Intersection point of two lines
(2 dimensions)

Written by Paul Bourke
April 1989

Sample source code
C++ contribution by Damian Coventry.
LISP implementation by Paul Reiners.
C version for Rockbox firmware by Karl Kurbjun.

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

P_a = P1 + u_a ( P2 - P1 )

P_b = P3 + u_b ( P4 - P3 )

Solving for the point where P_a = P_b gives the following two equations in two unknowns (u_a and u_b)

x1 + u_a (x2 - x1) = x3 + u_b (x4 - x3)
and
y1 + u_a (y2 - y1) = y3 + u_b (y4 - y3)

Solving gives the following expressions for u_a and u_b

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 + u_a (x2 - x1)

y = y1 + u_a (y2 - y1)

Notes:

The denominators for the equations for u_a and u_b are the same.
If the denominator for the equations for u_a and u_b is 0 then the two lines are parallel.
If the denominator and numerator for the equations for u_a and u_b 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.

Intersection point of two lines(2 dimensions)