Generalized conic curves and their applications in curve approximation

Abstract

The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation equivalent curves: (i) conic curves, including parabolas, hyperbolas and ellipses; (ii) generalized monomial curves, including curves of the form x=yγ, γ∈R, γ≠0,1, in the x-y Cartesian coordinate system; (iii) exponential spiral curves of the form ρ(θ{symbol})=Aeγθ{symbol}, A>0, γ≠0, in the ρ-θ{symbol} polar coordinate system. This type of curves has many important properties such as convexity, approximation property, effective numerical computation property and the subdivision property etc. Applications of these curves in both interpolation and approximations using piecewise generalized conic segment are also developed. It is shown that these generalized conic splines are very similar to the cubic polynomial splines and the best error of approximation is O(h5) or at least O(h4) in general provided appropriate procedures are used. Finally some numerical examples of interpolation and approximations with generalized conic splines are given. © 1997 Springer.