Many engineering and computer graphics applications require computing surface deformations minimizing an energy or solving an equation of motion. This type of deformations are used to model free-form surfaces in computer-aided design systems, to create animated characters, to simulate cloth or analyze stresses in a car body.
Complex surfaces are commonly represented by meshes, that is, piecewise-linear functions which cannot be differentiated directly. At the same time, the equations that we need to solve often involved derivatives of order four or higher. Approximating high-order derivatives on meshes with sufficient accuracy is difficult, and often requires costly computations. These computations may be prohibitively expensive in the context of interactive modeling and simulation. In many cases, cheap, but inaccurate approximations are available, resulting in faster algorithms, but less reliable results.
In this talk, I will discuss how mesh deformations can be computed efficiently while maintaining accuracy, and demonstrate several applications in geometric modeling and animation.
I will review several complimentary approaches that we have explored, in particular, taking advantage of geometric relations to simplify the equations we need to solve, decomposing higher-order problems into several low-order problems, and representing the solution of a general problem as a combination of solutions of special-case simpler problems.
Denis Zorin is an associate professor in the Computer Science Department of the Courant Institute of Mathematical Sciences at New York University. He received a PhD in Computer Science from the California Institute of Technology. Before joining the faculty at NYU he was a postdoctoral researcher at the Computer Science Department of Stanford University. He was a Sloan Foundation Fellow in 2000-2002; he received the NSF CAREER award in 2001, and IBM Faculty Partnership Award in 2001-2004 and 2006. His primary research interests are geometric modeling and scientific computing.