Line-Line Intersection

The intersection of two lines and in two dimensions with, containing the points and , and containing the points and , is given by

			(1)
			(2)

where denotes a determinant. This corresponds to simultaneously solving

			(3)
			(4)

for and . Other treatments are given by Antonio (1992) and Hill (1994).

The intersections of two lines given in trilinear coordinates as

			(5)
			(6)

is

(7)

Pseudocode for segment intersection is given by de Berg et al. (2000).

Three lines in trilinear coordinates

			(8)
			(9)
			(10)

concur if their trilinear coordinates satisfy

(11)

in which case the point is

(12)

Three lines in Cartesian coordinates concur if the coefficients of the lines

			(13)
			(14)
			(15)

satisfy

(16)

In three dimensions, the algebra becomes more complicated. The intersection of two lines containing the points and , and and , respectively, can also be found directly by simultaneously solving

			(17)
			(18)

together with the condition that the four points be coplanar (i.e., the lines are not skew),

(19)

for , eliminating and . This set of equations can be solved for to yield

(20)

where

			(21)
			(22)
			(23)

(Hill 1994).

The point of intersection can then be immediately found by plugging back in for to obtain

(24)

A slightly more symmetrical and concise form can obtained by additionally defining

			(25)
			(26)
			(27)

where denotes a unit vector, then

(28)

(Goldman 1990).

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