The intersection point of two lines in 3D

Question

How to calculate the intersection point of two lines in 3D?
source code!

Answer 1

this is actually a rather easy operation. lets look at some basic properties of lines / vectors in 3D space:
1. If two lines intersect at a single point, then there must be exactly one plane in which the two lines are co-planar. This is pretty trivial to prove, but I won't bother here, because you can google to find out that two vectors that extend from a common point define a single plane.
2. If you take the cross-product of two vectors, the resulting vector is guaranteed to be perpendicular to both lines (this is a basic property of a cross-product). If the two lines intersect, then this cross-product will be the normal vector to the plane in which both lines lie.
3. Now transform (rotate) yours space such that your z-axis becomes parallel to the normal vector of that plane.
Now, the parametric equation for a line in 3D space is this:
(x, y, z) = (xo, yo, zo) + k(xd, yd, zd)
After you have transformed your space, you will notice that the equation of each line will now be
(x, y, z) = (xo, yo, zo) + k(xd, yd, 0)
If zo is equal for both lines, then the two lines intersect. Otherwise, the two lines do not intersect.
if the intersection exists, then to solve for it, you ignore the z coordinate, and treat it like a two dimensional system:
(x, y) = (xo, yo) + k(xd, yd)
you can use the basic 2D formula y - yo = m(x - xo), or y = m(x - xo) + yo
you can calculate m = yd / xd.
which simplifies to y = mx -mxo + yo, or 0 = mx - y + (yo - mxo)
you now have two equations of the form Ax + By + C = 0.
From there, you can use Cramer's rule to find the intersection of the two lines.
It sounds kind of complicated when you read it as I have presented it, but it's actually extremely easy to write a program for, because each step is simple mathematic operation with basically no decisions involved (hence no if-structures, except at the one point where you actually need to determine if the lines intersect).
So to reiterate, the steps required are:
1. Check if lines are parallel. If yes, check if they are the same line, and then stop.
2. If not parallel, cross-multiply the direction vectors of each line.
3. Rotate your space such that z is parallel with cross-product of lines
i. dot-product of z axis (0, 0, 1) with cross-product of lines gives angle*
ii. cross-product of z axis (0, 0, 1) with cross-product of lines gives axis of rotation.
iii. rotate your lines (i.e. point and direction vector) about that axis by negative of ange determined in (i).
4. if z value of points from both lines is the different, no solution. stop.
5. otherwise, apply Cramer's rule.
6. done.
* obtaining the angle of rotation with the dot product relies on the angle being less than 90 degrees, so check if z value of normal vector is + or -, and then perform dot product and cross-product on appropriate + or - z vector.
- Adam