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 6: More Linear Transformations
Having considered the rigid body motions, reflection, and symmetry, we now explore the rest of the linear transformations. These are the more general affine transformations and the projection transformations. We'll consider the affine transformations of dilation and shear, where parallel lines remain parallel, and a variety of projections, where parallelism is not invariant but at least straight lines remain straight. We revisit appropriate Cartesian equations for these transformations presented in Chapter 2 and then express them as equivalent vector-matrix equations.
6.1 Isotropic Dilation
Uniform expansion or contraction about a fixed point, isotropic dilation fixing the origin, uniform expansion, centerless uniform expansion, isotropic dilation fixing an arbitrary point, the product of two dilations with the same center, the product of two dilations with different centers
An isotropic dilation is a uniform expansion or contraction of the plane or space about some fixed point or center (Figure 6.1). If this point is the origin, then the transformation is a homogeneous isotropic dilation. (We will also study anisotropic dilation and shear.) Although an isotropic dilation is an affine transformation and preserves angle-size, it is not an isometry. However, the product of such a dilation and an isometry is a similarity. Recall that a similarity transformation is a one-to-one mapping in which all distances or lengths are multiplied by the same number, k, where k is the ratio of the similarity transformation. Physicists and engineers often refer to this ratio as strain. Also, recall that two figures are similar independent of their...
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