| Abstract: | The shape of a smooth surface S in R^3 is governed by its Gaussian curvature K. In particular, S is strictly convex if and only if K is positive. In this paper, the notion of the GC signature of S is introduced. This scalar-valued function f has the same sign as K and becomes a polynomial if S is a parametric polynomial surface. Hence, the study of convexity of such surfaces S reduces to the study of positivity of the corresponding polynomials f. Efficient computational schemes are developed for expressing the Bézier coefficients of the Bernstein-Bézier formulation of f in terms of those of the parametric polynomial b that defines S. This approach differs from the usual consideration of positive definiteness of the Hessian matrix corresponding to the parametric polynomial b. |
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