The book is the first of its kind; no other known work has attempted this kind of unified presentation of geometrical methods taken from a wide range of disciplines.
As well as stimulating further the revival of interest in geometry, it is intended as a first step towards the future development of an adequate science of morphology.
It contains over 200 diagrams and an extensive bibliography.
Author(s): E. A. Lord, C. B. Wilson
Series: Ellis Horwood Series: Mathematics and its applications
Publisher: John Wiley & Sons
Year: 1984
Language: English
Commentary: Fully bookmarked
Pages: 260
Tags: Mathematics; Geometry; Topology
Contents
1. Introduction
2. Geometry: An outline of its historical Development
- Congruence and the Euclidean group. Cartesian coordinates. Plane curves. Three dimensions. Space curves. Gaussian surface theory. Gaussian and mean curvature. Curvilinear coordinates and non-Euclidean geometry. Projective geometry. Klein's Erlangen program. Transformations.
3. Topological Spaces
- Homeomorphisms. Polygonal symbols. Classification of surfaces.
4. Mappings
- Submersion and immersion. Description of mappings in terms of coordinates. The design of boat hulls. Map projections. Illumination engineering. Deformations in Euclidean space. Conformal mappings in Euclidean space. Infinitesimal deformations. Time-dependent deformations.
5. Singularities of Analytic mappings
- Curves in two dimensions. Space curves. Scalar field in two dimensions. Mappings from two dimensions to two dimensions. Surfaces in three-space. Critical points of algebraic curves and surfaces. Envelopes. Removal of critical points and multiple points. Blum's medial axis description. Thorn's catastrophe theory. Vector field in two dimensions. Vector field in three dimensions. Boundaries in fluid flow.
6. Mappings that are not analytic
- Isolated exceptional points. Branch points. Lines of exceptional points. Edges and corners. Polyhedra.
7. Minimising Principles
- Deformation of a curve. Fermat's principle. The catenary. Flexible bars and river meanders. Minimal surfaces.
8. Generation of Shapes
- Shape grammars. Ulam's modular patterns. Trees and river systems. Symmetry. Regular patterns in a plane. Crystallography. Tessellations and space filling. Spiral phyllotaxis. Spiral forms in three dimensions. Generation of surfaces.
9. Discrete Spaces
- Finite spaces. Graph theory. Planar graphs. Euler's formula.
10. Fitting of Curves and Surfaces
- Interpolation. Bivariate interpolation. Coons' surface patches. Approximate methods of curve fitting. Smoothing. Surface modelling. Linear operators.
11. The discretised euclidean plane
- Scalar functions on a square lattice. Finite difference methods. Binary patterns. Lattice description of curves.
12. Fourier Methods
- The Fourier series. Fourier series in two dimensions. Walsh functions. The Fourier integral. Bandlimiting. The discretised Fourier transform.
Bibliography and References
Index
