Curve Fitting with Conic Splines

Abstract

Conic splines are formed by arcs of conics, each defined by its endpoints and the tangents at them plus an intermediate point. Instead of the common general equation that depends on five parameters, an equation with a single parameter is used, thus simplifying significantly the curve fitting problem. The resulting guided conics resemble Bezier polynomials and for parabolas are identical to them. Such splines can be used conveniently both for interactive design and for automatic curve fitting. They allow circular, elliptical, and hyperbolic arcs to be included in the spline family, while the common forms using a B-spline basis allow the inclusion of parabolic arcs only. Conic splines are described either in a rational parametric or in algebraic form f(x, y) = 0. A simple estimate for the distance of a point from such a curve is given and is used to test the quality of approximations. The data to be fitted are first approximated by a polygon, and then simple heuristics are used to decide which sequences of vertices should be approximated by conics. The conics found by the applications of the heuristics are usually close approximations of the data and need no further adjustments. When adjustments are needed, the interval is split and a conic is fitted on each part. It is shown theoretically that exact knot placement at the optimal locations is less important for higher order splines than for polygons. Examples of application of the method to the fitting of font and other contours are given. Comparisons with other methods suggest that conic splines require no more knots than cubic splines for similar quality of approximation. © 1983, ACM. All rights reserved.