Find the Distance from a 3d Point to a Line Segment

Question

I have a 3d point P and a line segment defined by A and B (A is the start point of the line segment, B the end).

I want to calculate the shortest distance between P and the line AB.

Calculating the distance of a point to an infinite line was easy as their was a solution on Wolfram Mathworld, and I have implemented that, but I need to do this for a line of finite length.

I have not managed to find a reliable solution for this in 3d after a lot of looking around.

I have implemented algorithms to calculate the dot product, cross product, magnitude and so on in C++ with a struct that contains floats x, y and z.

Pseudo code, links, or code in pretty much any language for this would be great.

Answer 1

Java function

/**
 * Calculates the euclidean distance from a point to a line segment.
 *
 * @param v     the point
 * @param a     start of line segment
 * @param b     end of line segment 
 * @return      distance from v to line segment [a,b]
 *
 * @author      Afonso Santos
 */
 public static
 double
 distanceToSegment( final R3 v, final R3 a, final R3 b )
 {
   final R3 ab  = b.sub( a ) ;
   final R3 av  = v.sub( a ) ;

   if (av.dot(ab) <= 0.0)           // Point is lagging behind start of the segment, so perpendicular distance is not viable.
     return av.modulus( ) ;         // Use distance to start of segment instead.

   final R3 bv  = v.sub( b ) ;

   if (bv.dot(ab) >= 0.0)           // Point is advanced past the end of the segment, so perpendicular distance is not viable.
     return bv.modulus( ) ;         // Use distance to end of the segment instead.

   return (ab.cross( av )).modulus() / ab.modulus() ;       // Perpendicular distance of point to segment.
}

gist of the whole (self contained) R3 3D algebra package: https://gist.github.com/reciprocum/4e3599a9563ec83ba2a63f5a6cdd39eb

part of the open source library https://sourceforge.net/projects/geokarambola/

Answer 2

This is fairly straight forward. First, treat your line segment as if it were an infinite and find the point R on the line where a perpendicular ray off the line at R passes through your point P. If R is between A and B on the line, then the shortest distance is PR. Otherwise, shorestest distance is lessor of PA and PB.