In the case of the complex projective line, also called Riemann sphere, the homographies are called Möbius transformations. They are mappings of the plane written as linear fractional transformations:
For more general projective spaces – of different dimensions or over different fields – "collineation" (meaning "maps lines to lines") is a more general notion, which includes both homographies and automorphic collineations (collineations induced by a field automorphism), as well as combinations of these.
The homography is periodic when the ring is Z(mod n) since then Arthur Cayley was interested in periodicity when he calculated iterates in 1879. In his review of a brute force approach to periodicity of homographies, Harold Coxeter gave this analysis:
- A real homography is involutory (of period 2) if and only if a + d = 0. If it is periodic with period n > 2, then it is elliptic, and no loss of generality occurs by assuming that ad − bc = 1. Since the characteristic roots are exp(± h π i/m), where (h,m) = 1, the trace is a + d = 2 cos(h π/m).
As every homography has a inverse mapping and the compostion of two homographies is another, the homographies on a given projective line form a group. The Mobius group is a well-known example. Another example is PSL(2,7) which is the homography group on the projective line over the Galois field with seven elements. Even the ring of integers has a projective line, and its homography group PSL(2,Z) is the modular group.
Michel Chasles advanced projective geometry by exploiting cross-ratios, something he also identified in work of Pappus of Alexandria. On the other hand, Karl von Staudt developed geometric relations on the basis of projective harmonic conjugates which are featured on a projective range. More recently cross-ratio has been seen as a certain homography determined by three points in a ring: Suppose h is a homography that takes a, b, c in A to U(0,1), U(1,1), U(1,0). Then the cross-ratio (w, a, b, c) = h(w). In terms of ring operations, the homography may be written as the linear fractional transformation
This transformation is composed of two elementary homographies:
- The separating homography that sends U(a,1) to U(0,1) and U(c,1) to ∞ = U(1,0), and
- The normalizing homography where which sends U(b,1) to U(1,1) and has U(0,1) and U(1,0) as fixed points.
In general, the fraction e may not be available in the ring. Then the construction of a homography may be approached as follows: Suppose p, q, r ∈ A with
- t = (r – p)−1 and v = (t + (q – r)−1)−1.
When these inverses t and v exist we say "p, q, and r are separated sufficiently". Up to sufficient separation, the group of homographies is 3-transitive:
The first two factors put r at U(1, 0) = ∞ where it stays. The third factor moves t, the image of p under the first two factors, to U(0, 1), or zero in the canonical embedding. Finally, the fourth factor has traced q through the first three factors and formation of the rotation with v places U(q, 1) at U(1, 1). Thus the composition displayed places the triple p,q,r at the triple 0,1,∞. Evidently it is the unique such homography considering the pivotal use of fixed points of generators to bring the triple to 0,1,∞.
Proposition: If the group of homographies is sharply 3-transitive, then there is a cross-ratio function which is invariant under the permutation of the projective line by homographies.
- proof: If s and t are two sufficiently separated triples then they correspond to homographies g and h respectively which map each of s and t to (0,1,∞). Thus the homography h−1 o g maps s to t .
- Denote by (x,p,q,r) the image of x under the homography determined by p,q,r as above. This function f(x) is the cross-ratio determined by p,q,r ∈ A. The uniqueness of this function (sharp transitivity) implies that when a single homography g ∈ G(A) is used to form another triple g(p), g(q), g(r) from the first one, then the new cross-ratio function h must agree with f o g. Hence h o g−1 = f so that
As the sharpness does not hold in non-commutative rings like quaternions and biquaternions, there are limits to usage of cross-ratios.
Thus an anti-homography is the composition of conjugation with an ordinary homography. For example, geometrically, the mapping amounts to circle inversion. The transformations of inversive geometry of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.
- Arthur Cayley (1879) "On the matrix and its connection with the function ", Messenger of Mathematics 9:104
- H.S.M. Coxeter, On periodicity in Mathematical Reviews
- Walter Benz, Hans-Joachim Samaga & Helmut Schaeffer (1981) "Cross Ratios and a Unifying Treatment of Von Staudt’s Notion of Reeler Zug", page 130 in Geometry – Von Staudt’s Point of View, P. Plaumann & K. Strambach editors, D. Reidel
- Patrick du Val (1964) Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, MR0169108 .
- Gunter Ewald (1971) Geometry: An Introduction, page 263, Belmont:Wadsworth Publishing ISBN 0-534-00034-7 .
- Bill Goldman (2005) Transformations in Circle Geometry, course notes from University of Maryland.
- Frank Morley and F.V. Morley (1933) Inversive Geometry, page 38, London: G. Bell and Sons.