A-splines: local interpolation and approximation using G(k)-continuous piecewise real algebraic curves

We provide sufficient conditions for the Bernstein-Bezier (BB) form of an i mplicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebrai c curve segment. We call a piecewise G(k)-continuous chain of such real alg ebraic curve segments in BE-form as an A-spline (short for algebraic spline ). We prove that the degree n A-splines can achieve in general G(2n-3) cont inuity by local fitting and still have degrees of freedom to achieve local data approximation. As examples, we show how to construct locally convex cu bic A-splines to interpolate and/or approximate the vertices of an arbitrar y planar polygon with up to G(4) continuity, to fit discrete points and der ivatives data, and approximate high degree parametric and implicitly define d curves. Additionally, we provide computable error bounds. (C) 1999 Elsevi er Science B.V. All rights reserved.