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Stressing the interplay between theory and its practice, this text presents the construction of linear models that satisfy geometric postulate systems and develops geometric topics in computer graphics. It includes a computer graphics utility library of specialized subroutines on a 3.5 disk, designed for use with Turbo PASCAL 4.0 (or later version) - an effective means of computer-aided instruction for writing graphics problems.;Providing instructors with maximum flexibility that allows for the mathematics or computer graphics sections to be taught independently, this book: reviews linear algebra and notation, focusing on ideas of geometric significance that are often omitted in general purpose linear algebra courses; develops symmetric bilinear forms through classical results, including the inertia theorem, Witt's cancellation theorem and the unitary diagonalization of symmetric matrices; examines the Klein Erlanger programm, constructing models of geometries, and studying associated transformation groups; clarifies how to construct geometries from groups, encompassing topological notions; and introduces topics in computer graphics, including geometric modeling, surface rendering and transformation groups.
Part 1 Preliminaries: fields; vector spaces; linear transformations; cosets of a vector space; invariant subspaces. Part 2 Symmetric bilinear forms: symmetric bilinear forms; congruence; orthogonal complements; orthogonal bases; Witt's cancellation theorem; isotropic and anisotropic spaces; functions on inner product spaces. Part 3 Plane geometries: the affine plane; the affine group; postulates for the Euclidean plane; inner product planes; projective planes; conic sections. Part 4 Homogeneous spaces in Rn: topological groups; homogeneous spaces; geometry on homogeneous spaces; the Riemann sphere; the Poincare upper half-plane; differentiable manifolds. Part 5 Topics in computer graphics: a first graphics programme; a computer graphics system overview; geometric mappings in a CG system; the line-drawing algorithm; the wing-edge object representation; the conic sections; Bezier curves and B-splines; hidden surface removal; texture mapping; quadric intermediate surfaces; Koch systems. Appendices: equivalence relations - basics; the Jordan canonical form - proof of Jordan's theorem; GraphLib documentation - types, procedures and functions.