Projective line over a ring
Such matrices represent a normal subgroup N of V. The homographies of P1(A) correspond to elements of the quotient group V / N. P1(A) is considered an extension of the ring A since it contains a copy of A due to the embedding E : a → U[a, 1].
Homographies on P1(A) are called linear-fractional transformations since Rings that are fields are most familiar: The projective line over GF(2) has three elements: U[0, 1], U[1, 0], and U[1, 1].
The homography group on this projective line has 12 elements, also described with matrices or as permutations.
[1]: 31 For a finite field GF(q), the projective line is the Galois geometry PG(1, q).
Modular arithmetic will confirm that, in each table, a given letter represents multiple points.
The projective line over a division ring results in a single auxiliary point ∞ = U[1, 0].
[6]: 149–153 Similarly, if A is a local ring, then P1(A) is formed by adjoining points corresponding to the elements of the maximal ideal of A.
[6]: 174–200 [7] The projective line over M may be called the Minkowski plane when characterized by behaviour of hyperbolas under homographic mapping.
In the case of the ring of rational integers Z, the module summand definition of P1(Z) narrows attention to the U[m, n], m coprime to n, and sheds the embeddings that are a principal feature of P1(A) when A is topological.
The 1981 article by W. Benz, Hans-Joachim Samaga, & Helmut Scheaffer mentions the direct summand definition.
The projective line is the set of orbits under GL(2, R) of the free cyclic submodule R(1, 0) of R × R. Extending the commutative theory of Benz, the existence of a right or left multiplicative inverse of a ring element is related to P1(R) and GL(2, R).
The lemma refers to sufficient conditions for the existence of h. One application of cross ratio defines the projective harmonic conjugate of a triple a, b, c, as the element x satisfying (x, a, b, c) = −1.
Four points lie on a chain if and only if their cross-ratio is in F. Karl von Staudt exploited this property in his theory of "real strokes" [reeler Zug].
Karl Wilhelm Feuerbach and Julius Plücker are also credited with originating the use of homogeneous coordinates.
Eduard Study in 1898, and Élie Cartan in 1908, wrote articles on hypercomplex numbers for German and French Encyclopedias of Mathematics, respectively, where they use these arithmetics with linear fractional transformations in imitation of those of Möbius.
In 1902 Theodore Vahlen contributed a short but well-referenced paper exploring some linear fractional transformations of a Clifford algebra.
[15] The ring of dual numbers D gave Josef Grünwald opportunity to exhibit P1(D) in 1906.
[17] In 1968 Isaak Yaglom's Complex Numbers in Geometry appeared in English, translated from Russian.
Between the two editions, Walter Benz (1973) published his book,[7] which included the homogeneous coordinates taken from M.
