Intersection point of two lines
(2 dimensions)

Written by Paul Bourke
April 1989

Sample source code
Contribution by Damian Coventry: example.cpp

This note describes the technique and algorithm for determining the intersection point of two lines (or line segments) in 2 dimensions.

The equations of the lines are

Pa = P1 + ua ( P2 - P1 )

Pb = P3 + ub ( P4 - P3 )

Solving for the point where Pa = Pb gives the following two equations in two unknowns (ua and ub)

x1 + ua (x2 - x1) = x3 + ub (x4 - x3)

and
y1 + ua (y2 - y1) = y3 + ub (y4 - y3)

Solving gives the following expressions for ua and ub

Substituting either of these into the corresponding equation for the line gives the intersection point. For example the intersection point (x,y) is

x = x1 + ua (x2 - x1)

y = y1 + ua (y2 - y1)

Notes:

  • The denominators for the equations for ua and ub are the same.

  • If the denominator for the equations for ua and ub is 0 then the two lines are parallel.

  • If the denominator and numerator for the equations for ua and ub are 0 then the two lines are coincident.

  • The equations apply to lines, if the intersection of line segments is required then it is only necessary to test if ua and ub lie between 0 and 1. Whichever one lies within that range then the corresponding line segment contains the intersection point. If both lie within the range of 0 to 1 then the intersection point is within both line segments.