(p1~x,p2~x)>max(q1~x,q2~x) |,
min(q1~x,q2~x)>max(p1~x,p2~x) Then
res=0
Else Do
res=1
End
End
Else
res=0
End
Otherwise Do
x=(db-da)/(ka-kb)
y=ka*x+da
Call o 'cross:' x y
res=between(p1~x,x,p2~x)
End
End
Return res
::method string
expose edge p1 p2
ol=p1~string'-'p2~string
return ol

::routine between         /* check if a number is between two others */
Use Arg a,x,b,inc
Call o 'between:' a x b
Parse Var a anom '/' adenom
Parse Var x xnom '/' xdenom
Parse Var b bnom '/' bdenom
If xdenom='' Then xdenom=1
If bdenom='' Then bdenom=1
aa=anom*xdenom*bdenom
If inc='I' Then
res=sign(xx-aa)<>sign(xx-bb)
Else
res=sign(xx-aa)<>sign(xx-bb) & (xx-aa)*(xx-bb)<>0
Call o a x b 'res='res
Return res

::routine show_g          /* show a straight line's forula           */
/*---------------------------------------------------------------------
* given slope, y-distance, and (special) x-value
* compute y=k*x+d or, if a vertical line, k='*'; x=c
*--------------------------------------------------------------------*/
Use Arg k,d,x
Select
When k='*' Then res='x='||x       /* vertical line               */
When k=0   Then res='y='d         /* horizontal line             */
Otherwise Do                      /* ordinary line               */
Select
When k=1  Then res='y=x'dd(d)
When k=-1 Then res='y=-x'dd(d)
Otherwise      res='y='k'*x'dd(d)
End
End
End
Return res

::routine dd  /* prepare a displacement for presenting it in show_g  */
/*---------------------------------------------------------------------
* prepare y-distance for display
*--------------------------------------------------------------------*/
Use Arg dd
Select
When dd=0 Then dd=''            /* omit dd if it's zero          */
When dd<0 Then dd=dd            /* use dd as is (-value)         */
Otherwise      dd='+'dd         /* prepend '+' to positive dd    */
End
Return dd

::routine o               /* debug output                            */
Use Arg txt,say
If say=1 Then
Say txt
oid='trioo.txt'
Return lineout(oid,txt)

::routine draw
Use Arg pixl
Return                       /* remove to see the triangle corners */
Say 'pixl='pixl
pix.=' '
Do While pixl<>''
Parse Var pixl x y c pixl
x=2*x+16; y=2*y+4
If pix.x.y=' ' Then
pix.x.y=c
Else
pix.x.y='+'
End
Do j= 20 To 0 By -1
ol=''
Do i=0 To 40
ol=ol||pix.i.j
End
Say ol
End
Return
::routine reversex
Use Arg list
n=words(list)
res=word(list,n)
Do i=n-1 to 1 By -1
res=res word(list,i)
End
Return res

{{out}}

0 0 4 0 0 4 1 1 2 1 1 2
Triangle: ABC:  (0,0) (4,0) (0,4)
Triangle: DEF:  (1,1) (2,1) (1,2)
(1,1) (2,1) (1,2) is fully contained in  (0,0) (4,0) (0,4)

2 1 1 2 1 1 4 0 0 4 0 0
Triangle: ABC:  (2,1) (1,2) (1,1)
Triangle: DEF:  (4,0) (0,4) (0,0)
(2,1) (1,2) (1,1) is fully contained in  (4,0) (0,4) (0,0)

1 2 2 1 1 1 0 4 4 0 0 0
Triangle: ABC:  (1,2) (2,1) (1,1)
Triangle: DEF:  (0,4) (4,0) (0,0)
(1,2) (2,1) (1,1) is fully contained in  (0,4) (4,0) (0,0)

0 0 0 4 4 0 1 1 1 2 2 1
Triangle: ABC:  (0,0) (0,4) (4,0)
Triangle: DEF:  (1,1) (1,2) (2,1)
(1,1) (1,2) (2,1) is fully contained in  (0,0) (0,4) (4,0)

1 1 1 2 2 1 0 0 0 4 4 0
Triangle: ABC:  (1,1) (1,2) (2,1)
Triangle: DEF:  (0,0) (0,4) (4,0)
(1,1) (1,2) (2,1) is fully contained in  (0,0) (0,4) (4,0)

0 0 0 4 4 0 1 1 1 2 2 1
Triangle: ABC:  (0,0) (0,4) (4,0)
Triangle: DEF:  (1,1) (1,2) (2,1)
(1,1) (1,2) (2,1) is fully contained in  (0,0) (0,4) (4,0)

0 0 0 6 8 3 8 0 8 8 0 3
Triangle: ABC:  (0,0) (0,6) (8,3)
Triangle: DEF:  (8,0) (8,8) (0,3)
Corner(s) that touch the other triangle:  (0,3) (8,3)
Triangles overlap and touch on  (0,3) (8,3)
3 0 8 8 0 8 3 8 6 0 0 0
Triangle: ABC:  (3,0) (8,8) (0,8)
Triangle: DEF:  (3,8) (6,0) (0,0)
Corner(s) that touch the other triangle:  (3,8) (3,0)
Triangles overlap and touch on  (3,8) (3,0)
0 3 8 8 8 0 8 3 0 6 0 0
Triangle: ABC:  (0,3) (8,8) (8,0)
Triangle: DEF:  (8,3) (0,6) (0,0)
Corner(s) that touch the other triangle:  (8,3) (0,3)
Triangles overlap and touch on  (8,3) (0,3)
0 0 6 0 3 8 0 8 8 8 3 0
Triangle: ABC:  (0,0) (6,0) (3,8)
Triangle: DEF:  (0,8) (8,8) (3,0)
Corner(s) that touch the other triangle:  (3,0) (3,8)
Triangles overlap and touch on  (3,0) (3,8)
0 8 8 8 3 0 0 0 6 0 3 8
Triangle: ABC:  (0,8) (8,8) (3,0)
Triangle: DEF:  (0,0) (6,0) (3,8)
Corner(s) that touch the other triangle:  (3,8) (3,0)
Triangles overlap and touch on  (3,8) (3,0)
0 0 6 0 3 8 0 8 8 8 3 0
Triangle: ABC:  (0,0) (6,0) (3,8)
Triangle: DEF:  (0,8) (8,8) (3,0)
Corner(s) that touch the other triangle:  (3,0) (3,8)
Triangles overlap and touch on  (3,0) (3,8)
0 0 0 2 2 0 0 0 4 0 0 6
Triangle: ABC:  (0,0) (0,2) (2,0)
Triangle: DEF:  (0,0) (4,0) (0,6)
Corner(s) that touch the other triangle:  (0,0) (0,2) (2,0)
Triangles overlap and touch on  (0,0) (0,2) (2,0)

6 0 0 4 0 0 0 2 2 0 0 0
Triangle: ABC:  (6,0) (0,4) (0,0)
Triangle: DEF:  (0,2) (2,0) (0,0)
Corner(s) that touch the other triangle:  (0,2) (2,0) (0,0)
Triangles overlap and touch on  (0,2) (2,0) (0,0)

0 6 4 0 0 0 2 0 0 2 0 0
Triangle: ABC:  (0,6) (4,0) (0,0)
Triangle: DEF:  (2,0) (0,2) (0,0)
Corner(s) that touch the other triangle:  (2,0) (0,2) (0,0)
Triangles overlap and touch on  (2,0) (0,2) (0,0)

0 0 2 0 0 2 0 0 0 4 6 0
Triangle: ABC:  (0,0) (2,0) (0,2)
Triangle: DEF:  (0,0) (0,4) (6,0)
Corner(s) that touch the other triangle:  (0,0) (2,0) (0,2)
Triangles overlap and touch on  (0,0) (2,0) (0,2)

0 0 0 4 6 0 0 0 2 0 0 2
Triangle: ABC:  (0,0) (0,4) (6,0)
Triangle: DEF:  (0,0) (2,0) (0,2)
Corner(s) that touch the other triangle:  (0,0) (2,0) (0,2)
Triangles overlap and touch on  (0,0) (2,0) (0,2)

0 0 2 0 0 2 0 0 0 4 6 0
Triangle: ABC:  (0,0) (2,0) (0,2)
Triangle: DEF:  (0,0) (0,4) (6,0)
Corner(s) that touch the other triangle:  (0,0) (2,0) (0,2)
Triangles overlap and touch on  (0,0) (2,0) (0,2)

0 0 5 0 0 5 0 0 5 0 0 6
Triangle: ABC:  (0,0) (5,0) (0,5)
Triangle: DEF:  (0,0) (5,0) (0,6)
Corner(s) that touch the other triangle:  (0,0) (5,0) (0,5)
Triangles have an edge in common:  (0,0) (5,0)

6 0 0 5 0 0 5 0 0 5 0 0
Triangle: ABC:  (6,0) (0,5) (0,0)
Triangle: DEF:  (5,0) (0,5) (0,0)
Corner(s) that touch the other triangle:  (5,0) (0,5) (0,0)
Triangles have an edge in common:  (0,5) (0,0)

0 6 5 0 0 0 0 5 5 0 0 0
Triangle: ABC:  (0,6) (5,0) (0,0)
Triangle: DEF:  (0,5) (5,0) (0,0)
Corner(s) that touch the other triangle:  (0,5) (5,0) (0,0)
Triangles have an edge in common:  (5,0) (0,0)

0 0 0 5 5 0 0 0 0 5 6 0
Triangle: ABC:  (0,0) (0,5) (5,0)
Triangle: DEF:  (0,0) (0,5) (6,0)
Corner(s) that touch the other triangle:  (0,0) (0,5) (5,0)
Triangles have an edge in common:  (0,0) (0,5)

0 0 0 5 6 0 0 0 0 5 5 0
Triangle: ABC:  (0,0) (0,5) (6,0)
Triangle: DEF:  (0,0) (0,5) (5,0)
Corner(s) that touch the other triangle:  (0,0) (0,5) (5,0)
Triangles have an edge in common:  (0,0) (0,5)

0 0 0 5 5 0 0 0 0 5 6 0
Triangle: ABC:  (0,0) (0,5) (5,0)
Triangle: DEF:  (0,0) (0,5) (6,0)
Corner(s) that touch the other triangle:  (0,0) (0,5) (5,0)
Triangles have an edge in common:  (0,0) (0,5)

0 0 0 5 5 0 0 0 0 5 5 0
Triangle: ABC:  (0,0) (0,5) (5,0)
Triangle: DEF:  (0,0) (0,5) (5,0)
Corner(s) that touch the other triangle:  (0,0) (0,5) (5,0)
Triangles are identical

0 5 5 0 0 0 0 5 5 0 0 0
Triangle: ABC:  (0,5) (5,0) (0,0)
Triangle: DEF:  (0,5) (5,0) (0,0)
Corner(s) that touch the other triangle:  (0,5) (5,0) (0,0)
Triangles are identical

5 0 0 5 0 0 5 0 0 5 0 0
Triangle: ABC:  (5,0) (0,5) (0,0)
Triangle: DEF:  (5,0) (0,5) (0,0)
Corner(s) that touch the other triangle:  (5,0) (0,5) (0,0)
Triangles are identical

0 0 5 0 0 5 0 0 5 0 0 5
Triangle: ABC:  (0,0) (5,0) (0,5)
Triangle: DEF:  (0,0) (5,0) (0,5)
Corner(s) that touch the other triangle:  (0,0) (5,0) (0,5)
Triangles are identical

0 0 5 0 0 5 0 0 5 0 0 5
Triangle: ABC:  (0,0) (5,0) (0,5)
Triangle: DEF:  (0,0) (5,0) (0,5)
Corner(s) that touch the other triangle:  (0,0) (5,0) (0,5)
Triangles are identical

0 0 5 0 0 5 0 0 5 0 0 5
Triangle: ABC:  (0,0) (5,0) (0,5)
Triangle: DEF:  (0,0) (5,0) (0,5)
Corner(s) that touch the other triangle:  (0,0) (5,0) (0,5)
Triangles are identical

0 0 5 0 0 5 -10 0 -5 0 -1 6
Triangle: ABC:  (0,0) (5,0) (0,5)
Triangle: DEF:  (-10,0) (-5,0) (-1,6)
(0,0) (5,0) (0,5) and  (-10,0) (-5,0) (-1,6) don't overlap

6 -1 0 -5 0 -10 5 0 0 5 0 0
Triangle: ABC:  (6,-1) (0,-5) (0,-10)
Triangle: DEF:  (5,0) (0,5) (0,0)
(6,-1) (0,-5) (0,-10) and  (5,0) (0,5) (0,0) don't overlap

-1 6 -5 0 -10 0 0 5 5 0 0 0
Triangle: ABC:  (-1,6) (-5,0) (-10,0)
Triangle: DEF:  (0,5) (5,0) (0,0)
(-1,6) (-5,0) (-10,0) and  (0,5) (5,0) (0,0) don't overlap

0 0 0 5 5 0 0 -10 0 -5 6 -1
Triangle: ABC:  (0,0) (0,5) (5,0)
Triangle: DEF:  (0,-10) (0,-5) (6,-1)
(0,0) (0,5) (5,0) and  (0,-10) (0,-5) (6,-1) don't overlap

0 -10 0 -5 6 -1 0 0 0 5 5 0
Triangle: ABC:  (0,-10) (0,-5) (6,-1)
Triangle: DEF:  (0,0) (0,5) (5,0)
(0,-10) (0,-5) (6,-1) and  (0,0) (0,5) (5,0) don't overlap

0 0 0 5 5 0 0 -10 0 -5 6 -1
Triangle: ABC:  (0,0) (0,5) (5,0)
Triangle: DEF:  (0,-10) (0,-5) (6,-1)
(0,0) (0,5) (5,0) and  (0,-10) (0,-5) (6,-1) don't overlap

0 0 5 0 2.5 5 0 4 2.5 -1 5 4
Triangle: ABC:  (0,0) (5,0) (2.5,5)
Triangle: DEF:  (0,4) (2.5,-1) (5,4)
(0,0) (5,0) (2.5,5) and  (0,4) (2.5,-1) (5,4) overlap

4 5 -1 2.5 4 0 5 2.5 0 5 0 0
Triangle: ABC:  (4,5) (-1,2.5) (4,0)
Triangle: DEF:  (5,2.5) (0,5) (0,0)
(4,5) (-1,2.5) (4,0) and  (5,2.5) (0,5) (0,0) overlap

5 4 2.5 -1 0 4 2.5 5 5 0 0 0
Triangle: ABC:  (5,4) (2.5,-1) (0,4)
Triangle: DEF:  (2.5,5) (5,0) (0,0)
(5,4) (2.5,-1) (0,4) and  (2.5,5) (5,0) (0,0) overlap

0 0 0 5 5 2.5 4 0 -1 2.5 4 5
Triangle: ABC:  (0,0) (0,5) (5,2.5)
Triangle: DEF:  (4,0) (-1,2.5) (4,5)
(0,0) (0,5) (5,2.5) and  (4,0) (-1,2.5) (4,5) overlap

4 0 -1 2.5 4 5 0 0 0 5 5 2.5
Triangle: ABC:  (4,0) (-1,2.5) (4,5)
Triangle: DEF:  (0,0) (0,5) (5,2.5)
(4,0) (-1,2.5) (4,5) and  (0,0) (0,5) (5,2.5) overlap

0 0 0 5 5 2.5 4 0 -1 2.5 4 5
Triangle: ABC:  (0,0) (0,5) (5,2.5)
Triangle: DEF:  (4,0) (-1,2.5) (4,5)
(0,0) (0,5) (5,2.5) and  (4,0) (-1,2.5) (4,5) overlap

0 0 1 1 0 2 2 1 3 0 3 2
Triangle: ABC:  (0,0) (1,1) (0,2)
Triangle: DEF:  (2,1) (3,0) (3,2)
(0,0) (1,1) (0,2) and  (2,1) (3,0) (3,2) don't overlap

2 3 0 3 1 2 2 0 1 1 0 0
Triangle: ABC:  (2,3) (0,3) (1,2)
Triangle: DEF:  (2,0) (1,1) (0,0)
(2,3) (0,3) (1,2) and  (2,0) (1,1) (0,0) don't overlap

3 2 3 0 2 1 0 2 1 1 0 0
Triangle: ABC:  (3,2) (3,0) (2,1)
Triangle: DEF:  (0,2) (1,1) (0,0)
(3,2) (3,0) (2,1) and  (0,2) (1,1) (0,0) don't overlap

0 0 1 1 2 0 1 2 0 3 2 3
Triangle: ABC:  (0,0) (1,1) (2,0)
Triangle: DEF:  (1,2) (0,3) (2,3)
(0,0) (1,1) (2,0) and  (1,2) (0,3) (2,3) don't overlap

1 2 0 3 2 3 0 0 1 1 2 0
Triangle: ABC:  (1,2) (0,3) (2,3)
Triangle: DEF:  (0,0) (1,1) (2,0)
(1,2) (0,3) (2,3) and  (0,0) (1,1) (2,0) don't overlap

0 0 1 1 2 0 1 2 0 3 2 3
Triangle: ABC:  (0,0) (1,1) (2,0)
Triangle: DEF:  (1,2) (0,3) (2,3)
(0,0) (1,1) (2,0) and  (1,2) (0,3) (2,3) don't overlap

0 0 1 1 0 2 2 1 3 -2 3 4
Triangle: ABC:  (0,0) (1,1) (0,2)
Triangle: DEF:  (2,1) (3,-2) (3,4)
(0,0) (1,1) (0,2) and  (2,1) (3,-2) (3,4) don't overlap

4 3 -2 3 1 2 2 0 1 1 0 0
Triangle: ABC:  (4,3) (-2,3) (1,2)
Triangle: DEF:  (2,0) (1,1) (0,0)
(4,3) (-2,3) (1,2) and  (2,0) (1,1) (0,0) don't overlap

3 4 3 -2 2 1 0 2 1 1 0 0
Triangle: ABC:  (3,4) (3,-2) (2,1)
Triangle: DEF:  (0,2) (1,1) (0,0)
(3,4) (3,-2) (2,1) and  (0,2) (1,1) (0,0) don't overlap

0 0 1 1 2 0 1 2 -2 3 4 3
Triangle: ABC:  (0,0) (1,1) (2,0)
Triangle: DEF:  (1,2) (-2,3) (4,3)
(0,0) (1,1) (2,0) and  (1,2) (-2,3) (4,3) don't overlap

1 2 -2 3 4 3 0 0 1 1 2 0
Triangle: ABC:  (1,2) (-2,3) (4,3)
Triangle: DEF:  (0,0) (1,1) (2,0)
(1,2) (-2,3) (4,3) and  (0,0) (1,1) (2,0) don't overlap

0 0 1 1 2 0 1 2 -2 3 4 3
Triangle: ABC:  (0,0) (1,1) (2,0)
Triangle: DEF:  (1,2) (-2,3) (4,3)
(0,0) (1,1) (2,0) and  (1,2) (-2,3) (4,3) don't overlap

0 0 1 0 0 1 1 0 2 0 1 1
Triangle: ABC:  (0,0) (1,0) (0,1)
Triangle: DEF:  (1,0) (2,0) (1,1)
Corner(s) that touch the other triangle:  (1,0)
Triangles touch on  (1,0)
we analyze further
(0,0) (1,0) (0,1) and  (1,0) (2,0) (1,1) don't overlap but touch on  (1,0)

1 1 0 2 0 1 1 0 0 1 0 0
Triangle: ABC:  (1,1) (0,2) (0,1)
Triangle: DEF:  (1,0) (0,1) (0,0)
Corner(s) that touch the other triangle:  (0,1)
Triangles touch on  (0,1)
we analyze further
(1,1) (0,2) (0,1) and  (1,0) (0,1) (0,0) don't overlap but touch on  (0,1)

1 1 2 0 1 0 0 1 1 0 0 0
Triangle: ABC:  (1,1) (2,0) (1,0)
Triangle: DEF:  (0,1) (1,0) (0,0)
Corner(s) that touch the other triangle:  (1,0)
Triangles touch on  (1,0)
we analyze further
(1,1) (2,0) (1,0) and  (0,1) (1,0) (0,0) overlap and touch on  (1,0)

0 0 0 1 1 0 0 1 0 2 1 1
Triangle: ABC:  (0,0) (0,1) (1,0)
Triangle: DEF:  (0,1) (0,2) (1,1)
Corner(s) that touch the other triangle:  (0,1)
Triangles touch on  (0,1)
we analyze further
(0,0) (0,1) (1,0) and  (0,1) (0,2) (1,1) don't overlap but touch on  (0,1)

0 1 0 2 1 1 0 0 0 1 1 0
Triangle: ABC:  (0,1) (0,2) (1,1)
Triangle: DEF:  (0,0) (0,1) (1,0)
Corner(s) that touch the other triangle:  (0,1)
Triangles touch on  (0,1)
we analyze further
(0,1) (0,2) (1,1) and  (0,0) (0,1) (1,0) don't overlap but touch on  (0,1)

0 0 0 1 1 0 0 1 0 2 1 1
Triangle: ABC:  (0,0) (0,1) (1,0)
Triangle: DEF:  (0,1) (0,2) (1,1)
Corner(s) that touch the other triangle:  (0,1)
Triangles touch on  (0,1)
we analyze further
(0,0) (0,1) (1,0) and  (0,1) (0,2) (1,1) don't overlap but touch on  (0,1)

Perl

Port of Lua

use strict;
use warnings;

sub det2D {
my \$p1 = shift or die "4 Missing first point\n";
my \$p2 = shift or die "Missing second point\n";
my \$p3 = shift or die "Missing third point\n";

return \$p1->{x} * (\$p2->{y} - \$p3->{y})
+ \$p2->{x} * (\$p3->{y} - \$p1->{y})
+ \$p3->{x} * (\$p1->{y} - \$p2->{y});
}

sub checkTriWinding {
my \$p1 = shift or die "14 Missing first point\n";
my \$p2 = shift or die "Missing second point\n";
my \$p3 = shift or die "Missing third point\n";
my \$allowReversed = shift;

my \$detTri = det2D(\$p1, \$\$p2, \$\$p3);
if (\$detTri < 0.0) {
if (\$allowReversed) {
my \$t = \$\$p3;
\$\$p3 = \$\$p2;
\$\$p2 = \$t;
} else {
die "triangle has wrong winding direction";
}
}
return undef;
}

sub boundaryCollideChk {
my \$p1 = shift or die "33 Missing first point\n";
my \$p2 = shift or die "Missing second point\n";
my \$p3 = shift or die "Missing third point\n";
my \$eps = shift;

return det2D(\$p1, \$p2, \$p3) < \$eps;
}

sub boundaryDoesntCollideChk {
my \$p1 = shift or die "42 Missing first point\n";
my \$p2 = shift or die "Missing second point\n";
my \$p3 = shift or die "Missing third point\n";
my \$eps = shift;

return det2D(\$p1, \$p2, \$p3) <= \$eps;
}

sub triTri2D {
my \$t1 = shift or die "Missing first triangle to calculate with\n";
my \$t2 = shift or die "Missing second triangle to calculate with\n";
my \$eps = shift;
my \$allowReversed = shift;
my \$onBoundary = shift;

# triangles must be expressed anti-clockwise
checkTriWinding(\$t1->, \\$t1->, \\$t1->, \$allowReversed);
checkTriWinding(\$t2->, \\$t2->, \\$t2->, \$allowReversed);

my \$chkEdge;
if (\$onBoundary) {
# points on the boundary are considered as colliding
\$chkEdge = \&boundaryCollideChk;
} else {
# points on the boundary are NOT considered as colliding
\$chkEdge = \&boundaryDoesntCollideChk;
}

# for edge E of triangle 1
foreach my \$i (0, 1, 2) {
my \$j = (\$i + 1) % 3;

# check all points of triangle 2 lay on the external side of edge E
# if they do, the triangles do not collide
if (\$chkEdge->(\$t1->[\$i], \$t1->[\$j], \$t2->, \$eps)
and \$chkEdge->(\$t1->[\$i], \$t1->[\$j], \$t2->, \$eps)
and \$chkEdge->(\$t1->[\$i], \$t1->[\$j], \$t2->, \$eps)) {
return 0; # false
}
}

# for edge E of triangle 2
foreach my \$i (0, 1, 2) {
my \$j = (\$i + 1) % 3;

# check all points of triangle 1 lay on the external side of edge E
# if they do, the triangles do not collide
if (\$chkEdge->(\$t2->[\$i], \$t2->[\$j], \$t1->, \$eps)
and \$chkEdge->(\$t2->[\$i], \$t2->[\$j], \$t1->, \$eps)
and \$chkEdge->(\$t2->[\$i], \$t2->[\$j], \$t1->, \$eps)) {
return 0; # false
}
}

return 1; # true
}

sub formatTri {
my \$t = shift or die "Missing triangle to format\n";
my \$p1 = \$t->;
my \$p2 = \$t->;
my \$p3 = \$t->;
return "Triangle: (\$p1->{x}, \$p1->{y}), (\$p2->{x}, \$p2->{y}), (\$p3->{x}, \$p3->{y})";
}

sub overlap {
my \$t1 = shift or die "Missing first triangle to calculate with\n";
my \$t2 = shift or die "Missing second triangle to calculate with\n";
my \$eps = shift;
my \$allowReversed = shift or 0; # false
my \$onBoundary = shift or 1; # true

unless (\$eps) {
\$eps = 0.0;
}

if (triTri2D(\$t1, \$t2, \$eps, \$allowReversed, \$onBoundary)) {
return "overlap\n";
} else {
return "do not overlap\n";
}
}

###################################################
# Main
###################################################

my @t1 = ({x=>0, y=>0}, {x=>5, y=>0}, {x=>0, y=>5});
my @t2 = ({x=>0, y=>0}, {x=>5, y=>0}, {x=>0, y=>6});
print formatTri(\@t1), " and\n", formatTri(\@t2), "\n", overlap(\@t1, \@t2), "\n";

@t1 = ({x=>0, y=>0}, {x=>0, y=>5}, {x=>5, y=>0});
@t2 = ({x=>0, y=>0}, {x=>0, y=>5}, {x=>5, y=>0});
print formatTri(\@t1), " and\n", formatTri(\@t2), "\n", overlap(\@t1, \@t2, 0.0, 1), "\n";

@t1 = ({x=>0, y=>0}, {x=>5, y=>0}, {x=>0, y=>5});
@t2 = ({x=>-10, y=>0}, {x=>-5, y=>0}, {x=>-1, y=>6});
print formatTri(\@t1), " and\n", formatTri(\@t2), "\n", overlap(\@t1, \@t2), "\n";

@t1 = ({x=>0, y=>0}, {x=>5, y=>0}, {x=>2.5, y=>5});
@t2 = ({x=>0, y=>4}, {x=>2.5, y=>-1}, {x=>5, y=>4});
print formatTri(\@t1), " and\n", formatTri(\@t2), "\n", overlap(\@t1, \@t2), "\n";

@t1 = ({x=>0, y=>0}, {x=>1, y=>1}, {x=>0, y=>2});
@t2 = ({x=>2, y=>1}, {x=>3, y=>0}, {x=>3, y=>2});
print formatTri(\@t1), " and\n", formatTri(\@t2), "\n", overlap(\@t1, \@t2), "\n";

@t1 = ({x=>0, y=>0}, {x=>1, y=>1}, {x=>0, y=>2});
@t2 = ({x=>2, y=>1}, {x=>3, y=>-2}, {x=>3, y=>4});
print formatTri(\@t1), " and\n", formatTri(\@t2), "\n", overlap(\@t1, \@t2), "\n";

# Barely touching
@t1 = ({x=>0, y=>0}, {x=>1, y=>0}, {x=>0, y=>1});
@t2 = ({x=>1, y=>0}, {x=>2, y=>0}, {x=>1, y=>1});
print formatTri(\@t1), " and\n", formatTri(\@t2), "\n", overlap(\@t1, \@t2, 0.0, 0, 1), "\n";

# Barely touching
@t1 = ({x=>0, y=>0}, {x=>1, y=>0}, {x=>0, y=>1});
@t2 = ({x=>1, y=>0}, {x=>2, y=>0}, {x=>1, y=>1});
print formatTri(\@t1), " and\n", formatTri(\@t2), "\n", overlap(\@t1, \@t2, 0.0, 0, 0), "\n";

{{out}}

Triangle: (0, 0), (5, 0), (0, 5) and
Triangle: (0, 0), (5, 0), (0, 6)
overlap

Triangle: (0, 0), (0, 5), (5, 0) and
Triangle: (0, 0), (0, 5), (5, 0)
overlap

Triangle: (0, 0), (5, 0), (0, 5) and
Triangle: (-10, 0), (-5, 0), (-1, 6)
do not overlap

Triangle: (0, 0), (5, 0), (2.5, 5) and
Triangle: (0, 4), (2.5, -1), (5, 4)
overlap

Triangle: (0, 0), (1, 1), (0, 2) and
Triangle: (2, 1), (3, 0), (3, 2)
do not overlap

Triangle: (0, 0), (1, 1), (0, 2) and
Triangle: (2, 1), (3, -2), (3, 4)
do not overlap

Triangle: (0, 0), (1, 0), (0, 1) and
Triangle: (1, 0), (2, 0), (1, 1)
overlap

Triangle: (0, 0), (1, 0), (0, 1) and
Triangle: (1, 0), (2, 0), (1, 1)
do not overlap

More Idiomatic

use strict;
use warnings;
use feature 'say';

sub det2D {
my(\$p1,\$p2,\$p3) = @_;
return \$p1-> * (\$p2-> - \$p3->)
+ \$p2-> * (\$p3-> - \$p1->)
+ \$p3-> * (\$p1-> - \$p2->);
}

# triangles must be expressed anti-clockwise
sub checkTriWinding {
my(\$p1,\$p2,\$p3,\$allowReversed) = @_;
my \$detTri = det2D(\$p1, \$\$p2, \$\$p3);
if (\$detTri < 0.0) {
if (\$allowReversed) { (\$\$p3,\$\$p2) = (\$\$p2,\$\$p3) }
else                { die "triangle has wrong winding direction" }
}
return undef;
}

sub check_edge {
our(\$t1,\$t2,\$eps,\$onBoundary) = @_;

# points on the boundary may be considered as colliding, or not
my \$chkEdge = \$onBoundary ?  \&boundaryCollideChk : \&boundaryDoesntCollideChk;
sub boundaryCollideChk       { return det2D(\$_, \$_, \$_) <  \$eps }
sub boundaryDoesntCollideChk { return det2D(\$_, \$_, \$_) <= \$eps }

# for edge E of triangle 1
foreach my \$i (0, 1, 2) {
my \$j = (\$i + 1) % 3;

# check all points of triangle 2 lay on the external side of edge E
# if they do, the triangles do not collide
if (\$chkEdge->(\$\$t1->[\$i], \$\$t1->[\$j], \$\$t2->, \$eps)
and \$chkEdge->(\$\$t1->[\$i], \$\$t1->[\$j], \$\$t2->, \$eps)
and \$chkEdge->(\$\$t1->[\$i], \$\$t1->[\$j], \$\$t2->, \$eps)) {
return 0; # false
}
}
return 1;
}

sub triTri2D {
my(\$t1,\$t2,\$eps,\$allowReversed,\$onBoundary) = @_;
checkTriWinding(\$\$t1->, \\$\$t1->, \\$\$t1->, \$allowReversed);
checkTriWinding(\$\$t2->, \\$\$t2->, \\$\$t2->, \$allowReversed);
return check_edge(\$t1,\$t2,\$eps,\$onBoundary) && check_edge(\$t2,\$t1,\$eps,\$onBoundary);
}

sub formatTri {
my \$t = shift;
my @pairs;
push @pairs, sprintf "%8s", '(' . \$\$_ . ',' . \$\$_ . ')' for @\$\$t;
join ', ', @pairs;
}

sub overlap {
my \$t1            = shift or die "Missing first triangle to calculate with\n";
my \$t2            = shift or die "Missing second triangle to calculate with\n";
my \$eps           = shift || 0;
my \$allowReversed = shift || 1;
my \$onBoundary    = shift || 1;

my \$triangles = formatTri(\$t1) . ' and ' . formatTri(\$t2);
if (triTri2D(\$t1, \$t2, \$eps, \$allowReversed, \$onBoundary)) {
return "       overlap:" . \$triangles;
} else {
return "do not overlap:" . \$triangles;
}
}

my @tests = (
[ [[0,0], [5,0],   [0,5]], [   [0,0],    [5,0],  [0,6]] ],
[ [[0,0], [0,5],   [5,0]], [   [0,0],    [0,5],  [5,0]] ],
[ [[0,0], [5,0],   [0,5]], [ [-10,0],   [-5,0], [-1,6]] ],
[ [[0,0], [5,0], [2.5,5]], [   [0,4], [2.5,-1],  [5,4]] ],
[ [[0,0], [1,1],   [0,2]], [   [2,1],    [3,0],  [3,2]] ],
[ [[0,0], [1,1],   [0,2]], [   [2,1],   [3,-2],  [3,4]] ],             # barely touching
[ [[0,0], [1,0],   [0,1]], [   [1,0],    [2,0],  [1,1]], 0.0, 0, 0 ]   # barely touching
);

say overlap(\\$_->, \\$_->, \$_->, \$_->, \$_->) for @tests;

{{out}}

overlap:   (0,0),    (5,0),    (0,5) and    (0,0),    (5,0),    (0,6)
overlap:   (0,0),    (0,5),    (5,0) and    (0,0),    (0,5),    (5,0)
do not overlap:   (0,0),    (5,0),    (0,5) and  (-10,0),   (-5,0),   (-1,6)
overlap:   (0,0),    (5,0),  (2.5,5) and    (0,4), (2.5,-1),    (5,4)
do not overlap:   (0,0),    (1,1),    (0,2) and    (2,1),    (3,0),    (3,2)
do not overlap:   (0,0),    (1,1),    (0,2) and    (2,1),   (3,-2),    (3,4)
overlap:   (0,0),    (1,0),    (0,1) and    (1,0),    (2,0),    (1,1)

Perl 6

First, check if any vertex is inside each other triangle (that should cover most overlapping cases including enclosures). Then see if an edge of triangle A intersects any of two edges of B (for shapes like Star of David [https://en.wikipedia.org/wiki/Star_of_David])

#!/usr/bin/env perl6

# Reference:
# https://stackoverflow.com/questions/2049582/how-to-determine-if-a-point-is-in-a-2d-triangle
# https://www.geeksforgeeks.org/check-if-two-given-line-segments-intersect/

use v6;

sub if-overlap (\$triangle-pair) {
my (\A,\B) = \$triangle-pair;
my Bool \$result = False;

sub sign (\T) {
return (T. - T) * (T - T) -
(T - T) * (T - T);
}

sub point-in-triangle (\pt, \Y --> Bool) {
my (\$d1, \$d2, \$d3);
my Bool (\$has_neg, \$has_pos);

\$d1 = sign (pt, Y., Y.);
\$d2 = sign (pt, Y., Y.);
\$d3 = sign (pt, Y., Y.);

\$has_neg = (\$d1 < 0) || (\$d2 < 0) || (\$d3 < 0);
\$has_pos = (\$d1 > 0) || (\$d2 > 0) || (\$d3 > 0);

return !(\$has_neg && \$has_pos);
}

sub orientation(\P, \Q, \R --> Int) {
my \val = (Q. - P.) * (R. - Q.) -
(Q. - P.) * (R. - Q.);

return 0 if val == 0;  # colinear

return (val > 0) ?? 1 !! 2; # clock or counterclock wise
}

sub onSegment(\P, \Q, \R --> Bool) {
Q. <= max(P., R.) && Q. >= min(P., R.) &&
Q. <= max(P., R.) && Q. >= min(P., R.)
?? True !! False;
}

sub intersect(\A,\B,\C,\D --> Bool) {
my \o1 = orientation A,C,D;
my \o2 = orientation B,C,D;
my \o3 = orientation A,B,C;
my \o4 = orientation A,B,D;

return True if o1 != o2 && o3 != o4;

return True if (o1 == 0 && onSegment(A, C, D)) ;
return True if (o2 == 0 && onSegment(B, C, D)) ;
return True if (o3 == 0 && onSegment(A, B, C)) ;
return True if (o4 == 0 && onSegment(A, B, D)) ;

return False;
}

for ^3 {
{\$result = True; last } if point-in-triangle A.[\$^i] , B or
point-in-triangle B.[\$^i] , A ;
}

unless \$result {
\$result = True if intersect A., A., B., B. or
intersect A., A., B., B.
}

say A, " and ", B, " do",  \$result ?? "" !! " NOT" , " overlap.";
}

my \DATA = [
[ [(0,0),(5,0),(0,5)]   ,  [(0,0),(5,0),(0,6)]     ],
[ [(0,0),(0,5),(5,0)]   ,  [(0,0),(0,5),(5,0)]     ],
[ [(0,0),(5,0),(0,5)]   ,  [(-10,0),(-5,0),(-1,6)] ],
[ [(0,0),(5,0),(2.5,5)] ,  [ (0,4),(2.5,-1),(5,4)] ],
[ [(0,0),(1,1),(0,2)]   ,  [(2,1),(3,0),(3,2)]     ],
[ [(0,0),(1,1),(0,2)]   ,  [(2,1),(3,-2),(3,4)]    ],
[ [(0,0),(1,0),(0,1)]   ,  [(1,0),(2,0),(1,1)]     ]
];

if-overlap \$_ for DATA ;

{{out}}

[(0 0) (5 0) (0 5)] and [(0 0) (5 0) (0 6)] do overlap.
[(0 0) (0 5) (5 0)] and [(0 0) (0 5) (5 0)] do overlap.
[(0 0) (5 0) (0 5)] and [(-10 0) (-5 0) (-1 6)] do NOT overlap.
[(0 0) (5 0) (2.5 5)] and [(0 4) (2.5 -1) (5 4)] do overlap.
[(0 0) (1 1) (0 2)] and [(2 1) (3 0) (3 2)] do NOT overlap.
[(0 0) (1 1) (0 2)] and [(2 1) (3 -2) (3 4)] do NOT overlap.
[(0 0) (1 0) (0 1)] and [(1 0) (2 0) (1 1)] do overlap.

Phix

{{libheader|pGUI}} {{trans|zkl}} Plus draw all eight pairs of triangles for visual confirmation.

include pGUI.e

Ihandle dlg, canvas
cdCanvas cddbuffer, cdcanvas

constant triangles = {{{{0,0},{5,0},{0,5}},{{0,0},{5,0},{0,6}}},
{{{0,0},{0,5},{5,0}},{{0,0},{0,5},{5,0}}},
{{{0,0},{5,0},{0,5}},{{-10,0},{-5,0},{-1,6}}},
{{{0,0},{5,0},{2.5,5}},{{0,4},{2.5,-1},{5,4}}},
{{{0,0},{1,1},{0,2}},{{2,1},{3,0},{3,2}}},
{{{0,0},{1,1},{0,2}},{{2,1},{3,-2},{3,4}}},
{{{0,0},{1,0},{0,1}},{{1,0},{2,0},{1,1}}},
{{{0,0},{1,0},{0,1}},{{1,0},{2,0},{1,1}}}}

procedure draw_triangle(sequence t, integer cx,cy, c)
cdCanvasSetForeground(cddbuffer, c)
cdCanvasBegin(cddbuffer,CD_CLOSED_LINES)
for c=1 to 3 do
atom {x,y} = t[c]
cdCanvasVertex(cddbuffer, cx+x*10, cy+y*10)
end for
cdCanvasEnd(cddbuffer)
end procedure

function det2D(sequence triangle)
atom {{p1x,p1y},{p2x,p2y},{p3x,p3y}} := triangle
return p1x*(p2y-p3y) + p2x*(p3y-p1y) + p3x*(p1y-p2y)
end function

bool bReversed
function checkWinding(sequence triangle, bool allowReversed)
atom detTri := det2D(triangle);
if detTri<0.0 then
if allowReversed then
bReversed = true
triangle = extract(triangle,{1,3,2})
else
throw("triangle has wrong winding direction")
end if
end if
return triangle
end function

function overlap(sequence t1, t2, atom epsilon=0.0, bool allowReversed=false, onBoundary=true)
-- Trangles must be expressed anti-clockwise
bReversed = false
t1 = checkWinding(t1, allowReversed)
t2 = checkWinding(t2, allowReversed)

for t=1 to 2 do                     -- check t1 then t2
for edge=1 to 3 do              -- check each edge
sequence p1 = t1[edge],
p2 = t1[mod(edge,3)+1]
-- Check all points of trangle 2 lay on the external side
-- of the edge E. If they do, the triangles do not collide.
integer onside = 0
for k=1 to 3 do
integer c = compare(det2D({p1,p2,t2[k]}),epsilon)
if onBoundary then
if not (c<0) then exit end if
else
if not (c<=0) then exit end if
end if
--              -- (the following incomprehensible one-liner is equivalent:)
--              if compare(det2D({p1,p2,t2[k]}),epsilon)>-onBoundary then exit end if
onside += 1
end for
if onside=3 then
return iff(onBoundary?"no overlap":"no overlap (no boundary)")
end if
end for
{t2,t1} = {t1,t2}           -- flip and re-test
end for
return iff(bReversed?"overlap (reversed)":"overlap")
end function

function redraw_cb(Ihandle /*ih*/, integer /*posx*/, integer /*posy*/)
cdCanvasActivate(cddbuffer)
integer cy = 200, cx = 100
for i=1 to length(triangles) do
sequence {t1,t2} = triangles[i]
draw_triangle(t1,cx,cy,CD_RED)
integer s = (i<=2)  -- (smudge tests[1..2] by one
--  pixel to show them better)
draw_triangle(t2,cx+s,cy+s,CD_BLUE)
cdCanvasSetForeground(cddbuffer, CD_BLACK)
cdCanvasText(cddbuffer,cx+10,cy-40,overlap(t1,t2,0,i=2,i!=8))
if i=4 then
cy = 100
cx = 100
else
cx += 300
end if
end for
cdCanvasFlush(cddbuffer)
return IUP_DEFAULT
end function

function map_cb(Ihandle ih)
cdcanvas = cdCreateCanvas(CD_IUP, ih)
cddbuffer = cdCreateCanvas(CD_DBUFFER, cdcanvas)
cdCanvasSetBackground(cddbuffer, CD_WHITE)
return IUP_DEFAULT
end function

procedure main()
IupOpen()

canvas = IupCanvas(NULL)
IupSetAttribute(canvas, "RASTERSIZE", "1250x300")
IupSetCallback(canvas, "MAP_CB", Icallback("map_cb"))

dlg = IupDialog(canvas,"DIALOGFRAME=YES")
IupSetAttribute(dlg, "TITLE", "Triangle overlap")
IupSetCallback(canvas, "ACTION", Icallback("redraw_cb"))
IupCloseOnEscape(dlg)

IupMap(dlg)
IupSetAttribute(canvas, "RASTERSIZE", NULL)
IupShowXY(dlg,IUP_CENTER,IUP_CENTER)
IupMainLoop()
IupClose()
end procedure

main()

Python

Using numpy:

from __future__ import print_function
import numpy as np

def CheckTriWinding(tri, allowReversed):
trisq = np.ones((3,3))
trisq[:,0:2] = np.array(tri)
detTri = np.linalg.det(trisq)
if detTri < 0.0:
if allowReversed:
a = trisq[2,:].copy()
trisq[2,:] = trisq[1,:]
trisq[1,:] = a
else: raise ValueError("triangle has wrong winding direction")
return trisq

def TriTri2D(t1, t2, eps = 0.0, allowReversed = False, onBoundary = True):
#Trangles must be expressed anti-clockwise
t1s = CheckTriWinding(t1, allowReversed)
t2s = CheckTriWinding(t2, allowReversed)

if onBoundary:
#Points on the boundary are considered as colliding
chkEdge = lambda x: np.linalg.det(x) < eps
else:
#Points on the boundary are not considered as colliding
chkEdge = lambda x: np.linalg.det(x) <= eps

#For edge E of trangle 1,
for i in range(3):
edge = np.roll(t1s, i, axis=0)[:2,:]

#Check all points of trangle 2 lay on the external side of the edge E. If
#they do, the triangles do not collide.
if (chkEdge(np.vstack((edge, t2s))) and
chkEdge(np.vstack((edge, t2s))) and
chkEdge(np.vstack((edge, t2s)))):
return False

#For edge E of trangle 2,
for i in range(3):
edge = np.roll(t2s, i, axis=0)[:2,:]

#Check all points of trangle 1 lay on the external side of the edge E. If
#they do, the triangles do not collide.
if (chkEdge(np.vstack((edge, t1s))) and
chkEdge(np.vstack((edge, t1s))) and
chkEdge(np.vstack((edge, t1s)))):
return False

#The triangles collide
return True

if __name__=="__main__":
t1 = [[0,0],[5,0],[0,5]]
t2 = [[0,0],[5,0],[0,6]]
print (TriTri2D(t1, t2), True)

t1 = [[0,0],[0,5],[5,0]]
t2 = [[0,0],[0,6],[5,0]]
print (TriTri2D(t1, t2, allowReversed = True), True)

t1 = [[0,0],[5,0],[0,5]]
t2 = [[-10,0],[-5,0],[-1,6]]
print (TriTri2D(t1, t2), False)

t1 = [[0,0],[5,0],[2.5,5]]
t2 = [[0,4],[2.5,-1],[5,4]]
print (TriTri2D(t1, t2), True)

t1 = [[0,0],[1,1],[0,2]]
t2 = [[2,1],[3,0],[3,2]]
print (TriTri2D(t1, t2), False)

t1 = [[0,0],[1,1],[0,2]]
t2 = [[2,1],[3,-2],[3,4]]
print (TriTri2D(t1, t2), False)

#Barely touching
t1 = [[0,0],[1,0],[0,1]]
t2 = [[1,0],[2,0],[1,1]]
print (TriTri2D(t1, t2, onBoundary = True), True)

#Barely touching
t1 = [[0,0],[1,0],[0,1]]
t2 = [[1,0],[2,0],[1,1]]
print (TriTri2D(t1, t2, onBoundary = False), False)

{{out}}

True True
True True
False False
True True
False False
False False
True True
/False False

Using shapely:

from __future__ import print_function
from shapely.geometry import Polygon

def PolyOverlaps(poly1, poly2):
poly1s = Polygon(poly1)
poly2s = Polygon(poly2)
return poly1s.intersects(poly2s)

if __name__=="__main__":
t1 = [[0,0],[5,0],[0,5]]
t2 = [[0,0],[5,0],[0,6]]
print (PolyOverlaps(t1, t2), True)

t1 = [[0,0],[0,5],[5,0]]
t2 = [[0,0],[0,6],[5,0]]
print (PolyOverlaps(t1, t2), True)

t1 = [[0,0],[5,0],[0,5]]
t2 = [[-10,0],[-5,0],[-1,6]]
print (PolyOverlaps(t1, t2), False)

t1 = [[0,0],[5,0],[2.5,5]]
t2 = [[0,4],[2.5,-1],[5,4]]
print (PolyOverlaps(t1, t2), True)

t1 = [[0,0],[1,1],[0,2]]
t2 = [[2,1],[3,0],[3,2]]
print (PolyOverlaps(t1, t2), False)

t1 = [[0,0],[1,1],[0,2]]
t2 = [[2,1],[3,-2],[3,4]]
print (PolyOverlaps(t1, t2), False)

#Barely touching
t1 = [[0,0],[1,0],[0,1]]
t2 = [[1,0],[2,0],[1,1]]
print (PolyOverlaps(t1, t2), "?")

{{out}}

True True
True True
False False
True True
False False
False False
True ?

QB64

DATA 0,0,5,0,0,5,0,0,5,0,0,6
DATA 0,0,0,5,5,0,0,0,0,5,5,0
DATA 0,0,5,0,0,5,-10,0,-5,0,-1,6
DATA 0,0,5,0,2.5,5,0,4,2.5,-1,5,4
DATA 0,0,1,1,0,2,2,1,3,0,3,2
DATA 0,0,1,1,0,2,2,1,3,-2,3,4

TYPE point
x AS INTEGER
y AS INTEGER
END TYPE

DIM coord(12, 3) AS point

workscreen = _NEWIMAGE(800, 800, 32)
backscreen = _NEWIMAGE(800, 800, 32)

SCREEN workscreen
FOR i = 1 TO 12 '12 triangles
FOR j = 1 TO 3 'with 3 coordinates for each
FixCoord coord(i, j)
NEXT

NEXT
_DELAY .5
_SCREENMOVE _MIDDLE

FOR i = 1 TO 12
_DEST workscreen
CLS
_DEST backscreen
_DONTBLEND
CLS , 0
PSET (coord(i, 1).x, coord(i, 1).y), _RGBA32(255, 255, 255, 128)
FOR j = 2 TO 3
LINE -(coord(i, j).x, coord(i, j).y), _RGBA32(255, 255, 255, 128)
NEXT
LINE -(coord(i, 1).x, coord(i, 1).y), _RGBA32(255, 255, 255, 128)
xinside = (coord(i, 1).x + coord(i, 2).x + coord(i, 3).x) / 3
yinside = (coord(i, 1).y + coord(i, 2).y + coord(i, 3).y) / 3
PAINT (xinside, yinside), _RGBA32(255, 255, 255, 128)
_BLEND
_PUTIMAGE , backscreen, 0
CLS , 0
_DONTBLEND
i = i + 1
PSET (coord(i, 1).x, coord(i, 1).y), _RGBA32(255, 0, 0, 128)
FOR j = 2 TO 3
LINE -(coord(i, j).x, coord(i, j).y), _RGBA32(255, 0, 0, 128)
NEXT
LINE -(coord(i, 1).x, coord(i, 1).y), _RGBA32(255, 0, 0, 128)
xinside = (coord(i, 1).x + coord(i, 2).x + coord(i, 3).x) / 3
yinside = (coord(i, 1).y + coord(i, 2).y + coord(i, 3).y) / 3
PAINT (xinside, yinside), _RGBA32(255, 0, 0, 128)
_BLEND
_PUTIMAGE , backscreen, 0
_DEST workscreen
_SOURCE workscreen
overlap = 0
FOR x = 0 TO 999
FOR y = 0 TO 999
IF POINT(x, y) = _RGBA32(190, 63, 63, 255) THEN overlap = -1: GOTO overlap
NEXT
NEXT
overlap:
IF overlap THEN PRINT "OVERLAP" ELSE PRINT "NO OVERLAP"
SLEEP
NEXT
SYSTEM

SUB FixCoord (p AS point)
p.x = (10 + p.x) * 30 + 100
p.y = (10 + p.y) * 30
END SUB

Racket

{{trans|Zkl}}

#lang racket

;; A triangle is a list of three pairs of points: '((x . y) (x . y) (x . y))
(define (to-tri x1 y1 x2 y2 x3 y3) `((,x1 . ,y1) (,x2 . ,y2) (,x3 . ,y3)))

(define det-2D
(match-lambda
[`((,x1 . ,y1) (,x2 . ,y2) (,x3 . ,y3)) (+ (* x1 (- y2 y3)) (* x2 (- y3 y1)) (* x3 (- y1 y2)))]))

(define (assert-triangle-winding triangle allow-reversed?)
(cond
[(>= (det-2D triangle) 0) triangle]
[allow-reversed? (match triangle [(list p1 p2 p3) (list p1 p3 p2)])]
[else (error 'assert-triangle-winding "triangle is wound in wrong direction")]))

(define (tri-tri-2d? triangle1 triangle2
#:ϵ (ϵ 0)
#:allow-reversed? (allow-reversed? #f)
#:on-boundary? (on-boundary? #t))
(define check-edge
(if on-boundary? ; Points on the boundary are considered as colliding
(λ (triangle) (< (det-2D triangle) ϵ))
(λ (triangle) (<= (det-2D triangle) ϵ))))

(define (inr t1 t2)
(for*/and ((i (in-range 3)))
;; Check all points of trangle 2 lay on the external side
;; of the edge E. If they do, the triangles do not collide.
(define t1.i (list-ref t1 i))
(define t1.j (list-ref t1 (modulo (add1 i) 3)))
(not (for/and ((k (in-range 3))) (check-edge (list (list-ref t2 k) t1.i t1.j))))))

(let (;; Trangles must be expressed anti-clockwise
(tri1 (assert-triangle-winding triangle1 allow-reversed?))
(tri2 (assert-triangle-winding triangle2 allow-reversed?)))
(and (inr tri1 tri2) (inr tri2 tri1))))

;; ---------------------------------------------------------------------------------------------------
(module+ test
(require rackunit)

(define triangleses ; pairs of triangles
(for/list ((a.b (in-list '(((0 0  5 0  0   5) (  0 0   5    0   0 6))
((0 0  0 5  5   0) (  0 0   0    5   5 0))
((0 0  5 0  0   5) (-10 0  -5    0  -1 6))
((0 0  5 0  2.5 5) (  0 4   2.5 -1   5 4))
((0 0  1 1  0   2) (  2 1   3    0   3 2))
((0 0  1 1  0   2) (  2 1   3   -2   3 4))))))
(map (curry apply to-tri) a.b)))

(check-equal?
(for/list ((t1.t2 (in-list triangleses)))
(define t1 (first t1.t2))
(define t2 (second t1.t2))
(define-values (r reversed?)
(with-handlers ([exn:fail? (λ (_) (values (tri-tri-2d? t1 t2 #:allow-reversed? #t) #t))])
(values (tri-tri-2d? t1 t2) #f)))
(cons r reversed?))
'((#t . #f) (#t . #t) (#f . #f) (#t . #f) (#f . #f) (#f . #f)))

(let ((c1 (to-tri 0 0  1 0  0 1)) (c2 (to-tri 1 0  2 0  1 1)))
(check-true (tri-tri-2d? c1 c2 #:on-boundary? #t))
(check-false (tri-tri-2d? c1 c2 #:on-boundary? #f))))

No output → all tests passed

REXX

Note: The triangles must be real triangles (no edge of length 0)

/* REXX */
Signal On Halt
Signal On Novalue
Signal On Syntax

fid='trio.in'
oid='trio.txt'; 'erase' oid

Call trio_test '0 0   5 0   0 5   0 0   5  0   0 6'
Call trio_test '0 0   0 5   5 0   0 0   0  5   5 0'
Call trio_test '0 0   5 0   0 5 -10 0  -5  0  -1 6'
Call trio_test '0 0   5 0 2.5 5   0 4 2.5 -1   5 4'
Call trio_test '0 0   1 1   0 2   2 1   3  0   3 2'
Call trio_test '0 0   1 1   0 2   2 1   3 -2   3 4'
Call trio_test '0 0   1 0   0 1   1 0   2  0   1 1'

Call trio_test '1 0   3 0   2 2   1 3   3  3   2 5'
Call trio_test '1 0   3 0   2 2   1 3   3  3   2 2'
Call trio_test '0 0   2 0   2 2   3 3   5  3   5 5'
Call trio_test '2 0   2 6   1 8   0 1   0  5   8 3'
Call trio_test '0 0   4 0   0 4   0 2   2  0   2 2'
Call trio_test '0 0   4 0   0 4   1 1   2  1   1 2'
Exit

trio_test:
parse Arg tlist
tlist=space(tlist)
Parse Arg ax ay bx by cx cy dx dy ex ey fx fy

Say 'ABC:' show_p(ax ay) show_p(bx by) show_p(cx cy)
Say 'DEF:' show_p(dx dy) show_p(ex ey) show_p(fx fy)

bordl=bord(tlist)  /* corners that are on the other triangle's edges */
If bordl<>'' Then
Say 'Corners on the other triangle''s edges:' bordl
wb=words(bordl)                     /* how many of them?             */
Select
When wb=3 Then Do                 /* all three match               */
If ident(ax ay,bx by,cx cy,dx dy,ex ey,fx fy) Then
Say 'Triangles are identical'
Else
Say 'Triangles overlap'
Say ''
Return
End
When wb=2 Then Do                 /* two of them match             */
Say 'Triangles overlap'
Say '  they have a common edge 'bordl
Say ''
Return
End
When wb=1 Then Do                 /* one of them match             */
Say 'Triangles touch on' bordl  /* other parts may overlap       */
Say '  we analyze further'
End
Otherwise                         /* we know nothing yet           */
Nop
End

trio_result=trio(tlist)             /* any other overlap?            */

Select
When trio_result=0 Then Do        /* none whatsoever               */
If wb=1 Then
Say 'Triangles touch (border case) at' show_p(bordl)
Else
Say 'Triangles don''t overlap'
End
When trio_result>0 Then           /* plain overlapping case        */
Say 'Triangles overlap'
End
Say ''
Return

trio:
/*---------------------------------------------------------------------
* Determine if two triangles overlap
*--------------------------------------------------------------------*/
parse Arg tlist
Parse Arg pax pay pbx pby pcx pcy pdx pdy pex pey pfx pfy

abc=subword(tlist,1,6)
def=subword(tlist,7,6)

Do i=1 To 3
s.i=subword(abc abc,i*2-1,4)
t.i=subword(def def,i*2-1,4)
End

abc_=''
def_=''

Do i=1 To 3
abc.i=subword(abc,i*2-1,2)   /* corners of ABC */
def.i=subword(def,i*2-1,2)   /* corners of DEF */
Parse Var abc.i x y; abc_=abc_ '('||x','y')'
Parse Var def.i x y; def_=def_ '('||x','y')'
End
Call o 'abc_='abc_
Call o 'def_='def_

over=0
Do i=1 To 3 Until over
Do j=1 To 3 Until over
If ssx(s.i t.j) Then Do       /* intersection of two edges     */
over=1
Leave
End
End
End

If over=0 Then Do                   /* no edge intersection found    */
Do ii=1 To 3 Until over           /* look for first possibility    */
Call o '    '  'abc.'ii'='abc.ii 'def='def
Call o 'ii='ii 'def.'ii'='def.ii 'abc='abc
If in_tri(abc.ii,def) Then Do   /* a corner of ABC is in DEF     */
Say abc.ii 'is within' def
over=1
End
Else If in_tri(def.ii,abc) Then Do  /* a corner of DEF is in ABC */
Say def.ii 'is within' abc
over=1
End
End
End

If over=0 Then rw='don''t '
Else rw=''

Call o 'Triangles' show_p(pax pay) show_p(pbx pby) show_p(pcx pcy),
'and' show_p(pdx pdy) show_p(pex pey) show_p(pfx pfy),
rw'overlap'
Call o ''
Return over

ssx: Procedure Expose oid bordl
/*---------------------------------------------------------------------
* Intersection of 2 line segments A-B and C-D
*--------------------------------------------------------------------*/
Parse Arg xa ya xb yb xc yc xd yd

d=ggx(xa ya xb yb xc yc xd yd)

Call o 'ssx:' arg(1) d
res=0
Select
When d='-' Then res=0
When d='I' Then Do
If xa<>xb Then Do
xab_min=min(xa,xb)
xcd_min=min(xc,xd)
xab_max=max(xa,xb)
xcd_max=max(xc,xd)
If xab_min>xcd_max |,
xcd_min>xab_max Then
res=0
Else Do
res=1
Select
When xa=xc & isb(xc,xb,xd)=0 Then Do; x=xa; y=ya; End
When xb=xc & isb(xc,xa,xd)=0 Then Do; x=xb; y=yb; End
When xa=xd & isb(xc,xb,xd)=0 Then Do; x=xa; y=ya; End
When xb=xd & isb(xc,xa,xd)=0 Then Do; x=xb; y=yb; End
Otherwise Do
x='*'
y=ya
End
End
Call o  'ssx:' x y
End
End
Else Do
yab_min=min(ya,yb)
ycd_min=min(yc,yd)
yab_max=max(ya,yb)
ycd_max=max(yc,yd)
If yab_min>ycd_max |,
ycd_min>yab_max Then
res=0
Else Do
res=1
x=xa
y='*'
Parse Var bordl x_bord '/' y_bord
If x=x_bord Then Do
Call o  xa'/* IGNORED'
res=0
End
End
End
End
Otherwise Do
Parse Var d x y
If is_between(xa,x,xb) &,
is_between(xc,x,xd) &,
is_between(ya,y,yb) &,
is_between(yc,y,yd) Then Do
If x'/'y<>bordl Then
res=1
End
End
End
If res=1 Then Do
Say 'Intersection of line segments: ('||x'/'y')'
Parse Var bordl x_bord '/' y_bord
If x=x_bord Then Do
res=0
Call o x'/'y 'IGNORED'
End
End
Else Call o  'ssx: -'
Return res

ggx: Procedure Expose oid bordl
/*---------------------------------------------------------------------
* Intersection of 2 (straight) lines
*--------------------------------------------------------------------*/
Parse Arg xa ya xb yb xc yc xd yd
res=''
If xa=xb Then Do
k1='*'
x1=xa
If ya=yb Then Do
res='Points A and B are identical'
rs='*'
End
End
Else Do
k1=(yb-ya)/(xb-xa)
d1=ya-k1*xa
End
If xc=xd Then Do
k2='*'
x2=xc
If yc=yd Then Do
res='Points C and D are identical'
rs='*'
End
End
Else Do
k2=(yd-yc)/(xd-xc)
d2=yc-k2*xc
End

If res='' Then Do
If k1='*' Then Do
If k2='*' Then Do
If x1=x2 Then Do
res='Lines AB and CD are identical'
rs='I'
End
Else Do
res='Lines AB and CD are parallel'
rs='-'
End
End
Else Do
x=x1
y=k2*x+d2
End
End
Else Do
If k2='*' Then Do
x=x2
y=k1*x+d1
End
Else Do
If k1=k2 Then Do
If d1=d2 Then Do
res='Lines AB and CD are identical'
rs='I'
End
Else Do
res='Lines AB and CD are parallel'
rs='-'
End
End
Else Do
x=(d2-d1)/(k1-k2)
y=k1*x+d1
End
End
End
End
If res='' Then Do
res='Intersection is ('||x'/'y')'
rs=x y
Call o 'line intersection' x y
End
Call o 'A=('xa'/'ya') B=('||xb'/'yb') C=('||xc'/'yc') D=('||xd'/'yd')' '-->' res
Return rs

isb: Procedure
Parse Arg a,b,c
Return sign(b-a)<>sign(b-c)

is_between: Procedure Expose oid
Parse Arg a,b,c
Return diff_sign(b-a,b-c)

diff_sign: Procedure
Parse Arg diff1,diff2
Return (sign(diff1)<>sign(diff2))|(sign(diff1)=0)

o:
/*y 'sigl='sigl */
Return lineout(oid,arg(1))

in_tri: Procedure Expose oid bordl
/*---------------------------------------------------------------------
* Determine if the point (px/py) is within the given triangle
*--------------------------------------------------------------------*/
Parse Arg px py,ax ay bx by cx cy
abc=ax ay bx by cx cy
res=0
maxx=max(ax,bx,cx)
minx=min(ax,bx,cx)
maxy=max(ay,by,cy)
miny=min(ay,by,cy)

If px>maxx|px<minx|py>maxy|py<miny Then
Return 0

Parse Value mk_g(ax ay,bx by) With k.1 d.1 x.1
Parse Value mk_g(bx by,cx cy) With k.2 d.2 x.2
Parse Value mk_g(cx cy,ax ay) With k.3 d.3 x.3
/*
say 'g1:' show_g(k.1,d.1,x.1)
say 'g2:' show_g(k.2,d.2,x.2)
say 'g3:' show_g(k.3,d.3,x.3)
Say px py '-' ax ay bx by cx cy
*/
Do i=1 To 3
Select
When k.i='*' Then
Call o 'g.'i':' 'x='||x.i
When k.i=0 Then
Call o 'g.'i':' 'y='d.i
Otherwise
Call o 'g.'i':' 'y=' k.i'*x'dd(d.i)
End
End

If k.1='*' Then Do
y2=k.2*px+d.2
y3=k.3*px+d.3
If is_between(y2,py,y3) Then
res=1
End
Else Do
kp1=k.1
dp1=py-kp1*px
If k.2='*' Then
x12=x.2
Else
x12=(d.2-dp1)/(kp1-k.2)
If k.3='*' Then
x13=x.3
Else
x13=(d.3-dp1)/(kp1-k.3)
If is_between(x12,px,x13) Then
res=1
End

If res=1 Then rr=' '
Else rr=' not '
If pos(px'/'py,bordl)>0 Then Do
ignored=' but is IGNORED'
res=0
End
Else
ignored=''
Say 'P ('px','py') is'rr'in' abc  ignored
Return res

bord:
/*---------------------------------------------------------------------
* Look for corners of triangles that are situated
* on the edges of the other triangle
*--------------------------------------------------------------------*/
parse Arg tlist
Parse Arg pax pay pbx pby pcx pcy pdx pdy pex pey pfx pfy
bordl=''
abc=subword(tlist,1,6)
def=subword(tlist,7,6)

Do i=1 To 3
s.i=subword(abc abc,i*2-1,4)
t.i=subword(def def,i*2-1,4)
End

abc_=''
def_=''
Do i=1 To 3
abc.i=subword(abc,i*2-1,2)
def.i=subword(def,i*2-1,2)
Parse Var abc.i x y; abc_=abc_ '('||x','y')'
Parse Var def.i x y; def_=def_ '('||x','y')'
End

Do i=1 To 3
i1=i+1
If i1=4 Then i1=1
Parse Value mk_g(abc.i,abc.i1) With k.1.i d.1.i x.1.i
Parse Value mk_g(def.i,def.i1) With k.2.i d.2.i x.2.i
End
Do i=1 To 3
Call o  show_g(k.1.i,d.1.i,x.1.i)
End
Do i=1 To 3
Call o  show_g(k.2.i,d.2.i,x.2.i)
End

pl=''
Do i=1 To 3
p=def.i
Do j=1 To 3
j1=j+1
If j1=4 Then j1=1
g='1.'j
If in_segment(p,abc.j,abc.j1) Then Do
pp=Translate(p,'/',' ')
If wordpos(pp,bordl)=0 Then
bordl=bordl pp
End
Call o  show_p(p) show_g(k.g,d.g,x.g) '->' bordl
End
End
Call o  'Points on abc:' pl

pl=''
Do i=1 To 3
p=abc.i
Do j=1 To 3
j1=j+1
If j1=4 Then j1=1
g='2.'j
If in_segment(p,def.j,def.j1)Then Do
pp=Translate(p,'/',' ')
If wordpos(pp,bordl)=0 Then
bordl=bordl pp
End
Call o  show_p(p) show_g(k.g,d.g,x.g) '->' bordl
End
End
Call o  'Points on def:' pl

Return bordl

in_segment: Procedure Expose g. sigl
/*---------------------------------------------------------------------
* Determine if point x/y is on the line segment ax/ay bx/by
*--------------------------------------------------------------------*/
Parse Arg x y,ax ay,bx by
Call show_p(x y) show_p(ax ay) show_p(bx by)
Parse Value mk_g(ax ay,bx by) With gk gd gx
Select
When gx<>'' Then
res=(x=gx & is_between(ay,y,by))
When gk='*' Then
res=(y=gd & is_between(ax,x,bx))
Otherwise Do
yy=gk*x+gd
res=(y=yy & is_between(ax,x,bx))
End
End
Return res

mk_g: Procedure Expose g.
/*---------------------------------------------------------------------
* given two points (a and b)
* compute y=k*x+d or, if a vertical line, k='*'; x=c
*--------------------------------------------------------------------*/
Parse Arg a,b                       /* 2 points                      */
Parse Var a ax ay
Parse Var b bx by
If ax=bx Then Do                    /* vertical line                 */
gk='*'                            /* special slope                 */
gx=ax                             /* x=ax is  the equation         */
gd='*'                            /* not required                  */
End
Else Do
gk=(by-ay)/(bx-ax)                /* compute slope                 */
gd=ay-gk*ax                       /* compute y-distance            */
gx=''                             /* not required                  */
End
Return gk gd gx

is_between: Procedure
Parse Arg a,b,c
Return diff_sign(b-a,b-c)

diff_sign: Procedure
Parse Arg diff1,diff2
Return (sign(diff1)<>sign(diff2))|(sign(diff1)=0)

show_p: Procedure
Call trace 'O'
Parse Arg x y
If pos('/',x)>0 Then
Parse Var x x '/' y
Return space('('||x'/'y')',0)

isb: Procedure Expose oid
Parse Arg a,b,c
Return sign(b-a)<>sign(b-c)

o: Call o  arg(1)
Return

show_g: Procedure
/*---------------------------------------------------------------------
* given slope, y-distance, and (special) x-value
* compute y=k*x+d or, if a vertical line, k='*'; x=c
*--------------------------------------------------------------------*/
Parse Arg k,d,x
Select
When k='*' Then res='x='||x       /* vertical line                 */
When k=0   Then res='y='d         /* horizontal line               */
Otherwise Do                      /* ordinary line                 */
Select
When k=1  Then res='y=x'dd(d)
When k=-1 Then res='y=-x'dd(d)
Otherwise      res='y='k'*x'dd(d)
End
End
End
Return res

dd: Procedure
/*---------------------------------------------------------------------
* prepare y-distance for display
*--------------------------------------------------------------------*/
Parse Arg dd
Select
When dd=0 Then dd=''            /* omit dd if it's zero          */
When dd<0 Then dd=dd            /* use dd as is (-value)         */
Otherwise      dd='+'dd         /* prepend '+' to positive dd    */
End
Return dd

ident: Procedure
/*---------------------------------------------------------------------
* Determine if the corners ABC match those of DEF (in any order)
*--------------------------------------------------------------------*/
cnt.=0
Do i=1 To 6
Parse Value Arg(i) With x y
cnt.x.y=cnt.x.y+1
End
Do i=1 To 3
Parse Value Arg(i) With x y
If cnt.x.y<>2 Then
Return 0
End
Return 1

Novalue:
Say  'Novalue raised in line' sigl
Say  sourceline(sigl)
Say  'Variable' condition('D')
Signal lookaround

Syntax:
Say  'Syntax raised in line' sigl
Say  sourceline(sigl)
Say  'rc='rc '('errortext(rc)')'

halt:
lookaround:
If fore() Then Do
Say  'You can look around now.'
Trace ?R
Nop
End
Exit 12

{{out}}

ABC: (0/0) (5/0) (0/5)
DEF: (0/0) (5/0) (0/6)
Corners on the other triangle's edges:  0/0 5/0 0/5
Triangles overlap

ABC: (0/0) (0/5) (5/0)
DEF: (0/0) (0/5) (5/0)
Corners on the other triangle's edges:  0/0 0/5 5/0
Triangles are identical

ABC: (0/0) (5/0) (0/5)
DEF: (-10/0) (-5/0) (-1/6)
Triangles don't overlap

ABC: (0/0) (5/0) (2.5/5)
DEF: (0/4) (2.5/-1) (5/4)
Intersection of line segments: (2/0)
Triangles overlap

ABC: (0/0) (1/1) (0/2)
DEF: (2/1) (3/0) (3/2)
Triangles don't overlap

ABC: (0/0) (1/1) (0/2)
DEF: (2/1) (3/-2) (3/4)
Triangles don't overlap

ABC: (0/0) (1/0) (0/1)
DEF: (1/0) (2/0) (1/1)
Corners on the other triangle's edges:  1/0
Triangles touch on  1/0
we analyze further
Intersection of line segments: (1/0)
P (1,0) is in 0 0 1 0 0 1  but is IGNORED
P (1,0) is in 1 0 2 0 1 1  but is IGNORED
P (1,1) is not in 0 0 1 0 0 1
Triangles touch (border case) at (1/0)

ABC: (1/0) (3/0) (2/2)
DEF: (1/3) (3/3) (2/5)
Triangles don't overlap

ABC: (1/0) (3/0) (2/2)
DEF: (1/3) (3/3) (2/2)
Corners on the other triangle's edges:  2/2
Triangles touch on  2/2
we analyze further
P (2,2) is in 1 3 3 3 2 2  but is IGNORED
P (2,2) is in 1 0 3 0 2 2  but is IGNORED
Triangles touch (border case) at (2/2)

ABC: (0/0) (2/0) (2/2)
DEF: (3/3) (5/3) (5/5)
Triangles don't overlap

ABC: (2/0) (2/6) (1/8)
DEF: (0/1) (0/5) (8/3)
Intersection of line segments: (2/4.50)
Triangles overlap

ABC: (0/0) (4/0) (0/4)
DEF: (0/2) (2/0) (2/2)
Corners on the other triangle's edges:  0/2 2/0 2/2
Triangles overlap

ABC: (0/0) (4/0) (0/4)
DEF: (1/1) (2/1) (1/2)
P (1,1) is in 0 0 4 0 0 4
1 1 is within 0 0 4 0 0 4
Triangles overlap

Scala

{{trans|Kotlin}}

object Overlap {
type Point = (Double, Double)

class Triangle(var p1: Point, var p2: Point, var p3: Point) {
override def toString: String = s"Triangle: \$p1, \$p2, \$p3"
}

def det2D(t: Triangle): Double = {
val (p1, p2, p3) = (t.p1, t.p2, t.p3)
p1._1 * (p2._2 - p3._2) +
p2._1 * (p3._2 - p1._2) +
p3._1 * (p1._2 - p2._2)
}

def checkTriWinding(t: Triangle, allowReversed: Boolean): Unit = {
val detTri = det2D(t)
if (detTri < 0.0) {
if (allowReversed) {
val a = t.p3
t.p3 = t.p2
t.p2 = a
} else throw new RuntimeException("Triangle has wrong winding direction")
}
}

def boundaryCollideChk(t: Triangle, eps: Double): Boolean = det2D(t) < eps

def boundaryDoesntCollideChk(t: Triangle, eps: Double): Boolean = det2D(t) <= eps

def triTri2D(t1: Triangle, t2: Triangle, eps: Double = 0.0, allowReversed: Boolean = false, onBoundary: Boolean = true): Boolean = {
//triangles must be expressed anti-clockwise
checkTriWinding(t1, allowReversed)
checkTriWinding(t2, allowReversed)
// 'onBoundary' determines whether points on boundary are considered as colliding or not
val chkEdge = if (onBoundary) Overlap.boundaryCollideChk _ else Overlap.boundaryDoesntCollideChk _
val lp1 = Array(t1.p1, t1.p2, t1.p3)
val lp2 = Array(t2.p1, t2.p2, t2.p3)

// for each edge E of t1
for (i <- 0 until 3) {
val j = (i + 1) % 3
// Check all points of t2 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge(new Triangle(lp1(i), lp1(j), lp2(0)), eps)
&& chkEdge(new Triangle(lp1(i), lp1(j), lp2(1)), eps)
&& chkEdge(new Triangle(lp1(i), lp1(j), lp2(2)), eps)) return false
}

// for each edge E of t2
for (i <- 0 until 3) {
val j = (i + 1) % 3
// Check all points of t1 lay on the external side of edge E.
// If they do, the triangles do not overlap.
if (chkEdge(new Triangle(lp2(i), lp2(j), lp1(0)), eps)
&& chkEdge(new Triangle(lp2(i), lp2(j), lp1(1)), eps)
&& chkEdge(new Triangle(lp2(i), lp2(j), lp1(2)), eps)) return false
}

// The triangles overlap
true
}

def main(args: Array[String]): Unit = {
var t1 = new Triangle((0.0, 0.0), (5.0, 0.0), (0.0, 5.0))
var t2 = new Triangle((0.0, 0.0), (5.0, 0.0), (0.0, 6.0))
println(s"\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

// need to allow reversed for this pair to avoid exception
t1 = new Triangle((0.0, 0.0), (0.0, 5.0), (5.0, 0.0))
t2 = t1
println(s"\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2, 0.0, allowReversed = true)) "overlap (reversed)" else "do not overlap")

t1 = new Triangle((0.0, 0.0), (5.0, 0.0), (0.0, 5.0))
t2 = new Triangle((-10.0, 0.0), (-5.0, 0.0), (-1.0, 6.0))
println(s"\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

t1.p3 = (2.5, 5.0)
t2 = new Triangle((0.0, 4.0), (2.5, -1.0), (5.0, 4.0))
println(s"\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

t1 = new Triangle((0.0, 0.0), (1.0, 1.0), (0.0, 2.0))
t2 = new Triangle((2.0, 1.0), (3.0, 0.0), (3.0, 2.0))
println(s"\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

t2 = new Triangle((2.0, 1.0), (3.0, -2.0), (3.0, 4.0))
println(s"\n\$t1 and\n\$t2")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

t1 = new Triangle((0.0, 0.0), (1.0, 0.0), (0.0, 1.0))
t2 = new Triangle((1.0, 0.0), (2.0, 0.0), (1.0, 1.1))
println(s"\n\$t1 and\n\$t2")
println("which have only a single corner in contact, if boundary points collide")
println(if (triTri2D(t1, t2)) "overlap" else "do not overlap")

println(s"\n\$t1 and\n\$t2")
println("which have only a single corner in contact, if boundary points do not collide")
println(if (triTri2D(t1, t2, onBoundary = false)) "overlap" else "do not overlap")
}
}

{{out}}

Triangle: (0.0,0.0), (5.0,0.0), (0.0,5.0) and
Triangle: (0.0,0.0), (5.0,0.0), (0.0,6.0)
overlap

Triangle: (0.0,0.0), (0.0,5.0), (5.0,0.0) and
Triangle: (0.0,0.0), (0.0,5.0), (5.0,0.0)
overlap (reversed)

Triangle: (0.0,0.0), (5.0,0.0), (0.0,5.0) and
Triangle: (-10.0,0.0), (-5.0,0.0), (-1.0,6.0)
do not overlap

Triangle: (0.0,0.0), (5.0,0.0), (2.5,5.0) and
Triangle: (0.0,4.0), (2.5,-1.0), (5.0,4.0)
overlap

Triangle: (0.0,0.0), (1.0,1.0), (0.0,2.0) and
Triangle: (2.0,1.0), (3.0,0.0), (3.0,2.0)
do not overlap

Triangle: (0.0,0.0), (1.0,1.0), (0.0,2.0) and
Triangle: (2.0,1.0), (3.0,-2.0), (3.0,4.0)
do not overlap

Triangle: (0.0,0.0), (1.0,0.0), (0.0,1.0) and
Triangle: (1.0,0.0), (2.0,0.0), (1.0,1.1)
which have only a single corner in contact, if boundary points collide
overlap

Triangle: (0.0,0.0), (1.0,0.0), (0.0,1.0) and
Triangle: (1.0,0.0), (2.0,0.0), (1.0,1.1)
which have only a single corner in contact, if boundary points do not collide
do not overlap

Visual Basic .NET

{{trans|C#}}

Module Module1

Class Triangle
Property P1 As Tuple(Of Double, Double)
Property P2 As Tuple(Of Double, Double)
Property P3 As Tuple(Of Double, Double)

Sub New(p1 As Tuple(Of Double, Double), p2 As Tuple(Of Double, Double), p3 As Tuple(Of Double, Double))
Me.P1 = p1
Me.P2 = p2
Me.P3 = p3
End Sub

Function Det2D() As Double
Return P1.Item1 * (P2.Item2 - P3.Item2) +
P2.Item1 * (P3.Item2 - P1.Item2) +
P3.Item1 * (P1.Item2 - P2.Item2)
End Function

Sub CheckTriWinding(allowReversed As Boolean)
Dim detTri = Det2D()
If detTri < 0.0 Then
If allowReversed Then
Dim a = P3
P3 = P2
P2 = a
Else
Throw New Exception("Triangle has wrong winding direction")
End If
End If
End Sub

Function BoundaryCollideChk(eps As Double) As Boolean
Return Det2D() < eps
End Function

Function BoundaryDoesntCollideChk(eps As Double) As Boolean
Return Det2D() <= eps
End Function

Public Overrides Function ToString() As String
Return String.Format("Triangle: {0}, {1}, {2}", P1, P2, P3)
End Function
End Class

Function TriTri2D(t1 As Triangle, t2 As Triangle, Optional eps As Double = 0.0, Optional alloweReversed As Boolean = False, Optional onBoundary As Boolean = True) As Boolean
'Triangles must be expressed anti-clockwise
t1.CheckTriWinding(alloweReversed)
t2.CheckTriWinding(alloweReversed)

'"onboundary" determines whether points on boundary are considered as colliding or not
Dim chkEdge = If(onBoundary, Function(t As Triangle) t.BoundaryCollideChk(eps), Function(t As Triangle) t.BoundaryDoesntCollideChk(eps))
Dim lp1 As New List(Of Tuple(Of Double, Double)) From {t1.P1, t1.P2, t1.P3}
Dim lp2 As New List(Of Tuple(Of Double, Double)) From {t2.P1, t2.P2, t2.P3}

'for each edge E of t1
For i = 0 To 2
Dim j = (i + 1) Mod 3
'Check all points of t2 lay on the external side of edge E.
'If they do, the triangles do not overlap.
If chkEdge(New Triangle(lp1(i), lp1(j), lp2(0))) AndAlso
chkEdge(New Triangle(lp1(i), lp1(j), lp2(1))) AndAlso
chkEdge(New Triangle(lp1(i), lp1(j), lp2(2))) Then
Return False
End If
Next

'for each edge E of t2
For i = 0 To 2
Dim j = (i + 1) Mod 3
'Check all points of t1 lay on the external side of edge E.
'If they do, the triangles do not overlap.
If chkEdge(New Triangle(lp2(i), lp2(j), lp1(0))) AndAlso
chkEdge(New Triangle(lp2(i), lp2(j), lp1(1))) AndAlso
chkEdge(New Triangle(lp2(i), lp2(j), lp1(2))) Then
Return False
End If
Next

'The triangles overlap
Return True
End Function

Sub Overlap(t1 As Triangle, t2 As Triangle, Optional eps As Double = 0.0, Optional allowReversed As Boolean = False, Optional onBoundary As Boolean = True)
If TriTri2D(t1, t2, eps, allowReversed, onBoundary) Then
Console.WriteLine("overlap")
Else
Console.WriteLine("do not overlap")
End If
End Sub

Sub Main()
Dim t1 = New Triangle(Tuple.Create(0.0, 0.0), Tuple.Create(5.0, 0.0), Tuple.Create(0.0, 5.0))
Dim t2 = New Triangle(Tuple.Create(0.0, 0.0), Tuple.Create(5.0, 0.0), Tuple.Create(0.0, 6.0))
Console.WriteLine("{0} and", t1)
Console.WriteLine("{0}", t2)
Overlap(t1, t2)
Console.WriteLine()

' need to allow reversed for this pair to avoid exception
t1 = New Triangle(Tuple.Create(0.0, 0.0), Tuple.Create(0.0, 5.0), Tuple.Create(5.0, 0.0))
t2 = t1
Console.WriteLine("{0} and", t1)
Console.WriteLine("{0}", t2)
Overlap(t1, t2, 0.0, True)
Console.WriteLine()

t1 = New Triangle(Tuple.Create(0.0, 0.0), Tuple.Create(5.0, 0.0), Tuple.Create(0.0, 5.0))
t2 = New Triangle(Tuple.Create(-10.0, 0.0), Tuple.Create(-5.0, 0.0), Tuple.Create(-1.0, 6.0))
Console.WriteLine("{0} and", t1)
Console.WriteLine("{0}", t2)
Overlap(t1, t2)
Console.WriteLine()

t1.P3 = Tuple.Create(2.5, 5.0)
t2 = New Triangle(Tuple.Create(0.0, 4.0), Tuple.Create(2.5, -1.0), Tuple.Create(5.0, 4.0))
Console.WriteLine("{0} and", t1)
Console.WriteLine("{0}", t2)
Overlap(t1, t2)
Console.WriteLine()

t1 = New Triangle(Tuple.Create(0.0, 0.0), Tuple.Create(1.0, 1.0), Tuple.Create(0.0, 2.0))
t2 = New Triangle(Tuple.Create(2.0, 1.0), Tuple.Create(3.0, 0.0), Tuple.Create(3.0, 2.0))
Console.WriteLine("{0} and", t1)
Console.WriteLine("{0}", t2)
Overlap(t1, t2)
Console.WriteLine()

t2 = New Triangle(Tuple.Create(2.0, 1.0), Tuple.Create(3.0, -2.0), Tuple.Create(3.0, 4.0))
Console.WriteLine("{0} and", t1)
Console.WriteLine("{0}", t2)
Overlap(t1, t2)
Console.WriteLine()

t1 = New Triangle(Tuple.Create(0.0, 0.0), Tuple.Create(1.0, 0.0), Tuple.Create(0.0, 1.0))
t2 = New Triangle(Tuple.Create(1.0, 0.0), Tuple.Create(2.0, 0.0), Tuple.Create(1.0, 1.1))
Console.WriteLine("{0} and", t1)
Console.WriteLine("{0}", t2)
Console.WriteLine("which have only a single corner in contact, if boundary points collide")
Overlap(t1, t2)
Console.WriteLine()

Console.WriteLine("{0} and", t1)
Console.WriteLine("{0}", t2)
Console.WriteLine("which have only a single corner in contact, if boundary points do not collide")
Overlap(t1, t2, 0.0, False, False)
End Sub

End Module

{{out}}

Triangle: (0, 0), (5, 0), (0, 5) and
Triangle: (0, 0), (5, 0), (0, 6)
overlap

Triangle: (0, 0), (0, 5), (5, 0) and
Triangle: (0, 0), (0, 5), (5, 0)
overlap

Triangle: (0, 0), (5, 0), (0, 5) and
Triangle: (-10, 0), (-5, 0), (-1, 6)
do not overlap

Triangle: (0, 0), (5, 0), (2.5, 5) and
Triangle: (0, 4), (2.5, -1), (5, 4)
overlap

Triangle: (0, 0), (1, 1), (0, 2) and
Triangle: (2, 1), (3, 0), (3, 2)
do not overlap

Triangle: (0, 0), (1, 1), (0, 2) and
Triangle: (2, 1), (3, -2), (3, 4)
do not overlap

Triangle: (0, 0), (1, 0), (0, 1) and
Triangle: (1, 0), (2, 0), (1, 1.1)
which have only a single corner in contact, if boundary points collide
overlap

Triangle: (0, 0), (1, 0), (0, 1) and
Triangle: (1, 0), (2, 0), (1, 1.1)
which have only a single corner in contact, if boundary points do not collide
do not overlap

zkl

{{trans|C++}}

// A triangle is three pairs of points: ( (x,y), (x,y), (x,y) )

fcn det2D(triangle){
p1,p2,p3 := triangle;
p1*(p2 - p3) + p2*(p3 - p1) + p3*(p1 - p2);
}
fcn checkTriWinding(triangle,allowReversed){ //-->triangle, maybe new
detTri:=det2D(triangle);
if(detTri<0.0){
if(allowReversed){ p1,p2,p3 := triangle; return(p1,p3,p2); }  // reverse
else throw(Exception.AssertionError(
"triangle has wrong winding direction"));
}
triangle	// no change
}
fcn triTri2D(triangle1,triangle2, eps=0.0, allowReversed=False, onBoundary=True){
// Trangles must be expressed anti-clockwise
triangle1=checkTriWinding(triangle1, allowReversed);
triangle2=checkTriWinding(triangle2, allowReversed);

chkEdge:=
if(onBoundary) // Points on the boundary are considered as colliding
fcn(triangle,eps){ det2D(triangle)<eps }
else           // Points on the boundary are not considered as colliding
fcn(triangle,eps){ det2D(triangle)<=eps };; // first ; terminates if

t1,t2 := triangle1,triangle2;	// change names to protect the typist
do(2){				// check triangle1 and then triangle2
foreach i in (3){	//For edge E of trangle 1,
j:=(i+1)%3;	// 1,2,0
// Check all points of trangle 2 lay on the external side
// of the edge E. If they do, the triangles do not collide.
if(chkEdge(T(t1[i],t1[j],t2), eps) and
chkEdge(T(t1[i],t1[j],t2), eps) and
chkEdge(T(t1[i],t1[j],t2), eps)) return(False);  // no collision
}
t2,t1 = triangle1,triangle2; // flip and re-test
}
True   // The triangles collide
}
fcn toTri(ax,ay,bx,by,cx,cy){ //-->( (ax,ay),(bx,by),(cx,cy) )
}
triangles:=T(	// pairs of triangles
T(toTri(0,0, 5,0, 0,  5), toTri(  0,0,  5,   0,  0,6)),
T(toTri(0,0, 0,5, 5,  0), toTri(  0,0,  0,   5 , 5,0)),
T(toTri(0,0, 5,0, 0,  5), toTri(-10,0, -5,   0, -1,6)),
T(toTri(0,0, 5,0, 2.5,5), toTri(  0,4,  2.5,-1,  5,4)),
T(toTri(0,0, 1,1, 0,  2), toTri(  2,1,  3,   0,  3,2)),
T(toTri(0,0, 1,1, 0,  2), toTri(  2,1,  3,  -2,  3,4))
);

// Expect: overlap, overlap (reversed), no overlap, overlap, no overlap, no overlap
foreach t1,t2 in (triangles){
reg r, reversed=False;
try{ r=triTri2D(t1,t2) }
catch(AssertionError){ r=triTri2D(t1,t2,0.0,True); reversed=True; }
print(t1,"\n",t2," ");
println(r and "overlap" or "no overlap", reversed and " (reversed)" or "");
println();
}

c1,c2 := toTri(0,0, 1,0, 0,1), toTri(1,0, 2,0, 1,1);
println("Corner case (barely touching): ",triTri2D(c1,c2,0.0,False,True));  // True
println("Corner case (barely touching): ",triTri2D(c1,c2,0.0,False,False)); // False

{{out}}

L(L(0,0),L(5,0),L(0,5))
L(L(0,0),L(5,0),L(0,6)) overlap

L(L(0,0),L(0,5),L(5,0))
L(L(0,0),L(0,5),L(5,0)) overlap (reversed)

L(L(0,0),L(5,0),L(0,5))
L(L(-10,0),L(-5,0),L(-1,6)) no overlap

L(L(0,0),L(5,0),L(2.5,5))
L(L(0,4),L(2.5,-1),L(5,4)) overlap

L(L(0,0),L(1,1),L(0,2))
L(L(2,1),L(3,0),L(3,2)) no overlap

L(L(0,0),L(1,1),L(0,2))
L(L(2,1),L(3,-2),L(3,4)) no overlap

Corner case (barely touching): True
Corner case (barely touching): False