⚠️ 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.

[[Category:Geometry]] [[Category:Collision detection]] {{draft task}}Finding the intersection of an infinite ray with a plane in 3D is an important topic in collision detection.

Find the point of intersection for the infinite ray with direction (0,-1,-1) passing through position (0, 0, 10) with the infinite plane with a normal vector of (0, 0, 1) and which passes through [0, 0, 5].

## C

Straightforward application of the intersection formula, prints usage on incorrect invocation.

```
#include<stdio.h>

typedef struct{
double x,y,z;
}vector;

vector addVectors(vector a,vector b){
return (vector){a.x+b.x,a.y+b.y,a.z+b.z};
}

vector subVectors(vector a,vector b){
return (vector){a.x-b.x,a.y-b.y,a.z-b.z};
}

double dotProduct(vector a,vector b){
return a.x*b.x + a.y*b.y + a.z*b.z;
}

vector scaleVector(double l,vector a){
return (vector){l*a.x,l*a.y,l*a.z};
}

vector intersectionPoint(vector lineVector, vector linePoint, vector planeNormal, vector planePoint){
vector diff = subVectors(linePoint,planePoint);

}

int main(int argC,char* argV[])
{
vector lV,lP,pN,pP,iP;

if(argC!=5)
printf("Usage : %s <line direction, point on line, normal to plane and point on plane given as (x,y,z) tuples separated by space>");
else{
sscanf(argV[1],"(%lf,%lf,%lf)",&lV.x,&lV.y,&lV.z);
sscanf(argV[3],"(%lf,%lf,%lf)",&pN.x,&pN.y,&pN.z);

if(dotProduct(lV,pN)==0)
printf("Line and Plane do not intersect, either parallel or line is on the plane");
else{
sscanf(argV[2],"(%lf,%lf,%lf)",&lP.x,&lP.y,&lP.z);
sscanf(argV[4],"(%lf,%lf,%lf)",&pP.x,&pP.y,&pP.z);

iP = intersectionPoint(lV,lP,pN,pP);

printf("Intersection point is (%lf,%lf,%lf)",iP.x,iP.y,iP.z);
}
}

return 0;
}

```

Invocation and output:

```
C:\rosettaCode>linePlane.exe (0,-1,-1) (0,0,10) (0,0,1) (0,0,5)
Intersection point is (0.000000,-5.000000,5.000000)

```

## C++

{{trans|Java}}

```#include <iostream>
#include <sstream>

class Vector3D {
public:
Vector3D(double x, double y, double z) {
this->x = x;
this->y = y;
this->z = z;
}

double dot(const Vector3D& rhs) const {
return x * rhs.x + y * rhs.y + z * rhs.z;
}

Vector3D operator-(const Vector3D& rhs) const {
return Vector3D(x - rhs.x, y - rhs.y, z - rhs.z);
}

Vector3D operator*(double rhs) const {
return Vector3D(rhs*x, rhs*y, rhs*z);
}

friend std::ostream& operator<<(std::ostream&, const Vector3D&);

private:
double x, y, z;
};

std::ostream & operator<<(std::ostream & os, const Vector3D &f) {
std::stringstream ss;
ss << "(" << f.x << ", " << f.y << ", " << f.z << ")";
return os << ss.str();
}

Vector3D intersectPoint(Vector3D rayVector, Vector3D rayPoint, Vector3D planeNormal, Vector3D planePoint) {
Vector3D diff = rayPoint - planePoint;
double prod1 = diff.dot(planeNormal);
double prod2 = rayVector.dot(planeNormal);
double prod3 = prod1 / prod2;
return rayPoint - rayVector * prod3;
}

int main() {
Vector3D rv = Vector3D(0.0, -1.0, -1.0);
Vector3D rp = Vector3D(0.0, 0.0, 10.0);
Vector3D pn = Vector3D(0.0, 0.0, 1.0);
Vector3D pp = Vector3D(0.0, 0.0, 5.0);
Vector3D ip = intersectPoint(rv, rp, pn, pp);

std::cout << "The ray intersects the plane at " << ip << std::endl;

return 0;
}
```

{{out}}

```The ray intersects the plane at (0, -5, 5)
```

## C#

```using System;

namespace FindIntersection {
class Vector3D {
private double x, y, z;

public Vector3D(double x, double y, double z) {
this.x = x;
this.y = y;
this.z = z;
}

public static Vector3D operator +(Vector3D lhs, Vector3D rhs) {
return new Vector3D(lhs.x + rhs.x, lhs.y + rhs.y, lhs.z + rhs.z);
}

public static Vector3D operator -(Vector3D lhs, Vector3D rhs) {
return new Vector3D(lhs.x - rhs.x, lhs.y - rhs.y, lhs.z - rhs.z);
}

public static Vector3D operator *(Vector3D lhs, double rhs) {
return new Vector3D(lhs.x * rhs, lhs.y * rhs, lhs.z * rhs);
}

public double Dot(Vector3D rhs) {
return x * rhs.x + y * rhs.y + z * rhs.z;
}

public override string ToString() {
return string.Format("({0:F}, {1:F}, {2:F})", x, y, z);
}
}

class Program {
static Vector3D IntersectPoint(Vector3D rayVector, Vector3D rayPoint, Vector3D planeNormal, Vector3D planePoint) {
var diff = rayPoint - planePoint;
var prod1 = diff.Dot(planeNormal);
var prod2 = rayVector.Dot(planeNormal);
var prod3 = prod1 / prod2;
return rayPoint - rayVector * prod3;
}

static void Main(string[] args) {
var rv = new Vector3D(0.0, -1.0, -1.0);
var rp = new Vector3D(0.0, 0.0, 10.0);
var pn = new Vector3D(0.0, 0.0, 1.0);
var pp = new Vector3D(0.0, 0.0, 5.0);
var ip = IntersectPoint(rv, rp, pn, pp);
Console.WriteLine("The ray intersects the plane at {0}", ip);
}
}
}
```

{{out}}

```The ray intersects the plane at (0.00, -5.00, 5.00)
```

## D

{{trans|Kotlin}}

```import std.stdio;

struct Vector3D {
private real x;
private real y;
private real z;

this(real x, real y, real z) {
this.x = x;
this.y = y;
this.z = z;
}

auto opBinary(string op)(Vector3D rhs) const {
static if (op == "+" || op == "-") {
mixin("return Vector3D(x" ~ op ~ "rhs.x, y" ~ op ~ "rhs.y, z" ~ op ~ "rhs.z);");
}
}

auto opBinary(string op : "*")(real s) const {
return Vector3D(s*x, s*y, s*z);
}

auto dot(Vector3D rhs) const {
return x*rhs.x + y*rhs.y + z*rhs.z;
}

void toString(scope void delegate(const(char)[]) sink) const {
import std.format;

sink("(");
formattedWrite!"%f"(sink, x);
sink(",");
formattedWrite!"%f"(sink, y);
sink(",");
formattedWrite!"%f"(sink, z);
sink(")");
}
}

auto intersectPoint(Vector3D rayVector, Vector3D rayPoint, Vector3D planeNormal, Vector3D planePoint) {
auto diff = rayPoint - planePoint;
auto prod1 = diff.dot(planeNormal);
auto prod2 = rayVector.dot(planeNormal);
auto prod3 = prod1 / prod2;
return rayPoint - rayVector * prod3;
}

void main() {
auto rv = Vector3D(0.0, -1.0, -1.0);
auto rp = Vector3D(0.0,  0.0, 10.0);
auto pn = Vector3D(0.0,  0.0,  1.0);
auto pp = Vector3D(0.0,  0.0,  5.0);
auto ip = intersectPoint(rv, rp, pn, pp);
writeln("The ray intersects the plane at ", ip);
}
```

{{out}}

```The ray intersects the plane at (0.000000,-5.000000,5.000000)
```

```open System

type Vector(x : double, y : double, z : double) =
member this.x = x
member this.y = y
member this.z = z
static member (-) (lhs : Vector, rhs : Vector) =
Vector(lhs.x - rhs.x, lhs.y - rhs.y, lhs.z - rhs.z)
static member (*) (lhs : Vector, rhs : double) =
Vector(lhs.x * rhs, lhs.y * rhs, lhs.z * rhs)
override this.ToString() =
String.Format("({0:F}, {1:F}, {2:F})", x, y, z)

let Dot (lhs:Vector) (rhs:Vector) =
lhs.x * rhs.x + lhs.y * rhs.y + lhs.z * rhs.z

let IntersectPoint rayVector rayPoint planeNormal planePoint =
let diff = rayPoint - planePoint
let prod1 = Dot diff planeNormal
let prod2 = Dot rayVector planeNormal
let prod3 = prod1 / prod2
rayPoint - rayVector * prod3

[<EntryPoint>]
let main _ =
let rv = Vector(0.0, -1.0, -1.0)
let rp = Vector(0.0, 0.0, 10.0)
let pn = Vector(0.0, 0.0, 1.0)
let pp = Vector(0.0, 0.0, 5.0)
let ip = IntersectPoint rv rp pn pp
Console.WriteLine("The ray intersects the plane at {0}", ip)

0 // return an integer exit code
```

{{out}}

```The ray intersects the plane at (0.00, -5.00, 5.00)
```

## FreeBASIC

```' version 11-07-2018
' compile with: fbc -s console

Type vector3d
Dim As Double x, y ,z
Declare Constructor ()
Declare Constructor (ByVal x As Double, ByVal y As Double, ByVal z As Double)
End Type

Constructor vector3d()
This.x = 0
This.y = 0
This.z = 0
End Constructor

Constructor vector3d(ByVal x As Double, ByVal y As Double, ByVal z As Double)
This.x = x
This.y = y
This.z = z
End Constructor

Operator + (lhs As vector3d, rhs As vector3d) As vector3d
Return Type(lhs.x + rhs.x, lhs.y + rhs.y, lhs.z + rhs.z)
End Operator

Operator - (lhs As vector3d, rhs As vector3d) As vector3d
Return Type(lhs.x - rhs.x, lhs.y - rhs.y, lhs.z - rhs.z)
End Operator

Operator * (lhs As vector3d, d As Double) As vector3d
Return Type(lhs.x * d, lhs.y * d, lhs.z * d)
End Operator

Function dot(lhs As vector3d, rhs As vector3d) As Double
Return lhs.x * rhs.x + lhs.y * rhs.y + lhs.z * rhs.z
End Function

Function tostring(vec As vector3d) As String
Return "(" + Str(vec.x) + ", " + Str(vec.y) + ", " + Str(vec.z) + ")"
End Function

Function intersectpoint(rayvector As vector3d, raypoint As vector3d, _
planenormal As vector3d, planepoint As vector3d) As vector3d

Dim As vector3d diff = raypoint - planepoint
Dim As Double prod1 = dot(diff, planenormal)
Dim As double prod2 = dot(rayvector, planenormal)
Return raypoint - rayvector * (prod1 / prod2)

End Function

' ------=< MAIN >=------

Dim As vector3d rv = Type(0, -1, -1)
Dim As vector3d rp = Type(0,  0, 10)
Dim As vector3d pn = Type(0,  0,  1)
Dim As vector3d pp = Type(0,  0,  5)
Dim As vector3d ip = intersectpoint(rv, rp, pn, pp)

print
Print "line intersects the plane at "; tostring(ip)

' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
```

{{out}}

```line intersects the plane at (0, -5, 5)
```

## Go

{{trans|Kotlin}}

```package main

import "fmt"

type Vector3D struct{ x, y, z float64 }

func (v *Vector3D) Add(w *Vector3D) *Vector3D {
return &Vector3D{v.x + w.x, v.y + w.y, v.z + w.z}
}

func (v *Vector3D) Sub(w *Vector3D) *Vector3D {
return &Vector3D{v.x - w.x, v.y - w.y, v.z - w.z}
}

func (v *Vector3D) Mul(s float64) *Vector3D {
return &Vector3D{s * v.x, s * v.y, s * v.z}
}

func (v *Vector3D) Dot(w *Vector3D) float64 {
return v.x*w.x + v.y*w.y + v.z*w.z
}

func (v *Vector3D) String() string {
return fmt.Sprintf("(%v, %v, %v)", v.x, v.y, v.z)
}

func intersectPoint(rayVector, rayPoint, planeNormal, planePoint *Vector3D) *Vector3D {
diff := rayPoint.Sub(planePoint)
prod1 := diff.Dot(planeNormal)
prod2 := rayVector.Dot(planeNormal)
prod3 := prod1 / prod2
return rayPoint.Sub(rayVector.Mul(prod3))
}

func main() {
rv := &Vector3D{0.0, -1.0, -1.0}
rp := &Vector3D{0.0, 0.0, 10.0}
pn := &Vector3D{0.0, 0.0, 1.0}
pp := &Vector3D{0.0, 0.0, 5.0}
ip := intersectPoint(rv, rp, pn, pp)
fmt.Println("The ray intersects the plane at", ip)
}
```

{{out}}

```
The ray intersects the plane at (0, -5, 5)

```

{{trans|Kotlin}} Note that V3 is implemented similarly in the external library [https://hackage.haskell.org/package/linear-1.20.7/docs/Linear-V3.html linear].

```import Control.Applicative (liftA2)
import Text.Printf (printf)

data V3 a = V3 a a a
deriving Show

instance Functor V3 where
fmap f (V3 a b c) = V3 (f a) (f b) (f c)

instance Applicative V3 where
pure a = V3 a a a
V3 a b c <*> V3 d e f = V3 (a d) (b e) (c f)

instance Num a => Num (V3 a) where
(+) = liftA2 (+)
(-) = liftA2 (-)
(*) = liftA2 (*)
negate = fmap negate
abs = fmap abs
signum = fmap signum
fromInteger = pure . fromInteger

dot ::Num a => V3 a -> V3 a -> a
dot a b = x + y + z
where
V3 x y z = a * b

intersect :: Fractional a => V3 a -> V3 a -> V3 a -> V3 a -> V3 a
intersect rayVector rayPoint planeNormal planePoint =
rayPoint - rayVector * pure prod3
where
diff = rayPoint - planePoint
prod1 = diff `dot` planeNormal
prod2 = rayVector `dot` planeNormal
prod3 = prod1 / prod2

main = printf "The ray intersects the plane at (%f, %f, %f)\n" x y z
where
V3 x y z = intersect rv rp pn pp :: V3 Double
rv = V3 0 (-1) (-1)
rp = V3 0 0 10
pn = V3 0 0 1
pp = V3 0 0 5
```

{{out}}

```The ray intersects the plane at (0.0, -5.0, 5.0)
```

## J

'''Solution:'''

```mp=: +/ .*                          NB. matrix product
p=: mp&{: %~ -~&{. mp {:@]          NB. solve
intersectLinePlane=: [ +/@:* 1 , p  NB. substitute
```

'''Example Usage:'''

```   Line=: 0 0 10 ,: 0 _1 _1   NB. Point, Ray
Plane=: 0 0 5 ,: 0 0 1     NB. Point, Normal
Line intersectLinePlane Plane
0 _5 5
```

## Java

{{trans|Kotlin}}

```public class LinePlaneIntersection {
private static class Vector3D {
private double x, y, z;

Vector3D(double x, double y, double z) {
this.x = x;
this.y = y;
this.z = z;
}

Vector3D plus(Vector3D v) {
return new Vector3D(x + v.x, y + v.y, z + v.z);
}

Vector3D minus(Vector3D v) {
return new Vector3D(x - v.x, y - v.y, z - v.z);
}

Vector3D times(double s) {
return new Vector3D(s * x, s * y, s * z);
}

double dot(Vector3D v) {
return x * v.x + y * v.y + z * v.z;
}

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

private static Vector3D intersectPoint(Vector3D rayVector, Vector3D rayPoint, Vector3D planeNormal, Vector3D planePoint) {
Vector3D diff = rayPoint.minus(planePoint);
double prod1 = diff.dot(planeNormal);
double prod2 = rayVector.dot(planeNormal);
double prod3 = prod1 / prod2;
return rayPoint.minus(rayVector.times(prod3));
}

public static void main(String[] args) {
Vector3D rv = new Vector3D(0.0, -1.0, -1.0);
Vector3D rp = new Vector3D(0.0, 0.0, 10.0);
Vector3D pn = new Vector3D(0.0, 0.0, 1.0);
Vector3D pp = new Vector3D(0.0, 0.0, 5.0);
Vector3D ip = intersectPoint(rv, rp, pn, pp);
System.out.println("The ray intersects the plane at " + ip);
}
}
```

{{out}}

```The ray intersects the plane at (0.000000, -5.000000, 5.000000)
```

## Julia

{{works with|Julia|0.6}} {{trans|Python}}

```function lineplanecollision(planenorm::Vector, planepnt::Vector, raydir::Vector, raypnt::Vector)
ndotu = dot(planenorm, raydir)
if ndotu ≈ 0 error("no intersection or line is within plane") end

w  = raypnt - planepnt
si = -dot(planenorm, w) / ndotu
ψ  = w .+ si .* raydir .+ planepnt
return ψ
end

# Define plane
planenorm = Float64[0, 0, 1]
planepnt  = Float64[0, 0, 5]

# Define ray
raydir = Float64[0, -1, -1]
raypnt = Float64[0,  0, 10]

ψ = lineplanecollision(planenorm, planepnt, raydir, raypnt)
println("Intersection at \$ψ")
```

{{out}}

```Intersection at [0.0, -5.0, 5.0]
```

## Kotlin

```// version 1.1.51

class Vector3D(val x: Double, val y: Double, val z: Double) {

operator fun plus(v: Vector3D) = Vector3D(x + v.x, y + v.y, z + v.z)

operator fun minus(v: Vector3D) = Vector3D(x - v.x, y - v.y, z - v.z)

operator fun times(s: Double) = Vector3D(s * x, s * y, s * z)

infix fun dot(v: Vector3D) = x * v.x + y * v.y + z * v.z

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

fun intersectPoint(
rayVector: Vector3D,
rayPoint: Vector3D,
planeNormal: Vector3D,
planePoint: Vector3D
): Vector3D {
val diff  = rayPoint - planePoint
val prod1 = diff dot planeNormal
val prod2 = rayVector dot planeNormal
val prod3 = prod1 / prod2
return rayPoint - rayVector * prod3
}

fun main(args: Array<String>) {
val rv = Vector3D(0.0, -1.0, -1.0)
val rp = Vector3D(0.0,  0.0, 10.0)
val pn = Vector3D(0.0,  0.0,  1.0)
val pp = Vector3D(0.0,  0.0,  5.0)
val ip = intersectPoint(rv, rp, pn, pp)
println("The ray intersects the plane at \$ip")
}
```

{{out}}

```
The ray intersects the plane at (0.0, -5.0, 5.0)

```

## Lua

```function make(xval, yval, zval)
return {x=xval, y=yval, z=zval}
end

function plus(lhs, rhs)
return make(lhs.x + rhs.x, lhs.y + rhs.y, lhs.z + rhs.z)
end

function minus(lhs, rhs)
return make(lhs.x - rhs.x, lhs.y - rhs.y, lhs.z - rhs.z)
end

function times(lhs, scale)
return make(scale * lhs.x, scale * lhs.y, scale * lhs.z)
end

function dot(lhs, rhs)
return lhs.x * rhs.x + lhs.y * rhs.y + lhs.z * rhs.z
end

function tostr(val)
return "(" .. val.x .. ", " .. val.y .. ", " .. val.z .. ")"
end

function intersectPoint(rayVector, rayPoint, planeNormal, planePoint)
diff = minus(rayPoint, planePoint)
prod1 = dot(diff, planeNormal)
prod2 = dot(rayVector, planeNormal)
prod3 = prod1 / prod2
return minus(rayPoint, times(rayVector, prod3))
end

rv = make(0.0, -1.0, -1.0)
rp = make(0.0, 0.0, 10.0)
pn = make(0.0, 0.0, 1.0)
pp = make(0.0, 0.0, 5.0)
ip = intersectPoint(rv, rp, pn, pp)
print("The ray intersects the plane at " .. tostr(ip))
```

{{out}}

```The ray intersects the plane at (0, -5, 5)
```

## Maple

```geom3d:-plane(P, [geom3d:-point(p1,0,0,5), [0,0,1]]);
geom3d:-line(L, [geom3d:-point(p2,0,0,10), [0,-1,-1]]);
geom3d:-intersection(px,L,P);
geom3d:-detail(px);
```

{{Out}}

```[["name of the object",px],["form of the object",point3d],["coordinates of the point",[0,-5,5]]]
```

## MATLAB

{{trans|Kotlin}}

```function point = intersectPoint(rayVector, rayPoint, planeNormal, planePoint)

pdiff = rayPoint - planePoint;
prod1 = dot(pdiff, planeNormal);
prod2 = dot(rayVector, planeNormal);
prod3 = prod1 / prod2;

point = rayPoint - rayVector * prod3;
```

{{out}}

``` intersectPoint([0 -1 -1], [0 0 10], [0 0 1], [0 0 5])

ans =

0    -5     5

```

```MODULE LinePlane;
FROM RealStr IMPORT RealToStr;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;

TYPE
Vector3D = RECORD
x,y,z : REAL;
END;

PROCEDURE Minus(lhs,rhs : Vector3D) : Vector3D;
VAR out : Vector3D;
BEGIN
RETURN Vector3D{lhs.x-rhs.x, lhs.y-rhs.y, lhs.z-rhs.z};
END Minus;

PROCEDURE Times(a : Vector3D; s : REAL) : Vector3D;
BEGIN
RETURN Vector3D{a.x*s, a.y*s, a.z*s};
END Times;

PROCEDURE Dot(lhs,rhs : Vector3D) : REAL;
BEGIN
RETURN lhs.x*rhs.x + lhs.y*rhs.y + lhs.z*rhs.z;
END Dot;

PROCEDURE ToString(self : Vector3D);
VAR buf : ARRAY[0..63] OF CHAR;
BEGIN
WriteString("(");
RealToStr(self.x,buf);
WriteString(buf);
WriteString(", ");
RealToStr(self.y,buf);
WriteString(buf);
WriteString(", ");
RealToStr(self.z,buf);
WriteString(buf);
WriteString(")");
END ToString;

PROCEDURE IntersectPoint(rayVector,rayPoint,planeNormal,planePoint : Vector3D) : Vector3D;
VAR
diff : Vector3D;
prod1,prod2,prod3 : REAL;
BEGIN
diff := Minus(rayPoint,planePoint);
prod1 := Dot(diff, planeNormal);
prod2 := Dot(rayVector, planeNormal);
prod3 := prod1 / prod2;
RETURN Minus(rayPoint, Times(rayVector, prod3));
END IntersectPoint;

VAR ip : Vector3D;
BEGIN
ip := IntersectPoint(Vector3D{0.0,-1.0,-1.0},Vector3D{0.0,0.0,10.0},Vector3D{0.0,0.0,1.0},Vector3D{0.0,0.0,5.0});

WriteString("The ray intersects the plane at ");
ToString(ip);
WriteLn;

END LinePlane.
```

## Perl

{{trans|Perl 6}}

```package  Line; sub new { my (\$c, \$a) = @_; my \$self = { P0 => \$a->{P0}, u => \$a->{u} } } # point / ray
package Plane; sub new { my (\$c, \$a) = @_; my \$self = { V0 => \$a->{V0}, n => \$a->{n} } } # point / normal

package main;

sub dot { my \$p; \$p    += \$_[0][\$_] * \$_[1][\$_] for 0..@{\$_[0]}-1; \$p } # dot product
sub vd  { my @v; \$v[\$_] = \$_[0][\$_] - \$_[1][\$_] for 0..@{\$_[0]}-1; @v } # vector difference
sub va  { my @v; \$v[\$_] = \$_[0][\$_] + \$_[1][\$_] for 0..@{\$_[0]}-1; @v } # vector addition
sub vp  { my @v; \$v[\$_] = \$_[0][\$_] * \$_[1][\$_] for 0..@{\$_[0]}-1; @v } # vector product

sub line_plane_intersection {
my(\$L, \$P) = @_;

my \$cos = dot(\$L->{u}, \$P->{n});     # cosine between normal & ray
return 'Vectors are orthogonol; no intersection or line within plane' if \$cos == 0;

my @W = vd(\$L->{P0},\$P->{V0});       # difference between P0 and V0
my \$SI = -dot(\$P->{n}, \@W) / \$cos;  # line segment where it intersets the plane

my @a = vp(\$L->{u},[(\$SI)x3]);
my @b = va(\$P->{V0},\@a);
va(\@W,\@b);
}

my \$L =  Line->new({ P0=>[0,0,10], u=>[0,-1,-1]});
my \$P = Plane->new({ V0=>[0,0,5 ], n=>[0, 0, 1]});
print 'Intersection at point: ', join(' ', line_plane_intersection(\$L, \$P)) . "\n";

```

{{out}}

```Intersection at point: 0 -5 5
```

## Perl 6

{{works with|Rakudo|2016.11}} {{trans|Python}}

```class Line {
has \$.P0; # point
has \$.u⃗;  # ray
}
class Plane {
has \$.V0; # point
has \$.n⃗;  # normal
}

sub infix:<∙> ( @a, @b where +@a == +@b ) { [+] @a «*» @b } # dot product

sub line-plane-intersection (\$𝑳, \$𝑷) {
my \$cos = \$𝑷.n⃗ ∙ \$𝑳.u⃗; # cosine between normal & ray
return 'Vectors are orthogonal; no intersection or line within plane'
if \$cos == 0;
my \$𝑊 = \$𝑳.P0 «-» \$𝑷.V0;      # difference between P0 and V0
my \$S𝐼 = -(\$𝑷.n⃗ ∙ \$𝑊) / \$cos;  # line segment where it intersects the plane
\$𝑊 «+» \$S𝐼 «*» \$𝑳.u⃗ «+» \$𝑷.V0; # point where line intersects the plane
}

say 'Intersection at point: ', line-plane-intersection(
Line.new( :P0(0,0,10), :u⃗(0,-1,-1) ),
Plane.new( :V0(0,0, 5), :n⃗(0, 0, 1) )
);
```

{{out}}

```Intersection at point: (0 -5 5)
```

## Phix

```function dot(sequence a, b) return sum(sq_mul(a,b)) end function

function intersection_point(sequence line_vector,line_point,plane_normal,plane_point)
atom a = dot(line_vector,plane_normal)
if a=0 then return "no intersection" end if
sequence diff = sq_sub(line_point,plane_point)
end function

?intersection_point({0,-1,-1},{0,0,10},{0,0,1},{0,0,5})
?intersection_point({3,2,1},{0,2,4},{1,2,3},{3,3,3})
?intersection_point({1,1,0},{0,0,1},{0,0,3},{0,0,0}) -- (parallel to plane)
?intersection_point({1,1,0},{1,1,0},{0,0,3},{0,0,0}) -- (line within plane)
```

{{out}}

```
{0,-5,5}
{0.6,2.4,4.2}
"no intersection"
"no intersection"

```

## Python

Based on the approach at geomalgorithms.comhttp://geomalgorithms.com/a05-_intersect-1.html

```#!/bin/python
from __future__ import print_function
import numpy as np

def LinePlaneCollision(planeNormal, planePoint, rayDirection, rayPoint, epsilon=1e-6):

ndotu = planeNormal.dot(rayDirection)
if abs(ndotu) < epsilon:
raise RuntimeError("no intersection or line is within plane")

w = rayPoint - planePoint
si = -planeNormal.dot(w) / ndotu
Psi = w + si * rayDirection + planePoint
return Psi

if __name__=="__main__":
#Define plane
planeNormal = np.array([0, 0, 1])
planePoint = np.array([0, 0, 5]) #Any point on the plane

#Define ray
rayDirection = np.array([0, -1, -1])
rayPoint = np.array([0, 0, 10]) #Any point along the ray

Psi = LinePlaneCollision(planeNormal, planePoint, rayDirection, rayPoint)
print ("intersection at", Psi)
```

{{out}}

```intersection at [ 0 -5  5]
```

## Racket

{{trans|Sidef}}

```#lang racket
;; {{trans|Sidef}}
;; vectors are represented by lists

(struct Line (P0 u⃗))

(struct Plane (V0 n⃗))

(define (· a b) (apply + (map * a b)))

(define (line-plane-intersection L P)
(match-define (cons (Line P0 u⃗) (Plane V0 n⃗)) (cons L P))
(define cos (· n⃗ u⃗))
(when (zero? cos) (error "vectors are orthoganal"))
(define W (map - P0 V0))
(define *SI (let ((SI (- (/ (· n⃗ W) cos)))) (λ (n) (* SI n))))
(map + W (map *SI u⃗) V0))

(module+ test
(require rackunit)
(check-equal?
(line-plane-intersection (Line '(0 0 10) '(0 -1 -1))
(Plane '(0 0 5) '(0 0 1)))
'(0 -5 5)))
```

{{out}} No output -- all tests passed!

## REXX

### version 1

This program does NOT handle the case when the line is parallel to or within the plane.

```/* REXX */
Parse Value '0 0 1' With n.1 n.2 n.3   /* Normal Vector of the plane */
Parse Value '0 0 5' With p.1 p.2 p.3   /* Point in the plane         */
Parse Value '0 0 10' With a.1 a.2 a.3  /* Point of the line          */
Parse Value '0 -1 -1' With v.1 v.2 v.3 /* Vector of the line         */

a=n.1
b=n.2
c=n.3
d=n.1*p.1+n.2*p.2+n.3*p.3  /* Parameter form of the plane */
Say a'*x +' b'*y +' c'*z =' d

t=(d-(a*a.1+b*a.2+c*a.3))/(a*v.1+b*v.2+c*v.3)

x=a.1+t*v.1
y=a.2+t*v.2
z=a.3+t*v.3

Say 'Intersection: P('||x','y','z')'
```

{{out}}

```0*x + 0*y + 1*z = 5
Intersection: P(0,-5,5)
```

### version 2

handle the case that the line is parallel to the plane or lies within it.

```/*REXX*/
Parse Value '1 2 3' With n.1 n.2 n.3
Parse Value '3 3 3' With p.1 p.2 p.3
Parse Value '0 2 4' With a.1 a.2 a.3
Parse Value '3 2 1' With v.1 v.2 v.3

a=n.1
b=n.2
c=n.3
d=n.1*p.1+n.2*p.2+n.3*p.3  /* Parameter form of the plane */
Select
When a=0 Then
pd=''
When a=1 Then
pd='x'
When a=-1 Then
pd='-x'
Otherwise
pd=a'*x'
End
pd=pd
yy=mk2('y',b)
Select
When left(yy,1)='-' Then
pd=pd '-' substr(yy,2)
When left(yy,1)='0' Then
Nop
Otherwise
pd=pd '+' yy
End
zz=mk2('z',c)
Select
When left(zz,1)='-' Then
pd=pd '-' substr(zz,2)
When left(zz,1)='0' Then
Nop
Otherwise
pd=pd '+' zz
End
pd=pd '=' d

Say 'Plane definition:' pd

ip=0
Do i=1 To 3
ip=ip+n.i*v.i
dd=n.1*a.1+n.2*a.2+n.3*a.3
End
If ip=0 Then Do
If dd=d Then
Say 'Line is part of the plane'
Else
Say 'Line is parallel to the plane'
Exit
End

t=(d-(a*a.1+b*a.2+c*a.3))/(a*v.1+b*v.2+c*v.3)

x=a.1+t*v.1
y=a.2+t*v.2
z=a.3+t*v.3

ld=mk('x',a.1,v.1) ';' mk('y',a.2,v.2) ';' mk('z',a.3,v.3)
Say 'Line definition:' ld

Say 'Intersection: P('||x','y','z')'
Exit

Mk: Procedure
/*---------------------------------------------------------------------
* build part of line definition
*--------------------------------------------------------------------*/
Parse Arg v,aa,vv
If aa<>0 Then
res=v'='aa
Else
res=v'='
Select
When vv=0 Then
res=res||'0'
When vv=-1 Then
res=res||'-t'
When vv<0 Then
res=res||vv'*t'
Otherwise Do
If res=v'=' Then Do
If vv=1 Then
res=res||'t'
Else
res=res||vv'*t'
End
Else Do
If vv=1 Then
res=res||'+t'
Else
res=res||'+'vv'*t'
End
End
End
Return res

mk2: Procedure
/*---------------------------------------------------------------------
* build part of plane definition
*--------------------------------------------------------------------*/
Parse Arg v,u
Select
When u=0 Then
res=''
When u=1 Then
res=v
When u=-1 Then
res='-'v
When u<0 Then
res=u'*'v
Otherwise Do
If pd<>'' Then
res='+'u'*'v
Else
res=u'*'v
End
End
Return res
```

{{out}}

```Plane definition: x+2*y+3*z=18
Line definition: x=3*t ; y=2+2*t ; z=4+t
Intersection: P(0.6,2.4,4.2)
```

## Rust

{{trans|Kotlin}}

```use std::ops::{Add, Div, Mul, Sub};

#[derive(Copy, Clone, Debug, PartialEq)]
struct V3<T> {
x: T,
y: T,
z: T,
}

impl<T> V3<T> {
fn new(x: T, y: T, z: T) -> Self {
V3 { x, y, z }
}
}

fn zip_with<F, T, U>(f: F, a: V3<T>, b: V3<T>) -> V3<U>
where
F: Fn(T, T) -> U,
{
V3 {
x: f(a.x, b.x),
y: f(a.y, b.y),
z: f(a.z, b.z),
}
}

impl<T> Add for V3<T>
where
T: Add<Output = T>,
{
type Output = Self;

fn add(self, other: Self) -> Self {
}
}

impl<T> Sub for V3<T>
where
T: Sub<Output = T>,
{
type Output = Self;

fn sub(self, other: Self) -> Self {
zip_with(<T>::sub, self, other)
}
}

impl<T> Mul for V3<T>
where
T: Mul<Output = T>,
{
type Output = Self;

fn mul(self, other: Self) -> Self {
zip_with(<T>::mul, self, other)
}
}

impl<T> V3<T>
where
T: Mul<Output = T> + Add<Output = T>,
{
fn dot(self, other: Self) -> T {
let V3 { x, y, z } = self * other;
x + y + z
}
}

impl<T> V3<T>
where
T: Mul<Output = T> + Copy,
{
fn scale(self, scalar: T) -> Self {
self * V3 {
x: scalar,
y: scalar,
z: scalar,
}
}
}

fn intersect<T>(
ray_vector: V3<T>,
ray_point: V3<T>,
plane_normal: V3<T>,
plane_point: V3<T>,
) -> V3<T>
where
T: Add<Output = T> + Sub<Output = T> + Mul<Output = T> + Div<Output = T> + Copy,
{
let diff = ray_point - plane_point;
let prod1 = diff.dot(plane_normal);
let prod2 = ray_vector.dot(plane_normal);
let prod3 = prod1 / prod2;
ray_point - ray_vector.scale(prod3)
}

fn main() {
let rv = V3::new(0.0, -1.0, -1.0);
let rp = V3::new(0.0, 0.0, 10.0);
let pn = V3::new(0.0, 0.0, 1.0);
let pp = V3::new(0.0, 0.0, 5.0);
println!("{:?}", intersect(rv, rp, pn, pp));
}

```

## Scala

```object LinePLaneIntersection extends App {
val (rv, rp, pn, pp) =
(Vector3D(0.0, -1.0, -1.0), Vector3D(0.0, 0.0, 10.0), Vector3D(0.0, 0.0, 1.0), Vector3D(0.0, 0.0, 5.0))
val ip = intersectPoint(rv, rp, pn, pp)

def intersectPoint(rayVector: Vector3D,
rayPoint: Vector3D,
planeNormal: Vector3D,
planePoint: Vector3D): Vector3D = {
val diff = rayPoint - planePoint
val prod1 = diff dot planeNormal
val prod2 = rayVector dot planeNormal
val prod3 = prod1 / prod2
rayPoint - rayVector * prod3
}

case class Vector3D(x: Double, y: Double, z: Double) {
def +(v: Vector3D) = Vector3D(x + v.x, y + v.y, z + v.z)
def -(v: Vector3D) = Vector3D(x - v.x, y - v.y, z - v.z)
def *(s: Double) = Vector3D(s * x, s * y, s * z)
def dot(v: Vector3D): Double = x * v.x + y * v.y + z * v.z
override def toString = s"(\$x, \$y, \$z)"
}

println(s"The ray intersects the plane at \$ip")
}
```

{{Out}}See it in running in your browser by [https://scalafiddle.io/sf/oLTlNZk/0 ScalaFiddle (JavaScript)].

## Sidef

{{trans|Perl 6}}

```struct Line {
P0,       # point
u⃗,        # ray
}

struct Plane {
V0,       # point
n⃗,        # normal
}

func dot_prod(a, b) { a »*« b -> sum }

func line_plane_intersection(𝑳, 𝑷) {
var cos = dot_prod(𝑷.n⃗, 𝑳.u⃗) ->
|| return 'Vectors are orthogonal'
var 𝑊 = (𝑳.P0 »-« 𝑷.V0)
var S𝐼 = -(dot_prod(𝑷.n⃗, 𝑊) / cos)
𝑊 »+« (𝑳.u⃗ »*» S𝐼) »+« 𝑷.V0
}

say ('Intersection at point: ', line_plane_intersection(
Line(P0: [0,0,10], u⃗: [0,-1,-1]),
Plane(V0: [0,0, 5], n⃗: [0, 0, 1]),
))
```

{{out}}

```Intersection at point: [0, -5, 5]
```

## Visual Basic .NET

{{trans|C#}}

```Module Module1

Class Vector3D
Private ReadOnly x As Double
Private ReadOnly y As Double
Private ReadOnly z As Double

Sub New(nx As Double, ny As Double, nz As Double)
x = nx
y = ny
z = nz
End Sub

Public Function Dot(rhs As Vector3D) As Double
Return x * rhs.x + y * rhs.y + z * rhs.z
End Function

Public Shared Operator +(ByVal a As Vector3D, ByVal b As Vector3D) As Vector3D
Return New Vector3D(a.x + b.x, a.y + b.y, a.z + b.z)
End Operator

Public Shared Operator -(ByVal a As Vector3D, ByVal b As Vector3D) As Vector3D
Return New Vector3D(a.x - b.x, a.y - b.y, a.z - b.z)
End Operator

Public Shared Operator *(ByVal a As Vector3D, ByVal b As Double) As Vector3D
Return New Vector3D(a.x * b, a.y * b, a.z * b)
End Operator

Public Overrides Function ToString() As String
Return String.Format("({0:F}, {1:F}, {2:F})", x, y, z)
End Function
End Class

Function IntersectPoint(rayVector As Vector3D, rayPoint As Vector3D, planeNormal As Vector3D, planePoint As Vector3D) As Vector3D
Dim diff = rayPoint - planePoint
Dim prod1 = diff.Dot(planeNormal)
Dim prod2 = rayVector.Dot(planeNormal)
Dim prod3 = prod1 / prod2
Return rayPoint - rayVector * prod3
End Function

Sub Main()
Dim rv = New Vector3D(0.0, -1.0, -1.0)
Dim rp = New Vector3D(0.0, 0.0, 10.0)
Dim pn = New Vector3D(0.0, 0.0, 1.0)
Dim pp = New Vector3D(0.0, 0.0, 5.0)
Dim ip = IntersectPoint(rv, rp, pn, pp)
Console.WriteLine("The ray intersects the plane at {0}", ip)
End Sub

End Module
```

{{out}}

```The ray intersects the plane at (0.00, -5.00, 5.00)
```

## zkl

{{trans|Perl 6}}{{trans|Python}}

```class Line { fcn init(pxyz, ray_xyz)   { var pt=pxyz, ray=ray_xyz;       } }
class Plane{ fcn init(pxyz, normal_xyz){ var pt=pxyz, normal=normal_xyz; } }

fcn dotP(a,b){ a.zipWith('*,b).sum(0.0); }  # dot product --> x
fcn linePlaneIntersection(line,plane){
cos:=dotP(plane.normal,line.ray); # cosine between normal & ray
_assert_((not cos.closeTo(0,1e-6)),
"Vectors are orthogonol; no intersection or line within plane");
w:=line.pt.zipWith('-,plane.pt); # difference between P0 and V0
si:=-dotP(plane.normal,w)/cos;   # line segment where it intersets the plane
# point where line intersects the plane:
//w.zipWith('+,line.ray.apply('*,si)).zipWith('+,plane.pt);  // or
w.zipWith('wrap(w,r,pt){ w + r*si + pt },line.ray,plane.pt);
}
```
```println("Intersection at point: ", linePlaneIntersection(
Line( T(0.0, 0.0, 10.0), T(0.0, -1.0, -1.0) ),
Plane(T(0.0, 0.0,  5.0), T(0.0,  0.0,  1.0) ))
);
```

{{out}}

```
Intersection at point: L(0,-5,5)

```

'''References'''