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## Laguerre formula

Absolute points have a surprising but important application: they can be used to determine the angle between two lines. To see how this works, let us assume that we have two lines and which intersect the ideal line at two points, say and . Then, the cross ratio between these two points and the two absolute points and yields the directed angle from the second line to the first:

which is known as the Laguerre formula.

To gain some intuition on why this formula is true, let us consider a simple example. Suppose we have two lines

in the affine plane. It is clear that these two lines can be represented as two vectors and in the Euclidean plane. The directed angle between the two lines is the directed angle between the two vectors and is given by:

Now in the projective plane these lines are represented as [a1, -1, 0]T and [a2, -1, 0]T, which are found by mapping points [x,y]T in the affine plane to points [x,y,1]T in the projective plane. The ideal line passing through and is given by . The two points of intersection between this line and the two original lines are given by [1, a1, 0]T and [1, a2, 0]T. The cross ratio of the four points is then given by:

Converting the complex numbers from rectangular to polar coordinates yields:

from which it follows that

which is the desired result.

Next: Projective Space Up: The Projective Plane Previous: Collineations
Stanley Birchfield
1998-04-23