Parameterizing n-holed Tori Using Hyperbolic Geometry
Many diverse applications in different areas of computer graphics, including geometric modeling, rendering and animation, require dealing with sets which cannot be easily represented with a single function on a simple domain in a Euclidean space: Examples include surfaces of nontrivial topology, environment maps, reflection/transmission functions, light fields, configuration spaces of animation skeletons, and others. In most cases these objects are described as collections of functions defined on multiple simple domains, with the functions satisfying various constraints (e.g., join smoothly). The unified mathematical view of many such structures is provided by the theory of smooth manifolds. While the concept is standard in mathematics, it is not broadly known in the graphics community and is often perceived as an impractical and complex abstraction. The goal of this half-day course is to present the basic concepts and definitions of manifold theory, demonstrate their computational nature and close connection to applications, and survey a variety of computer graphics applications in which manifolds appear, with a focus on modeling of surfaces and functions on surfaces.