Intersection point of two lines
(2 dimensions)

Written by Paul Bourke
April 1989

Sample source code
Original C code by Paul Bourke.
C++ contribution by Damian Coventry.
LISP implementation by Paul Reiners.
C version for Rockbox firmware by Karl Kurbjun.
C# version by Olaf Rabbachin.
VB.net version by Olaf Rabbachin.

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 u_a and u_b 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)

Intersection point of two lines
(2 dimensions)