The algorithm to find the point of intersection of two 3D line segment

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

Most 3D lines do not intersect. A reliable method is to find the shortest line between two 3D lines. If the shortest line has a length of zero (or distance less than whatever tolerance you specify) then you know that the two original lines intersect.

A method for finding the shortest line between two 3D lines, written by Paul Bourke is summarized / paraphrased as follows:

In what follows a line will be defined by two points lying on it, a point on line "a" defined by points P1 and P2 has an equation

Pa = P1 + mua (P2 - P1) 

similarly a point on a second line "b" defined by points P4 and P4 will be written as

Pb = P3 + mub (P4 - P3) 

The values of mua and mub range from negative to positive infinity. The line segments between P1 P2 and P3 P4 have their corresponding mu between 0 and 1.

There are two approaches to finding the shortest line segment between lines "a" and "b".

Approach one:

The first is to write down the length of the line segment joining the two lines and then find the minimum. That is, minimise the following

|| Pb - Pa ||^2 

Substituting the equations of the lines gives

|| P1 - P3 + mua (P2 - P1) - mub (P4 - P3) ||^2 

The above can then be expanded out in the (x,y,z) components.

There are conditions to be met at the minimum, the derivative with respect to mua and mub must be zero. ...the above function only has one minima and no other minima or maxima. These two equations can then be solved for mua and mub, the actual intersection points found by substituting the values of mu into the original equations of the line.

Approach two:

An alternative approach but one that gives the exact same equations is to realise that the shortest line segment between the two lines will be perpendicular to the two lines. This allows us to write two equations for the dot product as

(Pa - Pb) dot (P2 - P1) = 0 (Pa - Pb) dot (P4 - P3) = 0 

Expanding these given the equation of the lines

( P1 - P3 + mua (P2 - P1) - mub (P4 - P3) ) dot (P2 - P1) = 0 ( P1 - P3 + mua (P2 - P1) - mub (P4 - P3) ) dot (P4 - P3) = 0 

Expanding these in terms of the coordinates (x,y,z) ... the result is as follows

d1321 + mua d2121 - mub d4321 = 0 d1343 + mua d4321 - mub d4343 = 0 

where

dmnop = (xm - xn)(xo - xp) + (ym - yn)(yo - yp) + (zm - zn)(zo - zp) 

Note that dmnop = dopmn

Finally, solving for mua gives

mua = ( d1343 d4321 - d1321 d4343 ) / ( d2121 d4343 - d4321 d4321 ) 

and back-substituting gives mub

mub = ( d1343 + mua d4321 ) / d4343 

This method was found on Paul Bourke's website which is an excellent geometry resource. The site has been reorganized, so scroll down to find the topic.

Answer 2

I tried @Bill answer and it actually does not work every time, which I can explain. Based on the link in his code.Let's have for example these two line segments AB and CD.

A=(2,1,5), B=(1,2,5) and C=(2,1,3) and D=(2,1,2)

when you try to get the intersection it might tell you It's the point A (incorrect) or there is no intersection (correct). Depending on the order you put those segments in.

x = A+(B-A)s
x = C+(D-C)t

Bill solved for s but never solved t. And since you want that intersection point to be on both line segments both s and t have to be from interval <0,1>. What actually happens in my example is that only s if from that interval and t is -2. A lies on line defined by C and D, but not on line segment CD.

var s = Vector3.Dot(Vector3.Cross(dc, db), Vector3.Cross(da, db)) / Norm2(Vector3.Cross(da, db));  var t = Vector3.Dot(Vector3.Cross(dc, da), Vector3.Cross(da, db)) / Norm2(Vector3.Cross(da, db)); 

where da is B-A, db is D-C and dc is C-A, I just preserved names provided by Bill.

Then as I said you have to check if both s and t are from <0,1> and you can calculate the result. Based on formula above.

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); } 

Also another problem with Bills answer is when two lines are collinear and there is more than one intersection point. There would be division by zero. You want to avoid that.