Date of Graduation

Fall 2012

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Xingping Sun

Keywords

Bernstein polynomials, Weierstrass theorem, Bezier curves, computer-aided geometric design, polynomial approximation, applied mathematics

Subject Categories

Mathematics

Abstract

Bernstein polynomials certainly serve as the mathematical foundation for moderncomputer-aided geometric design (CAGD). Popular programs, like Adobe's Illustratorand Flash, and font imaging systems such as Postscript, utilize Bernstein polynomials toform what are known as Bezier curves. Introduced in the 1960's, Bezier curves were first implemented in automobile design but quickly gained popularity in other areas of CAGD with the expansion of the digital age. The aforementioned polynomials, however, came about long before the invention of computers. Russian mathematician, Sergei Natanovich Bernstein, first introduced the Bernstein polynomial basis 100 years ago, as a means to prove the Weierstrass Approximation Theorem. These polynomials possess distinct properties that are responsible for both their obscurity for 50 years and their nowwidespread application in CAGD. This thesis is devoted to a thorough study of Bernstein polynomials. After brief look at the history of Bernstein and his work, we define the basis polynomials and discuss their unique properties. We next consider the application of Bernstein polynomials in approximating continuous functions, and employ them to approximate discontinuous functions as well, making some interesting observations. We also survey the algorithms and software applications in the field of CAGD in which Bernstein polynomials play a prominent role, and finally, we address a few of the new applications of Bernstein polynomials in approximation theory and probability theory. Throughout this thesis we establish that the Bernstein polynomial form remains quite relevant in areas applied mathematics, and we provide many options by which one may extend their own research.

Copyright

© Cody Ryann Lawson

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