The shortest line between two lines in 3D

Two lines in 3 dimensions generally don't intersect at a point, they may be parallel (no intersections) or they may be coincident (infinite intersections) but most often only their projection onto a plane intersect.. When they don't exactly intersect at a point they can be connected by a line segment, the shortest line segment is unique and is often considered to be their intersection in 3D.

The following will show how to compute this shortest line segment that joins two lines in 3D, it will as a biproduct identify parrallel lines. 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". 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

Substituting the equations of the lines gives

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 mu_a and mu_b must be zero. Note: it is easy to convince oneself that the above function only has one minima and no other minima or maxima. These two equations can then be solved for mu_a and mu_b, the actual intersection points found by substituting the values of mu into the original equations of the line.

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

Expanding these given the equation of the lines

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

where

Note that d_mnop = d_opmn

Finally, solving for mu_a gives

and backsubstituting gives mu_b

C source

typedef struct {
   double x,y,z;
} XYZ;

/*
   Calculate the line segment PaPb that is the shortest route between
   two lines P1P2 and P3P4. Calculate also the values of mua and mub where
      Pa = P1 + mua (P2 - P1)
      Pb = P3 + mub (P4 - P3)
   Return FALSE if no solution exists.
*/
int LineLineIntersect(
   XYZ p1,XYZ p2,XYZ p3,XYZ p4,XYZ *pa,XYZ *pb,
   double *mua, double *mub)
{
   XYZ p13,p43,p21;
   double d1343,d4321,d1321,d4343,d2121;
   double numer,denom;

   p13.x = p1.x - p3.x;
   p13.y = p1.y - p3.y;
   p13.z = p1.z - p3.z;
   p43.x = p4.x - p3.x;
   p43.y = p4.y - p3.y;
   p43.z = p4.z - p3.z;
   if (ABS(p43.x)  < EPS && ABS(p43.y)  < EPS && ABS(p43.z)  < EPS)
      return(FALSE);
   p21.x = p2.x - p1.x;
   p21.y = p2.y - p1.y;
   p21.z = p2.z - p1.z;
   if (ABS(p21.x)  < EPS && ABS(p21.y)  < EPS && ABS(p21.z)  < EPS)
      return(FALSE);

   d1343 = p13.x * p43.x + p13.y * p43.y + p13.z * p43.z;
   d4321 = p43.x * p21.x + p43.y * p21.y + p43.z * p21.z;
   d1321 = p13.x * p21.x + p13.y * p21.y + p13.z * p21.z;
   d4343 = p43.x * p43.x + p43.y * p43.y + p43.z * p43.z;
   d2121 = p21.x * p21.x + p21.y * p21.y + p21.z * p21.z;

   denom = d2121 * d4343 - d4321 * d4321;
   if (ABS(denom) < EPS)
      return(FALSE);
   numer = d1343 * d4321 - d1321 * d4343;

   *mua = numer / denom;
   *mub = (d1343 + d4321 * (*mua)) / d4343;

   pa->x = p1.x + *mua * p21.x;
   pa->y = p1.y + *mua * p21.y;
   pa->z = p1.z + *mua * p21.z;
   pb->x = p3.x + *mub * p43.x;
   pb->y = p3.y + *mub * p43.y;
   pb->z = p3.z + *mub * p43.z;

   return(TRUE);
}