Boris Odehnal
University of Applied Arts, Vienna
Bezier curves of degree n allow for rational or even polynomial parametrizations and are, therefore, subsets of algebraic curves of degree n. As such, they are of genus zero and exhibit either the maximum number of singularities or carry compound singularities whose delta invariants sum up to the maximal possible genus (n-1)(n-2)/2.
Whenever Bezier curves are designed for application purposes, nobody takes care of singularities or even consideres them as present. In fact, they are, and, provided the control structure is chosen improperly, even the standard Bezier curve segment parametrized over the unit unterval contains singular points, the latter rather being singular points of the algebraic variety then on the parametrized curve.
We shall demonstrate that singularities with high delta invariants occur on Bezier curves. Further, we illustrate a constructive approach towards Bezier curves with compound singularities. For several reasons, we shall restrict our study of singularities on Bezier curve by limiting the degree of the curves to five.