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Over the last four decades GC has been developed as a universal geometric language for mathematics and physics. This can be regarded as the culmination of an R & D program innaugurated by Hermann Grassmann in 1844 [G44, H96]. Here, we draw on this rich source of concepts, tools and methods to enrich classical geometry by integrating it more fully into the whole system.
This chapter provides a synopsis of basic tools in Geometric Algebra to set the stage for further elaboration and applications in subsequent chapters. To make the synopsis compact, proofs are omitted. Geometric interpretation is emphasized, as it is essential for practical applications.
In classical geometry the primitive elements are points, and geometric objects are point sets with properties. The properties are of two main types: structural and transformational. Objects are characterized by structural relations and compared by transformations. In his Erlanger program, Felix Klein [K08] classified geometries by the transformations used to compare objects (for example, similarities, projectivities, affinities, etc). Geometric Algebra provides a unified algebraic framework for both kinds of properties and any kind of geometry.
Our "new model" has its origins in the work of F. A. Wachter (1792-1817), a student of Gauss. He showed that a certain type of surface in hyperbolic geometry known as a horosphere is metrically equivalent to Euclidean space, so it constitutes a non-Euclidean model of Euclidean geometry. Without knowledge of this model, the technique of comformal and projective splits needed to incorporate it into geometric algebra were developed by Hestenes in [H91]. The conformal split was developed to linearize the conformal group and simplify the connection to its spin representation. The projective split was developed to incorporate all the advantages of homogeneous coordinates in a "coordinate-free" representation of geometrical points by vectors.
Andraes Dress and Timothy Havel [DH93] recognized the relation of the conformal split to Wachter's model as well as to classical work on distance geometry by Menger [M31], Blumenthal [B53, 61] and Seidel [S52, 55]. They also stressed connections to classical invariant theory, for which the basics have been incorporated into geometric algebra in [HZ91] and [HS84]. The present work synthesizes all these developments and integrates conformal and projective splits into a powerful algebraic formalism for representing and manipulating geometric concepts. We demonstrate this power in an explicit construction of the new homogeneous model of E^n, the characterization of geometric objects therein, and in the proofs of geometric theorems.
The truly new thing about our model is the algebraic formalism in which it is embedded. This integrates the representational simplicity of synthetic geometry with the computational capabilities of analytic geometry. As in synthetic geometry we designate points by letters a, b, .... but we also give them algebraic properties. Thus, the outer product a \wedge b represents the line determined by a and b. This notion was invented by Hermann Grassmann [G1844] and applied to projective geometry, but it was incorporated into geometric algebra only recently [HZ91]. To this day, however, it has not been used in Euclidean geometry, owing to a subtle defect that is corrected by our homogeneous model. We show that in our model a \wedge b \wedge c represents the circle through the three points. If one of these points is a null vector e representing the point at infinity, then a \wedge b \wedge e represents the straight line through a and b as a circle through infinity. This representation was not available to Grassmann, because he did not have the concept of null vector.
Our model also solves another problem that perplexed Grassmann thoughout his life. He was finally forced to conclude that it is impossible to define a geometrically meaningful inner product between points. The solution eluded him because it requires the concept of indefinite metric that accompanies the concept of null vector. Our model supplies an inner product a \cdot b that directly represents the Euclidean distance between the points. This is a boon to distance geometry, because it greatly facilitates computation of distances among many points. Havel [H98] has used this in applications of geometric algebra to the theory of molecular conformations. The present work provides a framework for significantly advancing such applications.
We believe that our homogeneous model provides the first ideal framework for computational Euclidean geometry. The concepts and theorems of synthetic geometry can be translated into algebraic form without the unnecessary complexities of coordinates or matrices. Constructions and proofs can be done by direct computations, as needed for practical applications in computer vision and similar fields. The spin representation of conformal transformations greatly facilitates their composition and application. We aim to develop the basics and examples in sufficient detail to make applications in Euclidean geometry fairly straightforward. As a starting point, we presume familiarity with the notations and results of Chapter 1.
We have confined our analysis to Euclidean geometry, because it has the widest applicability. However, the algebraic and conceptual framework applies to geometrics of any signature. In particular, it applies to modeling spacetime geometry, but that is a matter for another time.
Although it is well known that the conformal groups of n-dimensional Euclidean and spherical spaces are isometric to each other, and are all isometric to the group of isometries of hyperbolic (n+1)-space [K1872], [K1873] spherical conformal geometry has its unique conformal transformations, and it can provide good understanding for hyperbolic conformal geometry. It is an indispensible part of the unification of all conformal geometries in the homogeneous model, which is addressed in the next chapter.
In the same paper Beltrami constructed two other Euclidean models of the hyperbolic plane, one on a disc and the other on a Euclidean half-plane. Both models are later generalized to n-dimensions by H. Poincare [P08], and are now associated with his name.
All three of the above models are built in Euclidean space, and the latter two are conformal in the sense that the metric is a point-to-point scaling of the Euclidean metric. In his 1878 paper [K1878], Killing described a hyperboloid model of hyperbolic geometry by constructing the stereographic projection of Beltrami's disc model onto the hyperbolic space. This hyperboloid model was generalized to n-dimensions by Poincare.
There is another model of hyperbolic geometry built in spherical space, called hemisphere model, which is also conformal. Altogether there are five well-known models for the n-dimensional hyperbolic geometry:
The theory of hyperbolic geometry can be built in a unified way within any of the models. With several models one can, so to speak, turn the object around and scrutinize it from different viewpoints. The connections among these models are largely established through stereographic projections. Because stereographic projections are conformal maps, the conformal groups of n-dimensional Euclidean, spherical, and hyperbolic spaces are isometric to each other, and are all isometric to the group of isometries of hyperbolic (n+1)-space, according to observations of Klein [K1872], [K1872]. It seems that everything is worked out for unified treatment of the three spaces. In this chapter we go further. We unify the three geometries, together with the stereographic projections, various models of hyperbolic geometry, in such a way that we need only one Minkowski space, where null vectors represent points or points at infinity in any of the three geometries and any of the models of hyperbolic space, where Minkowski subspaces represent spheres and hyperplanes in any of the three geometries, and where stereographic projections are simply rescaling of null vectors. We call this construction the homogeneous model. It serves as a sixth analytic model for hyperbolic geometry.
We constructed homogeneous models for Euclidean and spherical geometries in previous chapters. There the models are constructed in Minkowski space by projective splits with respect to a fixed vector of null or negative signature. Here we show that a projective split with respect to a fixed vector of positive signature produces the homogeneous model of hyperbolic geometry.
Because the three geometries are obtained by interpreting null vectors of the same Minkowski space differently, natural correspondences exist among geometric entities and constraints of these geometries. In particular, there are correspondences among theorems on conformal properties of the three geometries. Every algebraic identity can be interpreted in three ways and therefore represents three theorems. In the last section we illustrate this feature with an example.
The homogeneous model has the significant advantage of simplifying geometric computations, because it employs the powerful language of Geometric Algebra. Geometric Algebra was applied to hyperbolic geometry by H. Li in [L97], stimulated by Iversen's book [I92] on the algebraic treatment of hyperbolic geometry and by the paper of Hestenes and Ziegler [HZ91] on projective geometry with Geometric Algebra.
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