distance from point to a line

The distanceMathworldPlanetmath from a point P with coordinates (xp,yp)2 to the line with equation ax+by+c=0 is given by |axp+byp+c|/a2+b2.

Proof Every point x,y on the line is at some distance (x-xp)2+(y-yp)2 from P. What we need to find is the minimum such distance. Our problem is


subject to


This problem is solvable using the Lagrange multiplier method. We minimize


Calculating the derivatives with respect to x,y and λ and setting them to zero we get three equations:

2x-2xp+λa=0 (1)
2y-2yp+λb=0 (2)
2ax+2by+2c=0 (3)

Solving these leads to xp-x=aaxp+byp+ca2+b2 and yp-y=baxp+byp+ca2+b2. We can now substitute these expressions into (x-xp)2+(y-yp)2 and we get (after some simplification) the desired result.

Title distance from point to a line
Canonical name DistanceFromPointToALine
Date of creation 2013-03-22 15:24:30
Last modified on 2013-03-22 15:24:30
Owner acastaldo (8031)
Last modified by acastaldo (8031)
Numerical id 7
Author acastaldo (8031)
Entry type Result
Classification msc 51N20
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