Geometric Transformations for 3D Modeling
Written from a mathematical standpoint accessible to students, teachers, and professionals, this text provides a comprehensive introduction to the theory and application of transformations.
TABLE OF CONTENTS Geometric Transformations for 3D Modeling
Chapter 7: Nonlinear Transformations
This chapter investigates transformations that are given by nonlinear functions. We have already discussed some examples of this, including certain map projections and general topological transformations in earlier chapters. Following a brief introductory section, we will study inversion in a circle, parametric mappings, and deformation transformations.
7.1 Linear and Nonlinear Equations
Comparing linear and nonlinear equations, reciprocal radii transformation
Linear and nonlinear transformations are very different conceptually. For linear transformations, the input conditions are independent of the output state and it is possible to superimpose various input conditions to produce an output equivalent to the sum of the separate outputs. For example, rotation is a linear transformation and successive rotations ? 1 and ? 2 produce the same final image as the single rotation ? 1 + ? 2 Furthermore, the result of a linear transformation can be broken down and ascribed to a combination of any number of independent input states. Continuing with the example of a rotation transformation, the rotation through some angle ? T can be achieved by a single rotation ? T, two successive rotations ? 1 and ? 2 where ? 1 + ? 2 = ? T, or three successive rotations where ? 1 + ? 2 + ? 3 = ? T, or and so forth. The same is not true for nonlinear transformations. Let's look at some linear and nonlinear functions that clearly demonstrate their differences.
We begin...
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