Error Bounded Regular Algebraic Spline Curves Extended Abstract

i=0 Pn n=0 bijB m i (u)B n j (v) of bi-degree (m;n) has total degree m + n, however, the class of Gmn(u; v) is a subset of polynomials of total degree m + n. G (resp G) continuity implies curve segments share the same tangent(curvature) at join points(knots). In each of the G and G constructions, we develop a spline curve family whose member satis es given interpolation conditions. Each family depends on one free parameter that is related linearly to coe cients of Gmn(u; v). Compared with A-spline segments de ned in triangular (barycentric) BB-form [2], these algebraic curve segments in tensor product form have the following distinct features: (a) They are easy to construct. The coe cients of the bivariate polynomial that de ne the curve are explicitly given. (b) There exist convenient geometric control handles to locally modify the shape of the curve, essential for interactive curve design. (c) The spline curves, for the rectangle scheme, are -error controllable where is the prespeci ed width of the rectangle. This feature is especially important for tting to \noisy" data with uncertainty. (d) These splines curves have a minimal number of in ection points. Each curve segment of the spline curve has either no in ection points if the corresponding edge is convex, or one in ection point otherwise, and the join points of the curve segments are not in ection points. (e). Since the required bi-degree (m;n) for G and G is low(in this paper, minfm;ng 2), the curve can be evaluated and displayed extremely fast. We explore both display via parameterization as well as recursive subdivision techniques(see [11]). (f) In the six spline families we discuss in sections 2 and 3, there are four cases with minfm;ng = 1. In these cases, rational parametric expressions are easily derived. Hence, for