3d line-intersection code not working properly - math

Question

I created this piece of code to get the intersection of two 3d line-segments.
Unfortunately the result of this code is inaccurate, the intersection-point is not always on both lines.
I am confused and unsure what I'm doing wrong.
Here is my code:
--dir = direction
--p1,p2 = represents the line
function GetIntersection(dirStart, dirEnd, p1, p2)
local s1_x, s1_y, s2_x, s2_y = dirEnd.x - dirStart.x, dirEnd.z - dirStart.z, p2.x - p1.x, p2.z - p1.z
local div = (-s2_x * s1_y) + (s1_x * s2_y)
if div == 0 then return nil end
local s = (-s1_y * (dirStart.x - p1.x) + s1_x * (dirStart.z - p1.z)) / div
local t = ( s2_x * (dirStart.z - p1.z) - s2_y * (dirStart.x - p1.x)) / div
if (s >= 0 and s <= 1 and t >= 0 and t <= 1) and (Vector(dirStart.x + (t * s1_x), 0, dirStart.z + (t * s1_y)) or nil) then
local v = Vector(dirStart.x + (t * s1_x),0,dirStart.z + (t * s1_y))
return v
end
end

Answer 1

This is example of Delphi code to find a distance between two skew lines in 3D. For your purposes it is necessary to check that result if small enough value (intersection does exist), check that s and t parameters are in range 0..1, then
calculate point using parameter s
Math of this approach is described in 'the shortest line...' section of Paul Bourke page
VecDiff if vector difference function, Dot id scalar product function
function LineLineDistance(const L0, L1: TLine3D; var s, t: Double): Double;
var
u: TPoint3D;
a, b, c, d, e, det, invdet:Double;
begin
u := VecDiff(L1.Base, L0.Base);
a := Dot(L0.Direction, L0.Direction);
b := Dot(L0.Direction, L1.Direction);
c := Dot(L1.Direction, L1.Direction);
d := Dot(L0.Direction, u);
e := Dot(L1.Direction, u);
det := a * c - b * b;
if det < eps then
Result := -1
else begin
invdet := 1 / det;
s := invdet * (b * e - c * d);
t := invdet * (a * e - b * d);
Result := Distance(PointAtParam(L0, s), PointAtParam(L1, t));
end;
end;

Answer 2

As far as I can tell your code is good. I've implemented this in javascript at https://jsfiddle.net/SalixAlba/kkrc9kcf/
and it seems to work for all the cases I can think of.
The only changes I've done is to change things to work in javascript rather than lua. The final condition was commented out
function GetIntersection(dirStart, dirEnd, p1, p2) {
var s1_x = dirEnd.x - dirStart.x;
var s1_y = dirEnd.z - dirStart.z;
var s2_x = p2.x - p1.x;
var s2_y = p2.z - p1.z;
var div = (-s2_x * s1_y) + (s1_x * s2_y);
if (div == 0)
return new Vector(0,0);
var s = (-s1_y * (dirStart.x - p1.x) + s1_x * (dirStart.z - p1.z)) / div;
var t = ( s2_x * (dirStart.z - p1.z) - s2_y * (dirStart.x - p1.x)) / div;
if (s >= 0 && s <= 1 && t >= 0 && t <= 1) {
//and (Vector(dirStart.x + (t * s1_x), 0, dirStart.z + (t * s1_y)) or nil) then
var v = new Vector(
dirStart.x + (t * s1_x),
dirStart.z + (t * s1_y));
return v;
}
return new Vector(0,0);
}
Mathmatically it makes sense. If A,B and C,D are your two lines. Let s1 = B-A, s2 = C-D. A point of the line AB is given by A + t s1 and a point on the line CD is given by C + s s2. For an intersection we require
A + t s1 = C + s s2
or
(A-C) + t s1 = s s2
You two formula for s, t are found by taking the 2D cross product with each of the vectors s1 and s2
(A-C)^s1 + t s1^s1 = s s2^s1
(A-C)^s2 + t s1^s2 = s s2^s2
recalling s1^s1=s2^s2=0 and s2^s1= - s1^s2 we get
(A-C)^s1 = s s2^s1
(A-C)^s2 + t s1^s2 = 0
which can be solved to get s and t. This matches your equations.