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Classical geometry has emerged from efforts to codify perception of space and motion. With roots in ancient times, the great flowering of classical geometry was in the 19th century, when Euclidean, non-Euclidean and projective geometries were given precise mathematical formulations and the rich properties of geometric objects were explored. Though fundamental ideas of classical geometry are permanently imbedded and broadly applied in mathematics and physics, the subject itself has practically disappeared from the modern mathematics curriculum. Many of its results have been consigned to the seldom-visited museum of mathematics history, in part, because they are expressed in splintered and arcane language. To make them readily accessible and useful, they need to be reexamined and integrated into a coherent mathematical system. Classical geometry has been making a comeback recently because it is useful in such fields as Computer-Aided Geometric Design (CAGD), CAD/CAM, computer graphics, computer vision and robotics. In all these fields there is a premium on computational efficiency in designing and manipulating geometric objects. Our purpose here is to introduce powerful new mathematical tools for meeting that objective and developing new insights within a unified algebraic framework. In this and subsequent chapters we show how classical geometry fits neatly into the broader mathematical system of

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.

The standard algebraic model for Euclidean space

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.

The recorded study of spheres dates back to the first century in the book

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.

The study of relations among Euclidean, spherical and hyperbolic geometries dates back to the beginning of last century. The attempt to prove Euclid's fifth postulate led C. F. Gauss to discover hyperbolic geometry in the 1820's. Only a few years passed before this geometry was rediscovered independently by N. Lobachevski (1829) and J. Bolyai (1832). The strongest evidence given by the founders for its consistency is the duality between hyperbolic and spherical trigonometries. This duality was first demonstrated by Lambert in his 1770 memoir [L1770]. Some theorems, for example the law of sines, can be stated in a form that is valid in spherical, Euclidean, and hyperbolic geometries [B1832]. To prove the consistency of hyperbolic geometry, people built various analytic models of hyperbolic geometry on the Euclidean plane. E. Beltrami [B1868] constructed a Euclidean model of the hyperbolic plane, and using differential geometry, showed that his model satisfies all the axioms of hyperbolic plane geometry. In 1871, F. Klein gave an interpretation of Beltrami's model in terms of projective geometry. Because of Klein's interpretation, Beltrami's model is later called Klein's disc model of the hyperbolic plane. The generalization of this model to

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 half-space model,
- the conformal ball model,
- the Klein ball model,
- the hemisphere model,
- the hyperboloid model.

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.

D. Hestenes, in E. Bayro-Corrochano and G. Scheuermann (eds.),

Conformal Geometric Algebraic (CGA) provides ideal mathematical tools for construction, analysis and integration of classical Euclidean, Inversive & Projective Geometries, with practical applications to computer science, engineering and physics. This paper is a comprehensive introduction to a CGA tool kit. Synthetic statements in classical geometry translate directly to coordinate-free algebraic forms. Invariant and covariant methods are coordinated by conformal splits, which are readily related to the literature using methods of matrix algebra, biquaternions and screw theory. Designs for a complete system of powerful tools for the mechanics of linked rigid bodies are presented.

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