Linear Figures
An important projective invariant, the cross-ratio of four collinear points, is defined. Some special configurations in two, three and four dimensions are introduced, including extensions of Desargues’ theorem, Sylvester’s ‘duads and synthemes’, desmic sy
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Linear Figures
Abstract An important projective invariant, the cross-ratio of four collinear points, is defined. Some special configurations in two, three and four dimensions are introduced, including extensions of Desargues’ theorem, Sylvester’s ‘duads and synthemes’, desmic systems of tetrahedra, and Baker’s remarkable configuration derived from six points in a complex four-dimensional space.
3.1 The Projective Line Even one-dimensional projective spaces are not trivial. By analogy from what we have said above about homogeneous coordinates, it is clear that a point in a onedimensional projective space can be assigned a pair of homogeneous coordinates (X Y ). Since all the coordinate pairs λ(X Y ) = (λX λY ), for any non-zero λ, refer to the same point, the ratio θ = X/Y is sufficient to identify a point. This is a homographic parameter for the points on the line. Projective transformations on a line are called homographies. We must include the point θ = ∞, corresponding to Y = 0, as the parameter of a valid point—otherwise, we would have an affine line rather than a projective line. (Alternatively, by analogy with the assignment of homogeneous coordinates for the lines in a projective plane, we could label points by ‘dual’ coordinates [l m] = [Y −X]. The condition for two points to coincide is then lX + mY = 0.) Just as the group of projective transformations on the real projective plane is SL(3, R), the group of projective transformation on the real projective line is the group SL(2, R) of all real unimodular 2 × 2 matrices: a b T= , c d satisfying |T| = 1. In terms of the homographic parameter, these transformations are θ → θ =
aθ + b , cθ + d
ad − bc = 1
(including ∞ → a/c and −d/c → ∞). Note that any three points A, B and C on a projective line can be chosen to be the reference points (10), (01) and (11); that is, E. Lord, Symmetry and Pattern in Projective Geometry, DOI 10.1007/978-1-4471-4631-5_3, © Springer-Verlag London 2013
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Linear Figures
Fig. 3.1 Reference points on a projective line
they can be assigned the homographic parameters ∞, 0 and 1. This is just a lowerdimensional analogue of what we showed in Fig. 2.1: it is a matter of establishing homogeneous coordinates on the line by choosing the axes and the point (11) in an affine plane, as in Fig. 3.1. A non-trivial homography leaves two points of the line fixed. (The ‘trivial’ homography is θ = θ , which leaves every point on the line fixed.) As already noted, a homography maps ∞ to a/c, and so, if c = 0, θ = ∞ is a fixed point and the other fixed point is θ = b/(d − a). If c = 0 the fixed points are given by the roots of cθ 2 + (d − a)θ − b = 0. They may be real and distinct, real and coincident, or conjugate complex, according to whether the discriminant (d − a)2 − 4bc = (d − a)2 − 4 is positive, zero or negative. The fixed points of a homography θ → θ = aθ+b cθ+d are real and distinct, real and coincident, or conjugate complex, according to whether the eigen values of T = ac db are real and distinct, real and coi
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