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

We provide su cient conditions for the Bernstein-B ezier (BB) form of an implicitly de ned bivariate polynomial over a triangle, such that the zero contour of the polynomial de nes a smooth and single sheeted real algebraic curve segment. We call a piecewise G-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 3 continuity by local tting and still have degrees of freedom to achieve local data approximation. As examples, we show how to construct locally convex cubic A-splines to interpolate and/or approximate the vertices of an arbitrary planar polygon with up to G continuity, to t discrete points and derivatives data, and approximate high degree parametric and implicitly de ned curves. Additionally, we provide computable error bounds.