A-Splines: Local Interpolation and Approximation using G^k-Continuous Piecewise Real Algebraic Curves

Abstract

We characterize of the Bernstein-Bezier (BB) form of an implicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebraic curve segment. We call a piecewise G k -continuous chain of such real algebraic curve segments in BB-form as an A-spline (short for algebraic spline). We prove that the degree n A-splines can achieve in general G 2n\Gamma3 continuity by local fitting. As examples, we use the A-splines to fit the discrete data, parametric curve and implicit algebraic curve and also show how to construct quadratic and cubic A-splines to locally interpolate and/or approximate the vertices of an arbitrary planar polygon with up to G 2 and G 4 continuity, respectively. Quadratic A-splines are always locally convex. We also prove that our cubic A-splines are always locally convex. Additionally, we derive evaluation formulas for the efficient display of all these A-splines and computable error bo..

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Last time updated on 22/10/2014

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