⚠️ Warning: This is a draft ⚠️

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

If you want to help to improve and eventually enable this page, please fork RosettaGit's repository and open a merge request on GitHub.

Find the points which form a [[wp:Convex hull|convex hull]] from a set of arbitrary two dimensional points.

For example, given the points (16,3), (12,17), (0,6), (-4,-6), (16,6), (16,-7), (16,-3), (17,-4), (5,19), (19,-8), (3,16), (12,13), (3,-4), (17,5), (-3,15), (-3,-9), (0,11), (-9,-3), (-4,-2) and (12,10) the convex hull would be (-9,-3), (-3,-9), (19,-8), (17,5), (12,17), (5,19) and (-3,15).

• http://www.geeksforgeeks.org/convex-hull-set-2-graham-scan/

## C

{{trans|C++}}

```
#include <stdio.h>
#include <stdlib.h>

typedef struct tPoint {
int x, y;
} Point;

typedef struct tNode {
Point data;
struct tNode *next;
} Node;

bool ccw(const Point *a, const Point *b, const Point *c) {
return (b->x - a->x) * (c->y - a->y)
> (b->y - a->y) * (c->x - a->x);
}

int comp(const void *lhs, const void *rhs) {
Point lp = *((Point *)lhs);
Point rp = *((Point *)rhs);
if (lp.x < rp.x) return -1;
if (rp.x < lp.x) return 1;
return 0;
}

void freeNode(Node *ptr) {
if (ptr == NULL) {
return;
}

freeNode(ptr->next);
ptr->next = NULL;
free(ptr);
}

Node* pushBack(Node *ptr, Point data) {
Node *tmp = ptr;

if (ptr == NULL) {
ptr = (Node*)malloc(sizeof(Node));
ptr->data = data;
ptr->next = NULL;
return ptr;
}

while (tmp->next != NULL) {
tmp = tmp->next;
}

tmp->next = (Node*)malloc(sizeof(Node));
tmp->next->data = data;
tmp->next->next = NULL;
return ptr;
}

Node* popBack(Node *ptr) {
Node *tmp = ptr;

if (ptr == NULL) {
return NULL;
}
if (ptr->next == NULL) {
free(ptr);
return NULL;
}

while (tmp->next->next != NULL) {
tmp = tmp->next;
}

free(tmp->next);
tmp->next = NULL;
return ptr;
}

void print(Node *ptr) {
printf("[");
if (ptr != NULL) {
printf("(%d, %d)", ptr->data.x, ptr->data.y);
ptr = ptr->next;
}
while (ptr != NULL) {
printf(", (%d, %d)", ptr->data.x, ptr->data.y);
ptr = ptr->next;
}
printf("]");
}

Node* convexHull(int len, Point p[]) {
Node *h = NULL;
Node *hptr = NULL;
size_t hLen = 0;
int i;

qsort(p, len, sizeof(Point), comp);

/* lower hull */
for (i = 0; i < len; ++i) {
while (hLen >= 2) {
hptr = h;
while (hptr->next->next != NULL) {
hptr = hptr->next;
}
if (ccw(&hptr->data, &hptr->next->data, &p[i])) {
break;
}

h = popBack(h);
hLen--;
}

h = pushBack(h, p[i]);
hLen++;
}

/* upper hull */
for (i = len - 1; i >= 0; i--) {
while (hLen >= 2) {
hptr = h;
while (hptr->next->next != NULL) {
hptr = hptr->next;
}
if (ccw(&hptr->data, &hptr->next->data, &p[i])) {
break;
}

h = popBack(h);
hLen--;
}

h = pushBack(h, p[i]);
hLen++;
}

popBack(h);
return h;
}

int main() {
Point points[] = {
{16,  3}, {12, 17}, { 0,  6}, {-4, -6}, {16,  6},
{16, -7}, {16, -3}, {17, -4}, { 5, 19}, {19, -8},
{ 3, 16}, {12, 13}, { 3, -4}, {17,  5}, {-3, 15},
{-3, -9}, { 0, 11}, {-9, -3}, {-4, -2}, {12, 10}
};

Node *hull = convexHull(sizeof(points) / sizeof(Point), points);
printf("Convex Hull: ");
print(hull);
printf("\n");

freeNode(hull);
hull = NULL;

return 0;
}
```

{{out}}

```Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
```

## C++

{{trans|D}}

```#include <algorithm>
#include <iostream>
#include <ostream>
#include <vector>
#include <tuple>

typedef std::tuple<int, int> point;

std::ostream& print(std::ostream& os, const point& p) {
return os << "(" << std::get<0>(p) << ", " << std::get<1>(p) << ")";
}

std::ostream& print(std::ostream& os, const std::vector<point>& v) {
auto it = v.cbegin();
auto end = v.cend();

os << "[";

if (it != end) {
print(os, *it);
it = std::next(it);
}
while (it != end) {
os << ", ";
print(os, *it);
it = std::next(it);
}

return os << "]";
}

// returns true if the three points make a counter-clockwise turn
bool ccw(const point& a, const point& b, const point& c) {
return ((std::get<0>(b) - std::get<0>(a)) * (std::get<1>(c) - std::get<1>(a)))
> ((std::get<1>(b) - std::get<1>(a)) * (std::get<0>(c) - std::get<0>(a)));
}

std::vector<point> convexHull(std::vector<point> p) {
if (p.size() == 0) return std::vector<point>();
std::sort(p.begin(), p.end(), [](point& a, point& b){
if (std::get<0>(a) < std::get<0>(b)) return true;
return false;
});

std::vector<point> h;

// lower hull
for (const auto& pt : p) {
while (h.size() >= 2 && !ccw(h.at(h.size() - 2), h.at(h.size() - 1), pt)) {
h.pop_back();
}
h.push_back(pt);
}

// upper hull
auto t = h.size() + 1;
for (auto it = p.crbegin(); it != p.crend(); it = std::next(it)) {
auto pt = *it;
while (h.size() >= t && !ccw(h.at(h.size() - 2), h.at(h.size() - 1), pt)) {
h.pop_back();
}
h.push_back(pt);
}

h.pop_back();
return h;
}

int main() {
using namespace std;

vector<point> points = {
make_pair(16, 3),  make_pair(12, 17), make_pair(0,  6),  make_pair(-4, -6), make_pair(16,  6),
make_pair(16, -7), make_pair(16, -3), make_pair(17, -4), make_pair(5, 19),  make_pair(19, -8),
make_pair(3, 16),  make_pair(12, 13), make_pair(3, -4),  make_pair(17,  5), make_pair(-3, 15),
make_pair(-3, -9), make_pair(0, 11),  make_pair(-9, -3), make_pair(-4, -2), make_pair(12, 10)
};

auto hull = convexHull(points);
auto it = hull.cbegin();
auto end = hull.cend();

cout << "Convex Hull: ";
print(cout, hull);
cout << endl;

return 0;
}
```

{{out}}

```Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
```

## C#

```using System;
using System.Collections.Generic;

namespace ConvexHull {
class Point : IComparable<Point> {
private int x, y;

public Point(int x, int y) {
this.x = x;
this.y = y;
}

public int X { get => x; set => x = value; }
public int Y { get => y; set => y = value; }

public int CompareTo(Point other) {
return x.CompareTo(other.x);
}

public override string ToString() {
return string.Format("({0}, {1})", x, y);
}
}

class Program {
private static List<Point> ConvexHull(List<Point> p) {
if (p.Count == 0) return new List<Point>();
p.Sort();
List<Point> h = new List<Point>();

// lower hull
foreach (var pt in p) {
while (h.Count >= 2 && !Ccw(h[h.Count - 2], h[h.Count - 1], pt)) {
h.RemoveAt(h.Count - 1);
}
}

// upper hull
int t = h.Count + 1;
for (int i = p.Count - 1; i >= 0; i--) {
Point pt = p[i];
while (h.Count >= t && !Ccw(h[h.Count - 2], h[h.Count - 1], pt)) {
h.RemoveAt(h.Count - 1);
}
}

h.RemoveAt(h.Count - 1);
return h;
}

private static bool Ccw(Point a, Point b, Point c) {
return ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X));
}

static void Main(string[] args) {
List<Point> points = new List<Point>() {
new Point(16, 3),
new Point(12, 17),
new Point(0, 6),
new Point(-4, -6),
new Point(16, 6),

new Point(16, -7),
new Point(16, -3),
new Point(17, -4),
new Point(5, 19),
new Point(19, -8),

new Point(3, 16),
new Point(12, 13),
new Point(3, -4),
new Point(17, 5),
new Point(-3, 15),

new Point(-3, -9),
new Point(0, 11),
new Point(-9, -3),
new Point(-4, -2),
new Point(12, 10)
};

List<Point> hull = ConvexHull(points);
Console.Write("Convex Hull: [");
for (int i = 0; i < hull.Count; i++) {
if (i > 0) {
Console.Write(", ");
}
Point pt = hull[i];
Console.Write(pt);
}
Console.WriteLine("]");
}
}
}
```

{{out}}

```Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
```

## D

{{trans|Kotlin}}

```import std.algorithm.sorting;
import std.stdio;

struct Point {
int x;
int y;

int opCmp(Point rhs) {
if (x < rhs.x) return -1;
if (rhs.x < x) return 1;
return 0;
}

void toString(scope void delegate(const(char)[]) sink) const {
import std.format;
sink("(");
formattedWrite(sink, "%d", x);
sink(",");
formattedWrite(sink, "%d", y);
sink(")");
}
}

Point[] convexHull(Point[] p) {
if (p.length == 0) return [];
p.sort;
Point[] h;

// lower hull
foreach (pt; p) {
while (h.length >= 2 && !ccw(h[\$-2], h[\$-1], pt)) {
h.length--;
}
h ~= pt;
}

// upper hull
auto t = h.length + 1;
foreach_reverse (i; 0..(p.length - 1)) {
auto pt = p[i];
while (h.length >= t && !ccw(h[\$-2], h[\$-1], pt)) {
h.length--;
}
h ~= pt;
}

h.length--;
return h;
}

/* ccw returns true if the three points make a counter-clockwise turn */
auto ccw(Point a, Point b, Point c) {
return ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x));
}

void main() {
auto points = [
Point(16,  3), Point(12, 17), Point( 0,  6), Point(-4, -6), Point(16,  6),
Point(16, -7), Point(16, -3), Point(17, -4), Point( 5, 19), Point(19, -8),
Point( 3, 16), Point(12, 13), Point( 3, -4), Point(17,  5), Point(-3, 15),
Point(-3, -9), Point( 0, 11), Point(-9, -3), Point(-4, -2), Point(12, 10)
];
auto hull = convexHull(points);
writeln("Convex Hull: ", hull);
}
```

{{out}}

```Convex Hull: [(-9,-3), (-3,-9), (19,-8), (17,5), (12,17), (5,19), (-3,15)]
```

## Go

```package main

import (
"fmt"
"image"
"sort"
)

// ConvexHull returns the set of points that define the
// convex hull of p in CCW order starting from the left most.
func (p points) ConvexHull() points {
// From https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain
// with only minor deviations.
sort.Sort(p)
var h points

// Lower hull
for _, pt := range p {
for len(h) >= 2 && !ccw(h[len(h)-2], h[len(h)-1], pt) {
h = h[:len(h)-1]
}
h = append(h, pt)
}

// Upper hull
for i, t := len(p)-2, len(h)+1; i >= 0; i-- {
pt := p[i]
for len(h) >= t && !ccw(h[len(h)-2], h[len(h)-1], pt) {
h = h[:len(h)-1]
}
h = append(h, pt)
}

return h[:len(h)-1]
}

// ccw returns true if the three points make a counter-clockwise turn
func ccw(a, b, c image.Point) bool {
return ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X))
}

type points []image.Point

func (p points) Len() int      { return len(p) }
func (p points) Swap(i, j int) { p[i], p[j] = p[j], p[i] }
func (p points) Less(i, j int) bool {
if p[i].X == p[j].X {
return p[i].Y < p[i].Y
}
return p[i].X < p[j].X
}

func main() {
pts := points{
{16, 3}, {12, 17}, {0, 6}, {-4, -6}, {16, 6},
{16, -7}, {16, -3}, {17, -4}, {5, 19}, {19, -8},
{3, 16}, {12, 13}, {3, -4}, {17, 5}, {-3, 15},
{-3, -9}, {0, 11}, {-9, -3}, {-4, -2}, {12, 10},
}
hull := pts.ConvexHull()
fmt.Println("Convex Hull:", hull)
}
```

{{out}}

```
Convex Hull: [(-9,-3) (-3,-9) (19,-8) (17,5) (12,17) (5,19) (-3,15)]

```

```import Data.List (sortBy, groupBy, maximumBy)
import Data.Ord (comparing)

(x, y) = ((!! 0), (!! 1))

compareFrom
:: (Num a, Ord a)
=> [a] -> [a] -> [a] -> Ordering
compareFrom o l r =
compare ((x l - x o) * (y r - y o)) ((y l - y o) * (x r - x o))

distanceFrom
:: Floating a
=> [a] -> [a] -> a
distanceFrom from to = ((x to - x from) ** 2 + (y to - y from) ** 2) ** (1 / 2)

convexHull
:: (Floating a, Ord a)
=> [[a]] -> [[a]]
convexHull points =
let o = minimum points
presorted = sortBy (compareFrom o) (filter (/= o) points)
collinears = groupBy (((EQ ==) .) . compareFrom o) presorted
outmost = maximumBy (comparing (distanceFrom o)) <\$> collinears
in dropConcavities [o] outmost

dropConcavities
:: (Num a, Ord a)
=> [[a]] -> [[a]] -> [[a]]
dropConcavities (left:lefter) (right:righter:rightest) =
case compareFrom left right righter of
LT -> dropConcavities (right : left : lefter) (righter : rightest)
EQ -> dropConcavities (left : lefter) (righter : rightest)
GT -> dropConcavities lefter (left : righter : rightest)
dropConcavities output lastInput = lastInput ++ output

main :: IO ()
main =
mapM_ print \$
convexHull
[ [16, 3]
, [12, 17]
, [0, 6]
, [-4, -6]
, [16, 6]
, [16, -7]
, [16, -3]
, [17, -4]
, [5, 19]
, [19, -8]
, [3, 16]
, [12, 13]
, [3, -4]
, [17, 5]
, [-3, 15]
, [-3, -9]
, [0, 11]
, [-9, -3]
, [-4, -2]
, [12, 10]
]
```

{{Out}}

```[-3.0,-9.0]
[19.0,-8.0]
[17.0,5.0]
[12.0,17.0]
[5.0,19.0]
[-3.0,15.0]
[-9.0,-3.0]
```

## J

Restated from the implementation at http://kukuruku.co/hub/funcprog/introduction-to-j-programming-language-2004 which in turn is a translation of http://dr-klm.livejournal.com/42312.html

```counterclockwise =: ({. , }. /: 12 o. }. - {.) @ /:~
crossproduct =: 11"_ o. [: (* +)/ }. - {.
removeinner =: #~ 1, 0 > 3 crossproduct\ ], 1:
hull =: [: removeinner^:_ counterclockwise
```

Example use:

```   hull 16j3 12j17 0j6 _4j_6 16j6 16j_7 16j_3 17j_4 5j19 19j_8 3j16 12j13 3j_4 17j5 _3j15 _3j_9 0j11 _9j_3 _4j_2 12j10
_9j_3 _3j_9 19j_8 17j5 12j17 5j19 _3j15
```

## Java

{{trans|Kotlin}}

```import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.stream.Collectors;

import static java.util.Collections.emptyList;

public class ConvexHull {
private static class Point implements Comparable<Point> {
private int x, y;

public Point(int x, int y) {
this.x = x;
this.y = y;
}

@Override
public int compareTo(Point o) {
return Integer.compare(x, o.x);
}

@Override
public String toString() {
return String.format("(%d, %d)", x, y);
}
}

private static List<Point> convexHull(List<Point> p) {
if (p.isEmpty()) return emptyList();
p.sort(Point::compareTo);
List<Point> h = new ArrayList<>();

// lower hull
for (Point pt : p) {
while (h.size() >= 2 && !ccw(h.get(h.size() - 2), h.get(h.size() - 1), pt)) {
h.remove(h.size() - 1);
}
}

// upper hull
int t = h.size() + 1;
for (int i = p.size() - 1; i >= 0; i--) {
Point pt = p.get(i);
while (h.size() >= t && !ccw(h.get(h.size() - 2), h.get(h.size() - 1), pt)) {
h.remove(h.size() - 1);
}
}

h.remove(h.size() - 1);
return h;
}

// ccw returns true if the three points make a counter-clockwise turn
private static boolean ccw(Point a, Point b, Point c) {
return ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x));
}

public static void main(String[] args) {
List<Point> points = Arrays.asList(new Point(16, 3),
new Point(12, 17),
new Point(0, 6),
new Point(-4, -6),
new Point(16, 6),

new Point(16, -7),
new Point(16, -3),
new Point(17, -4),
new Point(5, 19),
new Point(19, -8),

new Point(3, 16),
new Point(12, 13),
new Point(3, -4),
new Point(17, 5),
new Point(-3, 15),

new Point(-3, -9),
new Point(0, 11),
new Point(-9, -3),
new Point(-4, -2),
new Point(12, 10));

List<Point> hull = convexHull(points);
System.out.printf("Convex Hull: %s\n", hull);
}
}
```

{{out}}

```Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
```

## Javascript

```
function convexHull(points) {
points.sort(comparison);
var L = [];
for (var i = 0; i < points.length; i++) {
while (L.length >= 2 && cross(L[L.length - 2], L[L.length - 1], points[i]) <= 0) {
L.pop();
}
L.push(points[i]);
}
var U = [];
for (var i = points.length - 1; i >= 0; i--) {
while (U.length >= 2 && cross(U[U.length - 2], U[U.length - 1], points[i]) <= 0) {
U.pop();
}
U.push(points[i]);
}
L.pop();
U.pop();
return L.concat(U);
}

function comparison(a, b) {
return a.x == b.x ? a.y - b.y : a.x - b.x;
}

function cross(a, b, o) {
return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
}

```

'''For usage''': convexhull.js

```
var points = [];
var hull = [];

function setup() {
createCanvas(1132, 700);
frameRate(10);

strokeWeight(4);
stroke(220);
}

function draw() {
background(40);
// draw points
for (i = 0; i < points.length; i++) {
point(points[i].x, points[i].y);
};
console.log(hull);
// draw hull
noFill();
beginShape();
for (i = 0; i < hull.length; i++) {
vertex(hull[i].x, hull[i].y);
};
endShape(CLOSE);
}

function mouseClicked() {
points.push(createVector(mouseX, mouseY));
hull = convexHull(points);
noFill();
//console.log(hull);
beginShape();
for (var i = 0; i < hull.length; i++) {
vertex(hull[i].x, hull[i].y);
}
endShape(CLOSE);
return false;
}

// https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain
function convexHull(points) {
points.sort(comparison);
var L = [];
for (var i = 0; i < points.length; i++) {
while (L.length >= 2 && cross(L[L.length - 2], L[L.length - 1], points[i]) <= 0) {
L.pop();
}
L.push(points[i]);
}
var U = [];
for (var i = points.length - 1; i >= 0; i--) {
while (U.length >= 2 && cross(U[U.length - 2], U[U.length - 1], points[i]) <= 0) {
U.pop();
}
U.push(points[i]);
}
L.pop();
U.pop();
return L.concat(U);
}

function comparison(a, b) {
return a.x == b.x ? a.y - b.y : a.x - b.x;
}

function cross(a, b, o) {
return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
}

```

convexhull.html

```
<html>

<script src="https://cdnjs.cloudflare.com/ajax/libs/p5.js/0.7.2/p5.js"></script>
<script src="convexhull.js"></script>

<body>
<table>
<tr>
<th><h1>Convex Hull</h4></th>
</tr>
</table>

</body>

</html>

```

## Julia

```# v1.0.4
# https://github.com/JuliaPolyhedra/Polyhedra.jl/blob/master/examples/operations.ipynb
using Polyhedra, CDDLib

A = vrep([[16,3],  [12,17], [0,6], [-4,-6], [16,6], [16,-7], [16,-3], [17,-4], [5,19], [19,-8], [3,16], [12,13], [3,-4], [17,5], [-3,15], [-3,-9], [0,11], [-9,-3], [-4,-2], [12,10]])
P = polyhedron(A, CDDLib.Library())
Pch = convexhull(P, P)
removevredundancy!(Pch)
println("\$Pch")
```

{{out}}

```convexhull([5.0, 19.0], [19.0, -8.0], [17.0, 5.0], [-3.0, 15.0], [-9.0, -3.0], [12.0, 17.0], [-3.0, -9.0])
```

## Kotlin

{{trans|Go}}

```// version 1.1.3

class Point(val x: Int, val y: Int) : Comparable<Point> {

override fun compareTo(other: Point) = this.x.compareTo(other.x)

override fun toString() = "(\$x, \$y)"
}

fun convexHull(p: Array<Point>): List<Point> {
if (p.isEmpty()) return emptyList()
p.sort()
val h = mutableListOf<Point>()

// lower hull
for (pt in p) {
while (h.size >= 2 && !ccw(h[h.size - 2], h.last(), pt)) {
h.removeAt(h.lastIndex)
}
}

// upper hull
val t = h.size + 1
for (i in p.size - 2 downTo 0) {
val pt = p[i]
while (h.size >= t && !ccw(h[h.size - 2], h.last(), pt)) {
h.removeAt(h.lastIndex)
}
}

h.removeAt(h.lastIndex)
return h
}

/* ccw returns true if the three points make a counter-clockwise turn */
fun ccw(a: Point, b: Point, c: Point) =
((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))

fun main(args: Array<String>) {
val points = arrayOf(
Point(16,  3), Point(12, 17), Point( 0,  6), Point(-4, -6), Point(16,  6),
Point(16, -7), Point(16, -3), Point(17, -4), Point( 5, 19), Point(19, -8),
Point( 3, 16), Point(12, 13), Point( 3, -4), Point(17,  5), Point(-3, 15),
Point(-3, -9), Point( 0, 11), Point(-9, -3), Point(-4, -2), Point(12, 10)
)
val hull = convexHull(points)
println("Convex Hull: \$hull")
}
```

{{out}}

```
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

```

## Lua

{{trans|C++}}

```function print_point(p)
io.write("("..p.x..", "..p.y..")")
return nil
end

function print_points(pl)
io.write("[")
for i,p in pairs(pl) do
if i>1 then
io.write(", ")
end
print_point(p)
end
io.write("]")
return nil
end

function ccw(a,b,c)
return (b.x - a.x) * (c.y - a.y) > (b.y - a.y) * (c.x - a.x)
end

function pop_back(ta)
table.remove(ta,#ta)
return ta
end

function convexHull(pl)
if #pl == 0 then
return {}
end
table.sort(pl, function(left,right)
return left.x < right.x
end)

local h = {}

-- lower hull
for i,pt in pairs(pl) do
while #h >= 2 and not ccw(h[#h-1], h[#h], pt) do
table.remove(h,#h)
end
table.insert(h,pt)
end

-- upper hull
local t = #h + 1
for i=#pl, 1, -1 do
local pt = pl[i]
while #h >= t and not ccw(h[#h-1], h[#h], pt) do
table.remove(h,#h)
end
table.insert(h,pt)
end

table.remove(h,#h)
return h
end

-- main
local points = {
{x=16,y= 3},{x=12,y=17},{x= 0,y= 6},{x=-4,y=-6},{x=16,y= 6},
{x=16,y=-7},{x=16,y=-3},{x=17,y=-4},{x= 5,y=19},{x=19,y=-8},
{x= 3,y=16},{x=12,y=13},{x= 3,y=-4},{x=17,y= 5},{x=-3,y=15},
{x=-3,y=-9},{x= 0,y=11},{x=-9,y=-3},{x=-4,y=-2},{x=12,y=10}
}
local hull = convexHull(points)

io.write("Convex Hull: ")
print_points(hull)
print()
```

{{out}}

```Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
```

```MODULE ConvexHull;
FROM FormatString IMPORT FormatString;
FROM Storage IMPORT ALLOCATE, DEALLOCATE;
FROM SYSTEM IMPORT TSIZE;

PROCEDURE WriteInt(n : INTEGER);
VAR buf : ARRAY[0..15] OF CHAR;
BEGIN
FormatString("%i", buf, n);
WriteString(buf);
END WriteInt;

TYPE
Point = RECORD
x, y : INTEGER;
END;

PROCEDURE WritePoint(pt : Point);
BEGIN
WriteString("(");
WriteInt(pt.x);
WriteString(", ");
WriteInt(pt.y);
WriteString(")");
END WritePoint;

TYPE
NextNode = POINTER TO PNode;
PNode = RECORD
value : Point;
next : NextNode;
END;

PROCEDURE WriteNode(it : NextNode);
BEGIN
IF it = NIL THEN
RETURN
END;
WriteString("[");

WritePoint(it^.value);
it := it^.next;

WHILE it # NIL DO
WriteString(", ");
WritePoint(it^.value);
it := it^.next
END;
WriteString("]")
END WriteNode;

PROCEDURE AppendNode(pn : NextNode; p : Point) : NextNode;
VAR it,nx : NextNode;
BEGIN
IF pn = NIL THEN
ALLOCATE(it,TSIZE(PNode));
it^.value := p;
it^.next := NIL;
RETURN it
END;

it := pn;
WHILE it^.next # NIL DO
it := it^.next
END;

ALLOCATE(nx,TSIZE(PNode));
nx^.value := p;
nx^.next := NIL;

it^.next := nx;
RETURN pn
END AppendNode;

PROCEDURE DeleteNode(VAR pn : NextNode);
BEGIN
IF pn = NIL THEN RETURN END;
DeleteNode(pn^.next);

DEALLOCATE(pn,TSIZE(PNode));
pn := NIL
END DeleteNode;

PROCEDURE SortNode(VAR pn : NextNode);
VAR
it : NextNode;
tmp : Point;
done : BOOLEAN;
BEGIN
REPEAT
done := TRUE;
it := pn;
WHILE (it # NIL) AND (it^.next # NIL) DO
IF it^.next^.value.x < it^.value.x THEN
tmp := it^.value;
it^.value := it^.next^.value;
it^.next^.value := tmp;
done := FALSE
END;
it := it^.next;
END
UNTIL done;
END SortNode;

PROCEDURE NodeLength(it : NextNode) : INTEGER;
VAR length : INTEGER;
BEGIN
length := 0;
WHILE it # NIL DO
INC(length);
it := it^.next;
END;
RETURN length
END NodeLength;

PROCEDURE ReverseNode(fp : NextNode) : NextNode;
VAR rp,tmp : NextNode;
BEGIN
IF fp = NIL THEN RETURN NIL END;

ALLOCATE(tmp,TSIZE(PNode));
tmp^.value := fp^.value;
tmp^.next := NIL;
rp := tmp;
fp := fp^.next;

WHILE fp # NIL DO
ALLOCATE(tmp,TSIZE(PNode));
tmp^.value := fp^.value;
tmp^.next := rp;
rp := tmp;
fp := fp^.next;
END;

RETURN rp
END ReverseNode;

(* ccw returns true if the three points make a counter-clockwise turn *)
PROCEDURE CCW(a,b,c : Point) : BOOLEAN;
BEGIN
RETURN ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))
END CCW;

PROCEDURE ConvexHull(p : NextNode) : NextNode;
VAR
hull,it,h1,h2 : NextNode;
t : INTEGER;
BEGIN
IF p = NIL THEN RETURN NIL END;
SortNode(p);
hull := NIL;

(* lower hull *)
it := p;
WHILE it # NIL DO
IF hull # NIL THEN
WHILE hull^.next # NIL DO
(* At least two points in the list *)
h2 := hull;
h1 := hull^.next;
WHILE h1^.next # NIL DO
h2 := h1;
h1 := h2^.next;
END;

IF CCW(h2^.value, h1^.value, it^.value) THEN
BREAK
ELSE
h2^.next := NIL;
DeleteNode(h1);
h1 := NIL
END
END
END;

hull := AppendNode(hull, it^.value);
it := it^.next;
END;

(* upper hull *)
t := NodeLength(hull) + 1;
p := ReverseNode(p);
it := p;
WHILE it # NIL DO
WHILE NodeLength(hull) >= t DO
h2 := hull;
h1 := hull^.next;
WHILE h1^.next # NIL DO
h2 := h1;
h1 := h2^.next;
END;

IF CCW(h2^.value, h1^.value, it^.value) THEN
BREAK
ELSE
h2^.next := NIL;
DeleteNode(h1);
h1 := NIL
END
END;

hull := AppendNode(hull, it^.value);
it := it^.next;
END;
DeleteNode(p);

h2 := hull;
h1 := h2^.next;
WHILE h1^.next # NIL DO
h2 := h1;
h1 := h1^.next;
END;
h2^.next := NIL;
DeleteNode(h1);
RETURN hull
END ConvexHull;

(* Main *)
VAR nodes,hull : NextNode;
BEGIN
nodes := AppendNode(NIL, Point{16, 3});
AppendNode(nodes, Point{12,17});
AppendNode(nodes, Point{ 0, 6});
AppendNode(nodes, Point{-4,-6});
AppendNode(nodes, Point{16, 6});
AppendNode(nodes, Point{16,-7});
AppendNode(nodes, Point{16,-3});
AppendNode(nodes, Point{17,-4});
AppendNode(nodes, Point{ 5,19});
AppendNode(nodes, Point{19,-8});
AppendNode(nodes, Point{ 3,16});
AppendNode(nodes, Point{12,13});
AppendNode(nodes, Point{ 3,-4});
AppendNode(nodes, Point{17, 5});
AppendNode(nodes, Point{-3,15});
AppendNode(nodes, Point{-3,-9});
AppendNode(nodes, Point{ 0,11});
AppendNode(nodes, Point{-9,-3});
AppendNode(nodes, Point{-4,-2});
AppendNode(nodes, Point{12,10});

hull := ConvexHull(nodes);
WriteNode(hull);
DeleteNode(hull);

DeleteNode(nodes);
END ConvexHull.
```

{{out}}

```[(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
```

## Perl 6

{{works with|Rakudo|2017.05}} {{trans|zkl}} Modified the angle sort method as the original could fail if there were multiple points on the same y coordinate as the starting point. Sorts on tangent rather than triangle area. Inexpensive since it still doesn't do any trigonometric math, just calculates the ratio of opposite over adjacent. The original returned the correct answer for the task example, but only by accident. If the points (14,-9), (1,-9) were added to the task example, it wouldn't give a correct answer. Now it does.

```class Point {
has Real \$.x is rw;
has Real \$.y is rw;
method gist { [~] '(', self.x,', ', self.y, ')' };
}

sub ccw (Point \$a, Point \$b, Point \$c) {
(\$b.x - \$a.x)*(\$c.y - \$a.y) - (\$b.y - \$a.y)*(\$c.x - \$a.x);
}

sub tangent (Point \$a, Point \$b) {
my \$opp = \$b.x - \$a.x;
my \$adj = \$b.y - \$a.y;
}

sub graham-scan (**@coords) {
# sort points by y, secondary sort on x
my @sp = @coords.map( { Point.new( :x(\$_[0]), :y(\$_[1]) ) } )
.sort: {.y, .x};

# need at least 3 points to make a hull
return @sp if +@sp < 3;

# first point on hull is minimum y point
my @h  = @sp.shift;

# re-sort the points by angle, secondary on x
@sp    = @sp.map( { \$++ => [tangent(@h[0], \$_), \$_.x] } )
.sort( {-\$_.value[0], \$_.value[1] } )
.map: { @sp[\$_.key] };

# first point of re-sorted list is guaranteed to be on hull
@h.push:  @sp.shift;

# check through the remaining list making sure that
# there is always a positive angle
for @sp -> \$point {
if ccw( |@h.tail(2), \$point ) >= 0 {
@h.push: \$point;
} else {
@h.pop;
redo;
}
}
@h
}

my @hull = graham-scan(
(16, 3), (12,17), ( 0, 6), (-4,-6), (16, 6), (16,-7), (16,-3),
(17,-4), ( 5,19), (19,-8), ( 3,16), (12,13), ( 3,-4), (17, 5),
(-3,15), (-3,-9), ( 0,11), (-9,-3), (-4,-2), (12,10)
);

say "Convex Hull ({+@hull} points): ", @hull;

@hull = graham-scan(
(16, 3), (12,17), ( 0, 6), (-4,-6), (16, 6), (16,-7), (16,-3),
(17,-4), ( 5,19), (19,-8), ( 3,16), (12,13), ( 3,-4), (17, 5),
(-3,15), (-3,-9), ( 0,11), (-9,-3), (-4,-2), (12,10), (14,-9), (1,-9)
);

say "Convex Hull ({+@hull} points): ", @hull;
```

{{out}}

```Convex Hull (7 points): [(-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]
Convex Hull (9 points): [(-3, -9) (1, -9) (14, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]

```

## Phix

{{trans|C}}

```enum x, y
function ccw(sequence a, b, c)
return (b[x] - a[x]) * (c[y] - a[y])
> (b[y] - a[y]) * (c[x] - a[x])
end function

function convex_hull(sequence points)
sequence h = {}
points = sort(points)

/* lower hull */
for i=1 to length(points) do
while length(h)>=2
and not ccw(h[\$-1], h[\$], points[i]) do
h = h[1..\$-1]
end while
h = append(h, points[i])
end for

/* upper hull */
for i=length(points) to 1 by -1 do
while length(h)>=2
and not ccw(h[\$-1],h[\$],points[i]) do
h = h[1..\$-1]
end while
h = append(h, points[i])
end for

h = h[1..\$-1]
return h
end function

constant points = {{16,  3}, {12, 17}, { 0,  6}, {-4, -6}, {16,  6},
{16, -7}, {16, -3}, {17, -4}, { 5, 19}, {19, -8},
{ 3, 16}, {12, 13}, { 3, -4}, {17,   5}, {-3, 15},
{-3, -9}, { 0, 11}, {-9, -3}, {-4, -2}, {12, 10}}
printf(1,"Convex Hull: %v\n",{convex_hull(points)})
```

{{out}}

```
Convex Hull: {{-9,-3},{-3,-9},{19,-8},{17,5},{12,17},{5,19},{-3,15}}

```

## Python

An approach that uses the shapely library:

```from __future__ import print_function
from shapely.geometry import MultiPoint

if __name__=="__main__":
pts = MultiPoint([(16,3), (12,17), (0,6), (-4,-6), (16,6), (16,-7), (16,-3), (17,-4), (5,19), (19,-8), (3,16), (12,13), (3,-4), (17,5), (-3,15), (-3,-9), (0,11), (-9,-3), (-4,-2), (12,10)])
print (pts.convex_hull)
```

{{out}}

```POLYGON ((-3 -9, -9 -3, -3 15, 5 19, 12 17, 17 5, 19 -8, -3 -9))
```

## Racket

Also an implementation of https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain (therefore kinda {{trans|Go}}

```#lang typed/racket
(define-type Point (Pair Real Real))
(define-type Points (Listof Point))

(: ⊗ (Point Point Point -> Real))
(define/match (⊗ o a b)
[((cons o.x o.y) (cons a.x a.y) (cons b.x b.y))
(- (* (- a.x o.x) (- b.y o.y)) (* (- a.y o.y) (- b.x o.x)))])

(: Point<? (Point Point -> Boolean))
(define (Point<? a b)
(cond [(< (car a) (car b)) #t] [(> (car a) (car b)) #f] [else (< (cdr a) (cdr b))]))

;; Input: a list P of points in the plane.
(define (convex-hull [P:unsorted : Points])
;; Sort the points of P by x-coordinate (in case of a tie, sort by y-coordinate).
(define P (sort P:unsorted Point<?))
;; for i = 1, 2, ..., n:
;;     while L contains at least two points and the sequence of last two points
;;             of L and the point P[i] does not make a counter-clockwise turn:
;;        remove the last point from L
;;     append P[i] to L
;; TB: U is identical with (reverse P)
(define (upper/lower-hull [P : Points])
(reverse
(for/fold ((rev : Points null))
((P.i (in-list P)))
(let u/l : Points ((rev rev))
(match rev
[(list p-2 p-1 ps ...) #:when (not (positive? (⊗ p-2 P.i p-1))) (u/l (list* p-1 ps))]
[(list ps ...) (cons P.i ps)])))))

;; Initialize U and L as empty lists.
;; The lists will hold the vertices of upper and lower hulls respectively.
(let ((U (upper/lower-hull (reverse P)))
(L (upper/lower-hull P)))
;; Remove the last point of each list (it's the same as the first point of the other list).
;; Concatenate L and U to obtain the convex hull of P.
(append (drop-right L 1) (drop-right U 1)))) ; Points in the result will be listed in CCW order.)

(module+ test
(require typed/rackunit)
(check-equal?
(convex-hull
(list '(16 . 3) '(12 . 17) '(0 . 6) '(-4 . -6) '(16 . 6) '(16 . -7) '(16 . -3) '(17 . -4)
'(5 . 19) '(19 . -8) '(3 . 16) '(12 . 13) '(3 . -4) '(17 . 5) '(-3 . 15) '(-3 . -9)
'(0 . 11) '(-9 . -3) '(-4 . -2) '(12 . 10)))
(list '(-9 . -3) '(-3 . -9) '(19 . -8) '(17 . 5) '(12 . 17) '(5 . 19) '(-3 . 15))))
```

{{out}}

silence implies tests pass (and output is as expected)

## REXX

### version 1

```/* REXX ---------------------------------------------------------------
* Compute the Convex Hull for a set of points
* Format of the input file:
* (16,3) (12,17) (0,6) (-4,-6) (16,6) (16,-7) (16,-3) (17,-4) (5,19)
* (19,-8) (3,16) (12,13) (3,-4) (17,5) (-3,15) (-3,-9) (0,11) (-9,-3)
* (-4,-2)
*--------------------------------------------------------------------*/
Signal On Novalue
Signal On Syntax
Parse Arg fid
If fid='' Then Do
fid='chullmin.in'                 /* miscellaneous test data       */
fid='chullx.in'
fid='chullt.in'
fid='chulla.in'
fid='chullxx.in'
fid='sq.in'
fid='tri.in'
fid='line.in'
fid='point.in'
fid='chull.in'                    /* data from task description    */
End
g.0debug=''
g.0oid=fn(fid)'.txt'; 'erase' g.0oid
x.=0
yl.=''
Parse Value '1000 -1000' With g.0xmin g.0xmax
Parse Value '1000 -1000' With g.0ymin g.0ymax
/*---------------------------------------------------------------------
* First read the input and store the points' coordinates
* x.0 contains the number of points, x.i contains the x.coordinate
* yl.x contains the y.coordinate(s) of points (x/y)
*--------------------------------------------------------------------*/
Do while lines(fid)>0
l=linein(fid)
Do While l<>''
Parse Var l '(' x ',' y ')' l
Call store x,y
End
End
Call lineout fid
Do i=1 To x.0                       /* loop over points              */
x=x.i
yl.x=sortv(yl.x)                  /* sort y-coordinates            */
End
Call sho

/*---------------------------------------------------------------------
* Now we look for special border points:
* lefthigh and leftlow: leftmost points with higheste and lowest y
* ritehigh and ritelow: rightmost points with higheste and lowest y
* yl.x contains the y.coordinate(s) of points (x/y)
*--------------------------------------------------------------------*/
leftlow=0
lefthigh=0
Do i=1 To x.0
x=x.i
If maxv(yl.x)=g.0ymax Then Do
If lefthigh=0 Then lefthigh=x'/'g.0ymax
ritehigh=x'/'g.0ymax
End
If minv(yl.x)=g.0ymin Then Do
ritelow=x'/'g.0ymin
If leftlow=0 Then leftlow=x'/'g.0ymin
End
End
Call o 'lefthigh='lefthigh
Call o 'ritehigh='ritehigh
Call o 'ritelow ='ritelow
Call o 'leftlow ='leftlow
/*---------------------------------------------------------------------
* Now we look for special border points:
* leftmost_n and leftmost_s: points with lowest x and highest/lowest y
* ritemost_n and ritemost_s: points with largest x and highest/lowest y
* n and s stand foNorth and South, respectively
*--------------------------------------------------------------------*/
x=g.0xmi; leftmost_n=x'/'maxv(yl.x)
x=g.0xmi; leftmost_s=x'/'minv(yl.x)
x=g.0xma; ritemost_n=x'/'maxv(yl.x)
x=g.0xma; ritemost_s=x'/'minv(yl.x)

/*---------------------------------------------------------------------
* Now we compute the paths from ritehigh to ritelow (n_end)
* and leftlow to lefthigh (s_end), respectively
*--------------------------------------------------------------------*/
x=g.0xma
n_end=''
Do i=words(yl.x) To 1 By -1
n_end=n_end x'/'word(yl.x,i)
End
Call o 'n_end='n_end
x=g.0xmi
s_end=''
Do i=1 To words(yl.x)
s_end=s_end x'/'word(yl.x,i)
End
Call o 's_end='s_end

n_high=''
s_low=''
/*---------------------------------------------------------------------
* Now we compute the upper part of the convex hull (nhull)
*--------------------------------------------------------------------*/
Call o 'leftmost_n='leftmost_n
Call o 'lefthigh  ='lefthigh
nhull=leftmost_n
res=mk_nhull(leftmost_n,lefthigh);
nhull=nhull res
Call o 'A nhull='nhull
Do While res<>lefthigh
res=mk_nhull(res,lefthigh); nhull=nhull res
Call o 'B nhull='nhull
End
res=mk_nhull(lefthigh,ritemost_n); nhull=nhull res
Call o 'C nhull='nhull
Do While res<>ritemost_n
res=mk_nhull(res,ritemost_n); nhull=nhull res
Call o 'D nhull='nhull
End

nhull=nhull n_end                /* attach the right vertical border */

/*---------------------------------------------------------------------
* Now we compute the lower part of the convex hull (shull)
*--------------------------------------------------------------------*/
res=mk_shull(ritemost_s,ritelow);
shull=ritemost_s res
Call o 'A shull='shull
Do While res<>ritelow
res=mk_shull(res,ritelow)
shull=shull res
Call o 'B shull='shull
End
res=mk_shull(ritelow,leftmost_s)
shull=shull res
Call o 'C shull='shull
Do While res<>leftmost_s
res=mk_shull(res,leftmost_s);
shull=shull res
Call o 'D shull='shull
End

shull=shull s_end

chull=nhull shull                 /* concatenate upper and lower part */
/* eliminate duplicates             */
/* too lazy to take care before :-) */
Parse Var chull chullx chull
have.=0
have.chullx=1
Do i=1 By 1 While chull>''
Parse Var chull xy chull
If have.xy=0 Then Do
chullx=chullx xy
have.xy=1
End
End
/* show the result                */
Say 'Points of convex hull in clockwise order:'
Say    chullx
/**********************************************************************
* steps that were necessary in previous attempts
/*---------------------------------------------------------------------
* Final polish: Insert points that are not yet in chullx but should be
* First on the upper hull going from left to right
*--------------------------------------------------------------------*/
i=1
Do While i<words(chullx)
xya=word(chullx,i)  ; Parse Var xya xa '/' ya
If xa=g.0xmax Then Leave
xyb=word(chullx,i+1); Parse Var xyb xb '/' yb
Do j=1 To x.0
If x.j>xa Then Do
If x.j<xb Then Do
xx=x.j
parse Value kdx(xya,xyb) With k d x
If (k*xx+d)=maxv(yl.xx) Then Do
chullx=subword(chullx,1,i) xx'/'maxv(yl.xx),
subword(chullx,i+1)
i=i+1
End
End
End
Else
i=i+1
End
End

Say    chullx

/*---------------------------------------------------------------------
* Final polish: Insert points that are not yet in chullx but should be
* Then on the lower hull going from right to left
*--------------------------------------------------------------------*/
i=wordpos(ritemost_s,chullx)
Do While i<words(chullx)
xya=word(chullx,i)  ; Parse Var xya xa '/' ya
If xa=g.0xmin Then Leave
xyb=word(chullx,i+1); Parse Var xyb xb '/' yb
Do j=x.0 To 1 By -1
If x.j<xa Then Do
If x.j>xb Then Do
xx=x.j
parse Value kdx(xya,xyb) With k d x
If (k*xx+d)=minv(yl.xx) Then Do
chullx=subword(chullx,1,i) xx'/'minv(yl.xx),
subword(chullx,i+1)
i=i+1
End
End
End
Else
i=i+1
End
End
Say chullx
**********************************************************************/
Call lineout g.0oid

Exit

store: Procedure Expose x. yl. g.
/*---------------------------------------------------------------------
* arrange the points in ascending order of x (in x.) and,
* for each x in ascending order of y (in yl.x)
* g.0xmin is the smallest x-value, etc.
* g.0xmi  is the x-coordinate
* g.0ymin is the smallest y-value, etc.
* g.0ymi  is the x-coordinate of such a point
*--------------------------------------------------------------------*/
Parse Arg x,y
Call o 'store' x y
If x<g.0xmin Then Do; g.0xmin=x; g.0xmi=x; End
If x>g.0xmax Then Do; g.0xmax=x; g.0xma=x; End
If y<g.0ymin Then Do; g.0ymin=y; g.0ymi=x; End
If y>g.0ymax Then Do; g.0ymax=y; g.0yma=x; End
Do i=1 To x.0
Select
When x.i>x Then
Leave
When x.i=x Then Do
yl.x=yl.x y
Return
End
Otherwise
Nop
End
End
Do j=x.0 To i By -1
ja=j+1
x.ja=x.j
End
x.i=x
yl.x=y
x.0=x.0+1
Return

sho: Procedure Expose x. yl. g.
Do i=1 To x.0
x=x.i
say  format(i,2) 'x='format(x,3) 'yl='yl.x
End
Say ''
Return

maxv: Procedure Expose g.
Call trace 'O'
Parse Arg l
res=-1000
Do While l<>''
Parse Var l v l
If v>res Then res=v
End
Return res

minv: Procedure Expose g.
Call trace 'O'
Parse Arg l
res=1000
Do While l<>''
Parse Var l v l
If v<res Then res=v
End
Return res

sortv: Procedure Expose g.
Call trace 'O'
Parse Arg l
res=''
Do Until l=''
v=minv(l)
res=res v
l=remove(v,l)
End
Return space(res)

lastword: return word(arg(1),words(arg(1)))

kdx: Procedure  Expose xy. g.
/*---------------------------------------------------------------------
* Compute slope and y-displacement of a straight line
* that is defined by two points:  y=k*x+d
* Specialty; k='*' x=xa if xb=xa
*--------------------------------------------------------------------*/
Call trace 'O'
Parse Arg xya,xyb
Parse Var xya xa '/' ya
Parse Var xyb xb '/' yb
If xa=xb Then
Parse Value '*' '-' xa With k d x
Else Do
k=(yb-ya)/(xb-xa)
d=yb-k*xb
x='*'
End
Return k d x

remove:
/*---------------------------------------------------------------------
* Remove a specified element (e) from a given string (s)
*--------------------------------------------------------------------*/
Parse Arg e,s
Parse Var s sa (e) sb
Return space(sa sb)

o: Procedure Expose g.
/*---------------------------------------------------------------------
* Write a line to the debug file
*--------------------------------------------------------------------*/
If arg(2)=1 Then say arg(1)
Return lineout(g.0oid,arg(1))

is_ok: Procedure Expose x. yl. g. sigl
/*---------------------------------------------------------------------
* Test if a given point (b) is above/on/or below a straight line
* defined by two points (a and c)
*--------------------------------------------------------------------*/
Parse Arg a,b,c,op
Call o    'is_ok' a b c op
Parse Value kdx(a,c) With k d x
Parse Var b x'/'y
If op='U' Then y=maxv(yl.x)
Else y=minv(yl.x)
Call o    y x (k*x+d)
If (abs(y-(k*x+d))<1.e-8) Then Return 0
If op='U' Then res=(y<=(k*x+d))
Else res=(y>=(k*x+d))
Return res

mk_nhull: Procedure Expose x. yl. g.
/*---------------------------------------------------------------------
* Compute the upper (north) hull between two points (xya and xyb)
* Move x from xyb back to xya until all points within the current
* range (x and xyb) are BELOW the straight line defined xya and x
* Then make x the new starting point
*--------------------------------------------------------------------*/
Parse Arg xya,xyb
Call o 'mk_nhull' xya xyb
If xya=xyb Then Return xya
Parse Var xya xa '/' ya
Parse Var xyb xb '/' yb
iu=0
iv=0
Do xi=1 To x.0
if x.xi>=xa & iu=0 Then iu=xi
if x.xi<=xb Then iv=xi
If x.xi>xb Then Leave
End
Call o    iu iv
xu=x.iu
xyu=xu'/'maxv(yl.xu)
Do h=iv To iu+1 By -1 Until good
Call o 'iv='iv,g.0debug
Call o ' h='h,g.0debug
xh=x.h
xyh=xh'/'maxv(yl.xh)
Call o    'Testing' xyu xyh,g.0debug
good=1
Do hh=h-1 To iu+1 By -1 While good
xhh=x.hh
xyhh=xhh'/'maxv(yl.xhh)
Call o 'iu hh iv=' iu hh h,g.0debug
If is_ok(xyu,xyhh,xyh,'U') Then Do
Call o    xyhh 'is under' xyu xyh,g.0debug
Nop
End
Else Do
good=0
Call o    xyhh 'is above' xyu xyh '-' xyh 'ist nicht gut'
End
End
End
Call o xyh 'is the one'

Return xyh

p: Return
Say arg(1)
Pull  .
Return

mk_shull: Procedure Expose x. yl. g.
/*---------------------------------------------------------------------
* Compute the lower (south) hull between two points (xya and xyb)
* Move x from xyb back to xya until all points within the current
* range (x and xyb) are ABOVE the straight line defined xya and x
* Then make x the new starting point
*-----
---------------------------------------------------------------*/
Parse Arg xya,xyb
Call o 'mk_shull' xya xyb
If xya=xyb Then Return xya
Parse Var xya xa '/' ya
Parse Var xyb xb '/' yb
iu=0
iv=0
Do xi=x.0 To 1 By -1
if x.xi<=xa & iu=0 Then iu=xi
if x.xi>=xb Then iv=xi
If x.xi<xb Then Leave
End
Call o iu iv '_' x.iu x.iv
Call o 'mk_shull iv iu' iv iu
xu=x.iu
xyu=xu'/'minv(yl.xu)
good=0
Do h=iv To iu-1 Until good
xh=x.h
xyh=xh'/'minv(yl.xh)
Call o    'Testing' xyu xyh   h iu
good=1
Do hh=h+1 To iu-1 While good
Call o 'iu hh h=' iu hh h
xhh=x.hh
xyhh=xhh'/'minv(yl.xhh)
If is_ok(xyu,xyhh,xyh,'O') Then Do
Call o xyhh 'is above' xyu xyh
Nop
End
Else Do
Call o xyhh 'is under' xyu xyh '-' xyh 'ist nicht gut'
good=0
End
End
End
Call o xyh 'is the one'
Return xyh

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:
Say 'You can look around now.'
Trace ?R
Nop
Exit 12
```

{{out}}

``` 1 x= -9 yl=-3
2 x= -4 yl=-6 -2
3 x= -3 yl=-9 15
4 x=  0 yl=6 11
5 x=  3 yl=-4 16
6 x=  5 yl=19
7 x= 12 yl=13 17
8 x= 16 yl=-7 -3 3 6
9 x= 17 yl=-4 5
10 x= 19 yl=-8

Points of convex hull in clockwise order:
-9/-3 -3/15 5/19 12/17 17/5 19/-8 -3/-9
```

### version 2

```/* REXX ---------------------------------------------------------------
* Compute the Convex Hull for a set of points
* Format of the input file:
* (16,3) (12,17) (0,6) (-4,-6) (16,6) (16,-7) (16,-3) (17,-4) (5,19)
* (19,-8) (3,16) (12,13) (3,-4) (17,5) (-3,15) (-3,-9) (0,11) (-9,-3)
* (-4,-2)
* Alternate (better) method using slopes
* 1) Compute path from lowest/leftmost to leftmost/lowest
* 2) Compute leftmost vertical border
* 3) Compute path from rightmost/highest to highest/rightmost
* 4) Compute path from highest/rightmost to rightmost/highest
* 5) Compute rightmost vertical border
* 6) Compute path from rightmost/lowest to lowest_leftmost point
*--------------------------------------------------------------------*/
Parse Arg fid
If fid='' Then Do
fid='line.in'
fid='point.in'
fid='chullmin.in'                 /* miscellaneous test data       */
fid='chullxx.in'
fid='chullx.in'
fid='chullt.in'
fid='chulla.in'
fid='sq.in'
fid='tri.in'
fid='z.in'
fid='chull.in'                    /* data from task description    */
End
g.0debug=''
g.0oid=fn(fid)'.txt'; 'erase' g.0oid
x.=0
yl.=''
Parse Value '1000 -1000' With g.0xmin g.0xmax
Parse Value '1000 -1000' With g.0ymin g.0ymax
/*---------------------------------------------------------------------
* First read the input and store the points' coordinates
* x.0 contains the number of points, x.i contains the x.coordinate
* yl.x contains the y.coordinate(s) of points (x/y)
*--------------------------------------------------------------------*/
Do while lines(fid)>0
l=linein(fid)
Do While l<>''
Parse Var l '(' x ',' y ')' l
Call store x,y
End
End
Call lineout fid
g.0xlist=''
Do i=1 To x.0                       /* loop over points              */
x=x.i
g.0xlist=g.0xlist x
yl.x=sortv(yl.x)                  /* sort y-coordinates            */
End
Call sho
If x.0<3 Then Do
Say 'We need at least three points!'
Exit
End
Call o 'g.0xmin='g.0xmin
Call o 'g.0xmi ='g.0xmi
Call o 'g.0ymin='g.0ymin
Call o 'g.0ymi ='g.0ymi

Do i=1 To x.0
x=x.i
If minv(yl.x)=g.0ymin Then Leave
End
lowest_leftmost=i

highest_rightmost=0
Do i=1 To x.0
x=x.i
If maxv(yl.x)=g.0ymax Then
highest_rightmost=i
If maxv(yl.x)<g.0ymax Then
If highest_rightmost>0 Then
Leave
End
Call o 'lowest_leftmost='lowest_leftmost
Call o 'highest_rightmost  ='highest_rightmost

x=x.lowest_leftmost
Call o 'We start at' from x'/'minv(yl.x)
path=x'/'minv(yl.x)
/*---------------------------------------------------------------------
* 1) Compute path from lowest/leftmost to leftmost/lowest
*--------------------------------------------------------------------*/
Call min_path lowest_leftmost,1
/*---------------------------------------------------------------------
* 2) Compute leftmost vertical border
*--------------------------------------------------------------------*/
Do i=2 To words(yl.x)
path=path x'/'word(yl.x,i)
End
cxy=x'/'maxv(yl.x)
/*---------------------------------------------------------------------
* 3) Compute path from rightmost/highest to highest/rightmost
*--------------------------------------------------------------------*/
Call max_path ci,highest_rightmost
/*---------------------------------------------------------------------
* 4) Compute path from highest/rightmost to rightmost/highest
*--------------------------------------------------------------------*/
Call max_path ci,x.0
/*---------------------------------------------------------------------
* 5) Compute rightmost vertical border
*--------------------------------------------------------------------*/
Do i=words(yl.x)-1 To 1 By -1
cxy=x'/'word(yl.x,i)
path=path cxy
End
/*---------------------------------------------------------------------
* 6) Compute path from rightmost/lowest to lowest_leftmost
*--------------------------------------------------------------------*/
Call min_path ci,lowest_leftmost

Parse Var path pathx path
have.=0
Do i=1 By 1 While path>''
Parse Var path xy path
If have.xy=0 Then Do
pathx=pathx xy
have.xy=1
End
End
Say 'Points of convex hull in clockwise order:'
Say pathx
Call lineout g.0oid
Exit

min_path:
Parse Arg from,tgt
ci=from
cxy=x.ci
Do Until ci=tgt
kmax=-1000
Do i=ci-1 To 1 By sign(tgt-from)
x=x.i
k=k(cxy'/'minv(yl.cxy),x'/'minv(yl.x))
If k>kmax Then Do
kmax=k
ii=i
End
End
ci=ii
cxy=x.ii
path=path cxy'/'minv(yl.cxy)
End
Return

max_path:
Parse Arg from,tgt
Do Until ci=tgt
kmax=-1000
Do i=ci+1 To tgt
x=x.i
k=k(cxy,x'/'maxv(yl.x))
If k>kmax Then Do
kmax=k
ii=i
End
End
x=x.ii
cxy=x'/'maxv(yl.x)
path=path cxy
ci=ii
End
Return

store: Procedure Expose x. yl. g.
/*---------------------------------------------------------------------
* arrange the points in ascending order of x (in x.) and,
* for each x in ascending order of y (in yl.x)
* g.0xmin is the smallest x-value, etc.
* g.0xmi  is the x-coordinate
* g.0ymin is the smallest y-value, etc.
* g.0ymi  is the x-coordinate of such a point
*--------------------------------------------------------------------*/
Parse Arg x,y
Call o 'store' x y
If x<g.0xmin Then Do; g.0xmin=x; g.0xmi=x; End
If x>g.0xmax Then Do; g.0xmax=x; g.0xma=x; End
If y<g.0ymin Then Do; g.0ymin=y; g.0ymi=x; End
If y>g.0ymax Then Do; g.0ymax=y; g.0yma=x; End
Do i=1 To x.0
Select
When x.i>x Then
Leave
When x.i=x Then Do
yl.x=yl.x y
Return
End
Otherwise
Nop
End
End
Do j=x.0 To i By -1
ja=j+1
x.ja=x.j
End
x.i=x
yl.x=y
x.0=x.0+1
Return

sho: Procedure Expose x. yl. g.
Do i=1 To x.0
x=x.i
say  format(i,2) 'x='format(x,3) 'yl='yl.x
End
Say ''
Return

maxv: Procedure Expose g.
Call trace 'O'
Parse Arg l
res=-1000
Do While l<>''
Parse Var l v l
If v>res Then res=v
End
Return res

minv: Procedure Expose g.
Call trace 'O'
Parse Arg l
res=1000
Do While l<>''
Parse Var l v l
If v<res Then res=v
End
Return res

sortv: Procedure Expose g.
Call trace 'O'
Parse Arg l
res=''
Do Until l=''
v=minv(l)
res=res v
l=remove(v,l)
End
Return space(res)

lastword: return word(arg(1),words(arg(1)))

k: Procedure
/*---------------------------------------------------------------------
* Compute slope of a straight line
* that is defined by two points:  y=k*x+d
* Specialty; k='*' x=xa if xb=xa
*--------------------------------------------------------------------*/
Call trace 'O'
Parse Arg xya,xyb
Parse Var xya xa '/' ya
Parse Var xyb xb '/' yb
If xa=xb Then
k='*'
Else
k=(yb-ya)/(xb-xa)
Return k

remove:
/*---------------------------------------------------------------------
* Remove a specified element (e) from a given string (s)
*--------------------------------------------------------------------*/
Parse Arg e,s
Parse Var s sa (e) sb
Return space(sa sb)

o: Procedure Expose g.
/*---------------------------------------------------------------------
* Write a line to the debug file
*--------------------------------------------------------------------*/
If arg(2)=1 Then say arg(1)
Return lineout(g.0oid,arg(1))
```

{{out}}

``` 1 x= -9 yl=-3
2 x= -4 yl=-6 -2
3 x= -3 yl=-9 15
4 x=  0 yl=6 11
5 x=  3 yl=-4 16
6 x=  5 yl=19
7 x= 12 yl=13 17
8 x= 16 yl=-7 -3 3 6
9 x= 17 yl=-4 5
10 x= 19 yl=-8

Points of convex hull in clockwise order:
-3/-9 -9/-3 -3/15 5/19 12/17 17/5 19/-8 -3/-9
```

## Rust

Calculates convex hull from list of points (f32, f32). This can be executed entirely in the Rust Playground.

```
#[derive(Debug, Clone)]
struct Point {
x: f32,
y: f32
}

fn calculate_convex_hull(points: &Vec<Point>) -> Vec<Point> {
//There must be at least 3 points
if points.len() < 3 { return points.clone(); }

let mut hull = vec![];

//Find the left most point in the polygon
let (left_most_idx, _) = points.iter()
.enumerate()
.min_by(|lhs, rhs| lhs.1.x.partial_cmp(&rhs.1.x).unwrap())
.expect("No left most point");

let mut p = left_most_idx;
let mut q = 0_usize;

loop {
//The left most point must be part of the hull
hull.push(points[p].clone());

q = (p + 1) % points.len();

for i in 0..points.len() {
if orientation(&points[p], &points[i], &points[q]) == 2 {
q = i;
}
}

p = q;

//Break from loop once we reach the first point again
if p == left_most_idx { break; }
}

return hull;
}

//Calculate orientation for 3 points
//0 -> Straight line
//1 -> Clockwise
//2 -> Counterclockwise
fn orientation(p: &Point, q: &Point, r: &Point) -> usize {
let val = (q.y - p.y) * (r.x - q.x) -
(q.x - p.x) * (r.y - q.y);

if val == 0. { return 0 };
if val > 0. { return 1; } else { return 2; }
}

fn main(){
let points = vec![pt(16,3), pt(12,17), pt(0,6), pt(-4,-6), pt(16,6), pt(16,-7), pt(16,-3), pt(17,-4), pt(5,19), pt(19,-8), pt(3,16), pt(12,13), pt(3,-4), pt(17,5), pt(-3,15), pt(-3,-9), pt(0,11), pt(-9,-3), pt(-4,-2), pt(12,10)];
let hull = calculate_convex_hull(&points);

hull.iter()
.for_each(|pt| println!("{:?}", pt));
}

fn pt(x: i32, y: i32) -> Point {
return Point {x:x as f32, y:y as f32};
}

```

{{out}}

```
Point { x: -9.0, y: -3.0 }
Point { x: -3.0, y: -9.0 }
Point { x: 19.0, y: -8.0 }
Point { x: 17.0, y: 5.0 }
Point { x: 12.0, y: 17.0 }
Point { x: 5.0, y: 19.0 }
Point { x: -3.0, y: 15.0 }

```

## Scala

Scala Implementation to find Convex hull of given points collection. Functional Paradigm followed

```
object convex_hull{
def get_hull(points:List[(Double,Double)], hull:List[(Double,Double)]):List[(Double,Double)] = points match{
case Nil  =>            join_tail(hull,hull.size -1)
}
def reduce(hull:List[(Double,Double)]):List[(Double,Double)] = hull match{
case p1::p2::p3::rest => {
if(check_point(p1,p2,p3))      hull
else                           reduce(p1::p3::rest)
}
case _ =>                          hull
}
def check_point(pnt:(Double,Double), p2:(Double,Double),p1:(Double,Double)): Boolean = {
val (x,y) = (pnt._1,pnt._2)
val (x1,y1) = (p1._1,p1._2)
val (x2,y2) = (p2._1,p2._2)
((x-x1)*(y2-y1) - (x2-x1)*(y-y1)) <= 0
}
def m(p1:(Double,Double), p2:(Double,Double)):Double = {
if(p2._1 == p1._1 && p1._2>p2._2)       90
else if(p2._1 == p1._1 && p1._2<p2._2)  -90
else if(p1._1<p2._1)                    180 - Math.toDegrees(Math.atan(-(p1._2 - p2._2)/(p1._1 - p2._1)))
else                                    Math.toDegrees(Math.atan((p1._2 - p2._2)/(p1._1 - p2._1)))
}
def join_tail(hull:List[(Double,Double)],len:Int):List[(Double,Double)] = {
if(m(hull(len),hull(0)) > m(hull(len-1),hull(0)))   join_tail(hull.slice(0,len),len-1)
else                                                hull
}
def main(args:Array[String]){
val points = List[(Double,Double)]((16,3), (12,17), (0,6), (-4,-6), (16,6), (16,-7), (16,-3), (17,-4), (5,19), (19,-8), (3,16), (12,13), (3,-4), (17,5), (-3,15), (-3,-9), (0,11), (-9,-3), (-4,-2), (12,10))
val sorted_points = points.sortWith(m(_,(0.0,0.0)) < m(_,(0.0,0.0)))
println(f"Points:\n" + points + f"\n\nConvex Hull :\n" +get_hull(sorted_points,List[(Double,Double)]()))
}
}

```

{{out}}

```
Points:
List((16.0,3.0), (12.0,17.0), (0.0,6.0), (-4.0,-6.0), (16.0,6.0), (16.0,-7.0), (16.0,-3.0), (17.0,-4.0), (5.0,19.0), (19.0,-8.0), (3.0,16.0), (12.0,13.0), (3.0,-4.0), (17.0,5.0), (-3.0,15.0), (-3.0,-9.0), (0.0,11.0), (-9.0,-3.0), (-4.0,-2.0), (12.0,10.0))

Convex Hull :
List((-3.0,-9.0), (-9.0,-3.0), (-3.0,15.0), (5.0,19.0), (12.0,17.0), (17.0,5.0), (19.0,-8.0))

```

## Sidef

{{trans|Perl 6}}

```class Point(Number x, Number y) {
method to_s {
"(#{x}, #{y})"
}
}

func ccw (Point a, Point b, Point c) {
(b.x - a.x)*(c.y - a.y) - (b.y - a.y)*(c.x - a.x)
}

func tangent (Point a, Point b) {
(b.x - a.x) / (b.y - a.y)
}

func graham_scan (*coords) {

## sort points by y, secondary sort on x
var sp = coords.map { |a|
Point(a...)
}.sort { |a,b|
(a.y <=> b.y) ||
(a.x <=> b.x)
}

# need at least 3 points to make a hull
if (sp.len < 3) {
return sp
}

# first point on hull is minimum y point
var h = [sp.shift]

# re-sort the points by angle, secondary on x
sp = sp.map_kv { |k,v|
Pair(k, [tangent(h[0], v), v.x])
}.sort { |a,b|
(b.value[0] <=> a.value[0]) ||
(a.value[1] <=> b.value[1])
}.map { |a|
sp[a.key]
}

# first point of re-sorted list is guaranteed to be on hull
h << sp.shift

# check through the remaining list making sure that
# there is always a positive angle
sp.each { |point|
loop {
if (ccw(h.last(2)..., point) >= 0) {
h << point
break
} else {
h.pop
}
}
}

return h
}

var hull = graham_scan(
[16, 3], [12,17], [ 0, 6], [-4,-6], [16, 6], [16,-7], [16,-3],
[17,-4], [ 5,19], [19,-8], [ 3,16], [12,13], [ 3,-4], [17, 5],
[-3,15], [-3,-9], [ 0,11], [-9,-3], [-4,-2], [12,10])

say("Convex Hull (#{hull.len} points): ", hull.join(" "))

hull = graham_scan(
[16, 3], [12,17], [ 0, 6], [-4,-6], [16, 6], [16,-7], [16,-3],
[17,-4], [ 5,19], [19,-8], [ 3,16], [12,13], [ 3,-4], [17, 5],
[-3,15], [-3,-9], [ 0,11], [-9,-3], [-4,-2], [12,10], [14,-9], [1,-9])

say("Convex Hull (#{hull.len} points): ", hull.join(" "))
```

{{out}}

```
Convex Hull (7 points): (-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)
Convex Hull (9 points): (-3, -9) (1, -9) (14, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)

```

## Visual Basic .NET

{{trans|C#}}

```Imports ConvexHull

Module Module1

Class Point : Implements IComparable(Of Point)
Public Property X As Integer
Public Property Y As Integer

Public Sub New(x As Integer, y As Integer)
Me.X = x
Me.Y = y
End Sub

Public Function CompareTo(other As Point) As Integer Implements IComparable(Of Point).CompareTo
Return X.CompareTo(other.X)
End Function

Public Overrides Function ToString() As String
Return String.Format("({0}, {1})", X, Y)
End Function
End Class

Function ConvexHull(p As List(Of Point)) As List(Of Point)
If p.Count = 0 Then
Return New List(Of Point)
End If
p.Sort()
Dim h As New List(Of Point)

' Lower hull
For Each pt In p
While h.Count >= 2 AndAlso Not Ccw(h(h.Count - 2), h(h.Count - 1), pt)
h.RemoveAt(h.Count - 1)
End While
Next

' Upper hull
Dim t = h.Count + 1
For i = p.Count - 1 To 0 Step -1
Dim pt = p(i)
While h.Count >= t AndAlso Not Ccw(h(h.Count - 2), h(h.Count - 1), pt)
h.RemoveAt(h.Count - 1)
End While
Next

h.RemoveAt(h.Count - 1)
Return h
End Function

Function Ccw(a As Point, b As Point, c As Point) As Boolean
Return ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X))
End Function

Sub Main()
Dim points As New List(Of Point) From {
New Point(16, 3),
New Point(12, 17),
New Point(0, 6),
New Point(-4, -6),
New Point(16, 6),
New Point(16, -7),
New Point(16, -3),
New Point(17, -4),
New Point(5, 19),
New Point(19, -8),
New Point(3, 16),
New Point(12, 13),
New Point(3, -4),
New Point(17, 5),
New Point(-3, 15),
New Point(-3, -9),
New Point(0, 11),
New Point(-9, -3),
New Point(-4, -2),
New Point(12, 10)
}

Dim hull = ConvexHull(points)
Dim it = hull.GetEnumerator()
Console.Write("Convex Hull: [")
If it.MoveNext() Then
Console.Write(it.Current)
End If
While it.MoveNext()
Console.Write(", {0}", it.Current)
End While
Console.WriteLine("]")
End Sub

End Module
```

{{out}}

```Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
```

## zkl

```// Use Graham Scan to sort points into a convex hull
// https://en.wikipedia.org/wiki/Graham_scan, O(n log n)
// http://www.geeksforgeeks.org/convex-hull-set-2-graham-scan/
// http://geomalgorithms.com/a10-_hull-1.html
fcn grahamScan(points){
N:=points.len();
# find the point with the lowest y-coordinate, x is tie breaker
p0:=points.reduce(fcn([(a,b)]ab,[(x,y)]xy){
if(b<y)ab else if(b==y and a<x)ab else xy });
#sort points by polar angle with p0, ie ccw from p0
points.sort('wrap(p1,p2){ ccw(p0,p1,p2)>0 });

# We want points[0] to be a sentinel point that will stop the loop.
points.insert(0,points[-1]);
M:=1; # M will denote the number of points on the convex hull.
foreach i in ([2..N]){
# Find next valid point on convex hull.
while(ccw(points[M-1], points[M], points[i])<=0){
if(M>1) M-=1;
else if(i==N) break;  # All points are collinear
else i+=1;
}
points.swap(M+=1,i); # Update M and swap points[i] to the correct place.
}
points[0,M]
}
# Three points are a counter-clockwise turn if ccw > 0, clockwise if
# ccw < 0, and collinear if ccw = 0 because ccw is a determinant that
# gives twice the signed  area of the triangle formed by p1, p2 and p3.
fcn ccw(a,b,c){  // a,b,c are points: (x,y)
((b[0] - a[0])*(c[1] - a[1])) - ((b[1] - a[1])*(c[0] - a[0]))
}
```
```pts:=List( T(16,3), T(12,17), T(0,6), T(-4,-6), T(16,6),
T(16, -7), T(16,-3),T(17,-4), T(5,19), T(19,-8),
T(3,16), T(12,13), T(3,-4), T(17,5), T(-3,15),
T(-3,-9), T(0,11), T(-9,-3), T(-4,-2), T(12,10), )
.apply(fcn(xy){ xy.apply("toFloat") }).copy();
hull:=grahamScan(pts);
println("Convex Hull (%d points): %s".fmt(hull.len(),hull.toString(*)));
```

{{out}}

```
Convex Hull (7 points): L(L(-3,-9),L(19,-8),L(17,5),L(12,17),L(5,19),L(-3,15),L(-9,-3))

```