Polynomial bases for quadratic and cubic polynomials which yield control points with small convex hulls

Abstract

Of the many useful properties of Bézier curves and surfaces, one used very often in computer graphic render operations is the convex hull property. That is, a Bézier curve or surface lies within the convex hull of its control points. However, this convex hull is a very poor approximation to the space occupied by the curve. We present here, a polynomial basis for the quadratic and cubic curves whose convex hulls contain the curve, but with a size of about 77% and 42% of the respective Bézier convex hulls. We then show how to convert between Bézier and this small convex hull representation, and how to subdivide these curves. Such curve representations may allow large savings in rendering techniques which depend on the convex hull of the control points.