Find the intersection of a line with a plane
Finding the intersection of an infinite ray with a plane in 3D is an important topic in collision detection.
- Task
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].
Perl 6
<lang perl6>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 orthoganol; 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 intersets 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) ) );</lang>
- Output:
Intersection at point: (0 -5 5)
Python
Based on the approach at geomalgorithms.com[1]
<lang python>#!/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)</lang>
- Output:
intersection at [ 0 -5 5]
Racket
<lang racket>#lang racket
- 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)))</lang>
- Output:
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. <lang rexx>/* 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')'</lang>
- Output:
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 withi it. <lang rexx>/*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' Otherwise pd=a'*x' End
pd=pd||mk2('y',b)||mk2('z',c)'='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||v1'*t'
Otherwise Do
If res='x=' Then Do
If v1=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<0 Then
res=u'*'v
Otherwise Do
If pd<> Then
res='+'u'*'v
Else
res=u'*'v
End
End
Return res</lang>
- Output:
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)
Sidef
<lang ruby>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]),
))</lang>
- Output:
Intersection at point: [0, -5, 5]
zkl
<lang zkl>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 orthoganol; 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);
}</lang> <lang zkl>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) ))
);</lang>
- Output:
Intersection at point: L(0,-5,5)