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intersecting 3D lines

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kitwallace

intersecting 3D lines

Sorry -being lazy- but is there code somewhere for computing the intersection, if any, of 2 3-d lines?

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Re: intersecting 3D lines

In case no one has code...

Not code but some formulas and videos can be found using Google "intersection of lines in 3D"

https://socratic.org/questions/how-do-i-find-the-intersection-of-two-lines-in-three-dimensional-space
https://math.stackexchange.com/questions/270767/find-intersection-of-two-3d-lines/271366

http://paulbourke.net/geometry/pointlineplane/calclineline.cs has some code in C.

I hope that this gets you started.

On 2021-02-26 7:08 a.m., kitwallace wrote:

Sorry -being lazy- but is there code somewhere for computing the intersection, if any, of 2 3-d lines?

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kitwallace

Re: intersecting 3D lines

Thanks - I used Pauls' code - here it is in OpenSCAD in case anyone looks :

function intersection (p1,p2,p3,p4,eps=0.0001) =
let (p13=p1-p3,
p43=p4-p3)
norm(p43) <eps
? undef
:
let (p21=p2-p1)
norm(p21) < eps
? undef
:
let (
d1343= p13*p43,
d4321= p43*p21,
d1321= p13*p21,
d4343= p43*p43,
d2121= p21*p21,
denom=d2121 * d4343 - d4321 * d4321)
abs(denom) < eps
? undef
: let (numer=d1343 * d4321 - d1321 * d4343,
mua = numer / denom,
mub = (d1343 + d4321 *mua ) / d4343)
[p1 + mua*p21,
p3 + mub*p43];

Ronaldo

Re: intersecting 3D lines

Anyway, here is my version that seems to be about the same:

function 3D_line_intersection( line1, line2, prec=1e-6) =
assert(norm(p1-p0)>prec) // assert well defined lines
assert(norm(q1-q0)>prec)
let(
p0 = line1[0],
p1 = line1[1],
q0 = line2[0],
q1 = line2[1],
d = cross( p1-p0, q1-q0 ) // normal to lines if non zero
)
norm(d)<prec // (nearly) coincident or parallel lines ?
? norm(cross( q1-p0, p1-p0 ))<=prec*norm(q1-p0)*norm(p1-p0)
? undef // coincident lines
: [] // disjoint parallel lines
: let( d = d/norm(d) )
abs(d*(p1-q1)) > prec // are the lines disjoint ?
? [] // disjoint lines
: let( // computes the crossing points
crxq = cross(d, q1-q0),
crxp = cross(d, p1-p0),
u = crxq*(q0-p0)/(crxq*(p1-p0)),
v = crxp*(p0-q0)/(crxp*(q1-q0))
)
[p0 + u*(p1-p0), q0 + v*(q1-q0)];

It returns undef when the lines are nearly coincident and [] if they are parallel. When crossing, it returns two points, one in each line, whose distance is less than prec.

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kitwallace

Re: intersecting 3D lines

Thanks Ronaldo - that's a bit bit more vectory - I should use assert more - although just to note that the first two asserts are too early :)