This note describes the technique and gives the solution to finding the shortest distance from a point to a line or line segment. The equation of a line defined through two points P1 (x1,y1) and P2 (x2,y2) is
The point P3 (x3,y3) is closest to the line at the tangent to the line which passes through P3, that is, the dot product of the tangent and line is 0, thus
Substituting the equation of the line gives
Solving this gives the value of u
Substituting this into the equation of the line gives the point of intersection (x,y) of the tangent as
The distance therefore between the point P3 and the line is the distance between (x,y) above and P3.
Notes