with five points. The homography group is a subgroup of the collineation group of the collineations of a projective space of dimension n. When the points and lines of the projective space are viewed as a block design, whose blocks are the sets of points contained in a line, it is common to call the collineation group the automorphism group of the design. Cross-ratio The cross-ratio of four collinear points is an invariant under the homography that is fundamental for the study of the homographies of the lines. Three distinct points , and on a projective line over a field form a projective frame of this line. There is therefore a unique homography of this line onto that maps to , to 0, and to 1. Given a fourth point on the same line, the cross-ratio of the four points , , and , denoted , is the element of . In other words, if has homogeneous coordinates over the projective frame , then . Over a ring Suppose A is a ring and U is its group of units. Homographies act on a projective line over A, written P(A), consisting of points with projective coordinates. The homographies on P(A) are described by matrix mappings When A is a commutative ring, the homography may be written but otherwise the linear fractional transformation is seen as an equivalence: The homography group of the ring of integers Z is modular group . Ring homographies have been used in quaternion analysis, and with dual quaternions to facilitate screw theory. The conformal group of spacetime can be represented with homographies where A is the composition algebra of biquaternions. Periodic homographies The homography is periodic when the ring is Z/nZ (the integers modulo 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, H. S. M. Coxeter gave this analysis: A real homography is involutory (of period 2) if and only if . If it is periodic with period , then it is elliptic, and no loss of generality occurs by assuming that . Since the characteristic roots are exp(±hπi/m), where , the trace is . See also W-curve Notes References , translated from the 1977 French original by M. Cole and S. Levy, fourth printing of the 1987 English translation Further reading Patrick du Val (1964) Homographies,