Inversive ring geometry

In mathematics, inversive ring geometry is the extension to the context of associative rings, of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio, concepts usually built upon rings that happen to be fields.

One begins with ordered pairs ("a", "b") in "A"×"A" where "A" is an (associative) ring with 1. Let "U" be the group of units of the ring. When there is "g" in "U" such that

:("ag", "bg") = ("u", "v"),

then we write

:("u", "v") ~ ("a", "b").

In other words, we identify orbits under the action of "U", and ~ is the corresponding equivalence relation.

Two elements of a ring are "relatively prime" if the ideal in "A" that they generate is the whole of "A". The projective line over A is the set of equivalence classes for ~ on pairs of relatively prime elements :

:"P"("A") = { "U"("a", "b") ∈ "A" × "A" / ~ : "A a" + "A b" = "A" }.

Examples with topological descriptions (≈ denotes homeomorphism):

*"A" = "C" complex plane : "P"("C") ≈ "S"² = Riemann sphere
*"A" = "H" quaternion ring : "P"("H") ≈ "S"⁴ = One-point compactification
*"A" = "D" dual number plane : "P"("D") = "D" ∪ { "U"(1, "x n"): "x" ∈ "R"}, "nn" = 0
*"A" = "M" split-complex plane : "P"("M") ≈ hyperboloid of one sheet. This description appeared in Russian in 1969 (Yaglom), in German in 1973 (Benz), and English in 1979 (Shenitzer translates Yaglom).

Affine and projective groups

The affine group on "A" is generated by the mappings "x" → "x" + "c" and "x" → "x u", "u" ∈ "U".

The group of projectivities on "P"("A") extends the affine group by including reciprocation "x" → "x"⁻¹ as follows:

Represent translations by "U"("x", 1) = "U"("x" + "c", 1).

Represent "rotations" by "U"("x", 1) = "U"("x u", 1).

Include reciprocation with "U"("x", "y") = "U"("y", "x").

Note that if "u" ∈ "U", then "U"(1, "u") = "U"("u"⁻¹, 1) = "U"("u", 1).

Composition of mappings is represented by matrix multiplication where the matrices are of the 2 × 2 type exhibited with entries taken from the ring "A". Call the set of them "M"("A", 2) so the group of projectivities "G"("A") ⊂ "M"("A", 2).For instance, in "G"("A") one finds the projectivity

:

Its action is "U"("x", 1) = "U"("xu", "u") = "U"("u"⁻¹ "xu", 1).

Thus the inner automorphism "x" → "u"⁻¹ "x u" of the group of units "U" ⊂ "A" arises as a projectivity on "P"("A") by an element of "G"("A"). For example, when "A" is the ring of quaternions then one obtains rotations of 3-space.In case "A" is the ring of biquaternions, the mappings include both the ordinary and hyperbolic rotations of the Lorentz group.

Cross-ratio theorems

Here we consider existence, uniqueness, matching triples, and invariance.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". Now look at

:

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 projectivity considering the pivotal use of fixed points of generators to bring the triple to 0,1,∞.

If "s" and "t" are two sufficiently separated triples then they correspond to projectivities "g" and "h" respectively which map each of "s" and "t" to (0,1,∞). Thus the projectivity "h"⁻¹ o "g" maps "s" to "t" .

Denote by ("x","p","q","r") the image of "x" under the projectivity 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 implies that when a single projectivity 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
("g"("x"), "g"("p"), "g"("q"), "g"("r") ) = ("x", "p", "q", "r").

Historical notes

August Ferdinand Möbius investigated the Möbius transformations between his book "Baricentric Calculus"(1827) and his 1855 paper "Theorie der Kreisverwandtschaft in rein geometrischer Darstellung". Karl Wilhelm Feuerbach and Julius Plucker are also credited with originating the use of homogeneous coordinates. Eduard Study in 1898, and Elie Cartan in 1908, wrote articles on hypercomplex numbers for German and French "Encyclopedias of Mathematics", respectively. These articles also suggested the functor "A" → "G"("A") developed above, but in their era one lacked the concepts of the category of rings and the benefits of the rigor in equivalence relations was not yet appreciated, so the attempts of Study and Cartan were premature. The ring of dual numbers "D" gave Josef Grunwald opportunity to exhibit "P"("D") in 1906. (Über duale Zahlen und ihre Anwendung in der Geometrie", Monatshefte fur Mathematik17,81-136.) In 1947 the construction was carried out on H by P.G. Gormley, "Stereographic projection and the linear fractional group of transformations of quaternions"(Proceedings of the Royal Irish Academy, Section A 51, 67-85). In 1968 I.M. Yaglom's "Complex Numbers in Geometry" appeared in English, translated from Russian, wherein he uses "P"("D") to describe line geometry in the Euclidean plane and "P"("M") to describe it for Lobachevski's plane. Yaglom's text "A Simple Non-Euclidean Geometry" appeared in English in 1979. There in pages 174 to 200 he develops "Minkowskian geometry" and describes "P"("M") as the "inversive Minkowski plane". The Russian original of Yaglom's text was published in 1969. Between the two editions, Walter Benz (1973) published "Vorlesungen über Geometrie der Algebren" which included the homogeneous coordinates taken from "M".