A-splines: local interpolation and approximation using Gk-continuous piecewise real algebraic curves

We characterize of the Bernstein-Bezier (BB) form of an implicitly deened bivariate polynomial over a triangle, such that the zero contour of the polynomial deenes 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?3 continuity by local tting. As examples, we use the A-splines to t 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 eecient display of all these A-splines and computable error bounds.