The Algorithm to Find the Point of Intersection of

Question

Finding the point of intersection for two 2D line segment is easy; the formula is straight forward. But finding the point of intersection for two 3D line segment is not, I afraid.

What is the algorithm, in C# preferably that finds the point of intersection of two 3D line segments?

I found a C++ implementation here. But I don't trust the solution because it makes preference of a certain plane ( look at the way perp is implemented under the implementation section, it assumes a preference for z plane. Any generic algorithm must not assume any plane orientation or preference).

Is there a better solution?

Answer 1

I found an answer!

in an answer from above, I found these equations:

Eq#1: var s = Vector3.Dot(Vector3.Cross(dc, db), Vector3.Cross(da, db)) / Norm2(Vector3.Cross(da, db));

Eq#2: var t = Vector3.Dot(Vector3.Cross(dc, da), Vector3.Cross(da, db)) / Norm2(Vector3.Cross(da, db));

Then I modified #3rd Equation:

Eq#3:

if ((s >= 0 && s <= 1) && (k >= 0 && k <= 1))
{
   Vector3 res = new Vector3(this.A.x + da.x * s, this.A.y + da.y * s, this.A.z + da.z * s);
}

And while keeping Eq#1 and Eq#2 just the same, I created this equations:

MyEq#1: Vector3f p0 = da.mul(s).add(A<vector>); MyEq#2: Vector3f p1 = db.mul(t).add(C<vector>);

then I took a wild guess at creating these three more equations:

MyEq#3: Vector3f p0z = projUV(da, p0).add(A<vector>); MyEq#4: Vector3f p1z = projUV(db, p1).add(C<vector>);

and finally to get the subtraction of the two magnitudes of the projUV(1, 2) gives you the margin of the error between 0 and 0.001f to find whether the two lines intersect.

MyEq#5: var m = p0z.magnitude() - p1z.magnitude();

Now I mind you, this was done in Java. This explanation is not java convention ready. Just put it to work from the above equations. (Tip: Don't transform to World Space yet so that both projection of UV equations fall exactly where you want them).

And these equations are visually correct in my program.