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Collineations

A collineation of is defined as a mapping from the plane to itself such that the collinearity of any set of points is preserved. Such a mapping can be achieved with matrix multiplication by a matrix T. Each point is transformed into a point :

We will use the terms transformation and collineation interchangeably. Since scaling is unimportant, only eight elements of T are independent. Therefore, since each point contains two independent values, four pairs of corresponding points are necessary to determine T.

To transform a line into a line , we note that collinearity must be preserved, that is, if a point lies on the line , then the corresponding point must lie on the corresponding line . Therefore,

which indicates that

From these results, it is not hard to show that a point conic C transforms to T-TCT-1, and a line conic |C|C-1 transforms to T|C|C-1TT.

Regarding transformations, recall that projective affine similarity Euclidean. Let's study the matrix T to uncover the relationships between these various geometries. First we will write out the elements of T, for reference:

The affine plane is just the projective plane minus the ideal line. Therefore, affine transformations must preserve the ideal line and the ideal points, that is, any point [X, Y, 0]T must be transformed into for some arbitrary scaling :

which implies that t31 = t32 = 0. The matrix for affine transformation, then, is

where once again only six of these parameters are independent, since scale is unimportant.

Unlike affine transformations, similarity transformations preserve angles and ratios of lengths. Delaying the derivation for a moment, we simply state the result:

 (3)

where is an arbitrary angle.

Under Euclidean transformation, scale is important, and therefore the point must first be converted to Euclidean coordinates by dividing by its third element. The transformation then is

In closing this section, we offer one final proposition, along with its proof:

Proposition 1   A transformation is a similarity transformation if and only if it preserves the absolute points, .

The only if'' is rather easy to see: The absolute point is transformed through equation (3) to the point , which is equivalent because the scale factor is ignored.

The if'' is a little more complicated, but still rather straightforward. Starting with the unrestricted equation for T,

the fact that [1, i, 0]T is preserved yields the following two equations:

Since the elements of T are constrained to be real, this leads to the following three constraints on the elements of T:

So the matrix of T looks like this:

Given two arbitrary numbers t11 and t12, we can always reparameterize them as and , where is an angle and k is a scalar. Multiplying the previous equation by 1/k (which is legal because we are working in homogeneous coordinates), we then get

which is seen to be the equation of a similarity transformation when compared with equation (3). (NOTE: Using the point [1,-i,0]T yields the same result.)

Next: Laguerre formula Up: The Projective Plane Previous: Absolute points
Stanley Birchfield
1998-04-23