The Fractal Nature of Bezier Curves and the Bezier Nature of Fractal Curves
Monday, November 17, 2014 - 4:00pm to 4:50pm KEC 1003 Ron Goldman Professor Department of Computer Science Rice University Abstract: Polynomials and fractals have very different geometric features. Polynomials are tame; fractals are wild. Fractals can be continuous everywhere, yet differentiable nowhere. Fractals are often self-similar curves with fractional dimension. And fractals are also attractors, fixed points of iterated function systems. In contrast, polynomials are one-dimensional curves that are differentiable everywhere. Polynomials can be represented in Bezier form where they have control points, polynomial coefficients that can be used to adjust the shape of the curve in an intuitive fashion. Moreover, unlike fractals, polynomial curves have simple parametrizations. Nevertheless, the goal of this talk is to unify polynomials and fractals: to demonstrate that polynomials and fractals share many fundamental geometric properties and procedures. We shall show that just like polynomials, fractals can be parametrized and fractals have control points that allow us to adjust the shape of the fractal i! n an intuitive manner. Moreover, just like fractals, polynomials are attractors, fixed points of iterated function systems. We shall show how to apply fractal algorithms to generate polynomial curves and polynomial algorithms to generate fractals. We shall conclude that polynomials and fractals are not really that different after all. Biography: Ron Goldman is a Professor of Computer Science at Rice University in Houston, Texas. Professor Goldman received his B.S. in Mathematics from the Massachusetts Institute of Technology in 1968 and his M.A. and Ph.D. in Mathematics from Johns Hopkins University in 1973. Professor Goldman's current research interests lie in the mathematical representation, manipulation, and analysis of shape using computers. His work includes research in computer aided geometric design, solid modeling, computer graphics, polynomials and splines. He is particularly interested in algorithms for polynomial and piecewise polynomial curves and surfaces, as well as in applications of algebraic and differential geometry to geometric modeling. He most recent focus is on the uses of quaternions and Clifford algebras in computer graphics. Dr. Goldman has published over a hundred articles in journals, books, and conference proceedings on these and related topics. He has also published two books on Computer Graphics and Geometric Modeling: Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, and An Integrated Introduction to Computer Graphics and Geometric Modeling. Dr. Goldman is currently an Associate Editor of Computer Aided Geometric Design. Before returning to academia, Dr. Goldman worked for ten years in industry solving problems in computer graphics, geometric modeling, and computer aided design. He served as a Mathematician at Manufacturing Data Systems Inc., where he helped to implement one of the first industrial solid modeling systems. Later he worked as a Senior Design Engineer at Ford Motor Company, enhancing the capabilities of their corporate graphics and computer aided design software. From Ford he moved on to Control Data Corporation, where he was a Principal Consultant for the development group devoted to computer aided design and manufacture. His responsibilities included data base design, algorithms, education, acquisitions, and research. Dr. Goldman left Control Data Corporation in 1987 to become an Associate Professor of Computer Science at the University of Waterloo in Ontario, Canada. He joined the faculty at Rice University in Houston, Texas as a Professor of Computer Science in July 1990.
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