Computer Graphics and Geometric Modeling
This volume begins with material from linear algebra and a discussion of the transformations in affine & projective geometry, followed by topics from advanced calculus & chapters on general topology, combinatorial topology, algebraic topology, differential topology, differential geometry, and finally algebraic geometry. Two important goals throughout were to explain the material thoroughly, and to make it self-contained.
This volume by itself would make a good mathematics reference book, in particular for practitioners in the field of geometric modelling. Due to its broad coverage and emphasis on explanation it could be used as a text for introductory mathematics courses on some of the covered topics, such as topology (general, combinatorial, algebraic, and differential) and geometry (differential & algebraic).
Computer Graphics and Geometric Modeling
Algebraic Topology (p. 358)
The central problem of algebraic topology is to classify spaces up to homeomorphism by means of computable algebraic invariants. In the last chapter we showed how two invariants, namely, the Euler characteristic and orientability, gave a complete classi- .cation of surfaces. Unfortunately, these invariants are quite inadequate to classify higher-dimensional spaces. However, they are simple examples of the much more general invariants that we shall discuss in this chapter.
The heart of this chapter is its introduction to homology theory. Section 7.2.1 de.nes the homology groups for simplicial complexes and polyhedra, and Section 7.2.2 shows how continuous maps induce homomorphisms of these groups. Section 7.2.3 describes a few immediate applications. In Section 7.2.4 we indicate how homology theory can be extended to cell complexes and how this can greatly simplify some computations dealing with homology groups. Along the way we de.ne CW complexes, which are really the spaces of choice in algebraic topology because one can get the most convenient description of a space with them. Section 7.2.5 de.nes the incidence matrices for simplicial complexes.
These are a fundamental tool for computing homology groups with a computer. Section 7.2.6 describes a useful extension of homology groups where one uses an arbitrary coef.cient group, in particular, Z2. After this overview of homology theory we move on to de.ne cohomology in Section 7.3. The cohomology groups are a kind of dual to the homology groups.
We then come to the other major classical topic in algebraic topology, namely, homotopy theory. We start in Sections 7.4.1 and 7.4.2 with a discussion of the fundamental group of a topological space and covering spaces. These topics have their roots in complex analysis. Section 7.4.3 sketches the de.nition of the higher-dimensional homotopy groups and concludes with some major theorems from homotopy theory. Section 7.5 is devoted to pseudomanifolds, the degree of a map, manifolds, and Poincaré duality (probably the single most important algebraic property of manifolds and the property that sets manifolds apart from other spaces).
We wrap up our overview of algebraic topology in Section 7.6 by telling the reader brie.y about important aspects that we did not have time for and indicate further topics to pursue. Finally, as one last example, Section 7.7 applies the theory developed in this chapter to our ever-interesting space Pn.
The reader is warned that this chapter may be especially hard going if he/she has not previously studied some abstract algebra. We shall not be using any really advanced ideas from abstract algebra, but if the reader is new to it and has no one for a guide, then, as usual, it will take a certain amount of time to get accustomed to thinking along these lines. Groups and homomorphism are quite a bit different from topics in calculus and basic linear algebra.
The author hopes the reader will persevere because in the end one will be rewarded with some beautiful theories. The next chapter will make essential use of what is developed here and apply it to the study of manifolds. Manifolds are the natural spaces for geometric modeling and getting an understanding of our universe.