The intersection of two lines and in two dimensions with, containing the points and , and containing the points and , is given by
(1)
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(2)
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where denotes a determinant. This corresponds to simultaneously solving
(3)
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(4)
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for and . Other treatments are given by Antonio (1992) and Hill (1994).
The intersections of two lines given in trilinear coordinates as
(5)
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(6)
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is
(7)
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Pseudocode for segment intersection is given by de Berg et al. (2000).
Three lines in trilinear coordinates
(8)
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(9)
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(10)
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concur if their trilinear coordinates satisfy
(11)
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in which case the point is
(12)
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Three lines in Cartesian coordinates concur if the coefficients of the lines
(13)
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(14)
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(15)
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satisfy
(16)
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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)
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(18)
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together with the condition that the four points be coplanar (i.e., the lines are not skew),
(19)
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for , eliminating and . This set of equations can be solved for to yield
(20)
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where
(21)
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(22)
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(23)
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(Hill 1994).
The point of intersection can then be immediately found by plugging back in for to obtain
(24)
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A slightly more symmetrical and concise form can obtained by additionally defining
(25)
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(26)
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(27)
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where denotes a unit vector, then
(28)
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(Goldman 1990).