Surface modeling and parameterization with manifolds

What do the configuration space of an animation skeleton, a subdivision surface and a panorama have in common? All of these are examples of manifolds. The goal of this course is to present an overview of manifold constructions which are useful for graphics applications, with a focus on two-dimensional manifolds (surfaces). Description 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. The course will be mostly self-contained. The only mathematical prerequisites are basic calculus, complex numbers, and vector and matrix algebra. General familiarity with graphics research is helpful, but not required. Intended Audience researchers from academia and industry interested in applying manifold-based techniques in their work; practitioners interested in applying latest graphics research using mani-fold ideas.