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

We provide sufficient conditions for the Bernstein–Bézier (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 Gk-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 G2n−3 continuity by local fitting 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 G4 continuity, to fit discrete points and derivatives data, and approximate high degree parametric and implicitly defined curves. Additionally, we provide computable error bounds.