Mathematical Structures for Computer Graphics

Chapter 10Other Paradigms

The core mathematics in computer graphics is based on vector geometry. From object modeling to shading techniques, vectors play a significant role and offer a very useful paradigm for constructing images. Yet, there are corners of the image gallery that cannot be reached conveniently with the standard geometric approaches. Gradually, other mathematical tools, structures, and methods have surfaced to solve particular graphics problems and to expand image generation possibilities. This chapter highlights three of those approaches that fall outside the main flow of vectors, transformations, and geometric optics. It is reasonable to consider these techniques as separate paradigms.

The first category of graphics problems focuses on the fact that all modeling and rendering efforts end with an array of pixels on the display screen. Resolutions continue to increase, but the pixel still is more a block of color than a colored point. Jagged edges in images are a common result of this coarseness. Coping with this and other artifacts requires a different take on the mathematics of images.

Nature is not particularly fond of Euclidean geometry, so a major challenge is to generate more natural organic forms. One approach here involves introducing noise into the process. It is a little odd that adding randomness expands the set of image-generating tools, but properly handled, noise produces many of the natural patterns we encounter.

A more structured approach to organic ...