Geometric Transformations for 3D Modelling

Written from a mathematical standpoint accessible to students, teachers, and professionals studying or practicing in engineering, mathematics, or physics, the new second edition is a comprehensive introduction to the theory and application of transformations. Presenting the more abstract foundation material in the first three chapters, Geometric Transformations in 3D Modeling reduces the clutter of theoretical derivation and development in the remainder of the text and introduces the operational and more application-oriented tools and concepts as the need arises. It assumes the reader has already taken analytic geometry and first-year calculus and has a working knowledge of basic matrix and vector algebra. This self-contained resource is sure to appeal to those working in 3D modeling, geometric modeling, computer graphics, animation, robotics, and kinematics.  Explores and develops the subject in much greater breadth and depth than other books, offering readers a better understanding of transformation theory, the role of invariants, the uses of various notation systems, and the relations between transformations.Describes how geometric objects may change position, orientation, or even shape when subjected to mathematical operations, while properties characterizing their geometric identity and integrity remain unchanged.Presents eigenvalues, eigenvectors, and tensors in a way that makes it easier for readers to understand.Contains revised and improved figures, with many in color to highlight important features.Provides exercises throughout nearly all of the chapters whose answers are found at the end of the book.GeometryWhat Is Geometry History Geometric ObjectsSpace Geometry Is… E Pluribus Unum – Transformation and Invariance Theory of TransformationsFunctions, Mappings, and Transformations Linear TransformationsGeometric Invariants Isometries Similarities Affinities Projectivities Topological Transformations Vector SpacesIntroduction to Linear Vector Spaces Basis Vectors Eigenvalues and Eigenvectors TensorsRigid-Body MotionTranslation Rotation Composite Motion Kinematics Reflection and SymmetryCentral Inversion Reflections in the Plane Reflections in Space Summary of Reflection Matrices Symmetry Basics Symmetry Groups Ornamental Groups Polygonal Symmetry and TilingPolyhedral Symmetry More Linear Transformations Isotropic Dilation Anisotropic Dilation Shear Projective Geometry Parallel Projection Central Projection Map Projections Display Projection Nonlinear TransformationsLinear and Nonlinear Equations Inversion in a Circle Curvilinear Coordinate SystemsDeformations Answers to Selected ExercisesIndex