Minimum Distance between a Point and a Line
Written by Paul Bourke October 1988
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
P = P1 + u ( P2 - P1 )
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
( P3 - P ) dot ( P2 - P1 ) = 0
Substituting the equation of the line gives
[ P3 - P1 - u( P2 - P1 )] dot ( P2 - P1 ) = 0
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
x = x1 + u (x2 - x1) y = y1 + u (y2 - y1)
The distance therefore between the point P3 and the line is the distance between (x,y) above and P3 .
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