Quadratic approximation to plane parametric curves and its application in approximate implicitization

PART I 2D-IPO's for constant speed Lines, Curves, NURBS

“…We round these points to 2 decimal points. We can use the formula of algorithm 4 of [35], but we obtain the same result when we match 2k 1 M − with the shoulder point The approximation was good and the results are {ω1, ω3, ω5, ω7, ω9}={1.88,1.55,2.27,1.55,1.88} f. Compute the extremes of NURB fnurb2, parallel to the x-axis and y-axis…”

“…This trident is 3 rd -degree NURB fnurb3, and we converted this NURB to a rational 2 nd -degree using the method of [31,32,33,34,35]. So, the 3 rd -degree NURB was approximated to a 2 nd -degree NURB fnurb2, and we can always do it when we apply algorithm 6 of [35] for a given maximum approximation error MAX  . The NURB fnurb2 is identical with the five composite rational Bézier curves CBk[t] (Appendix 4.1, Convert NURB to Conics).…”

Approximation by Conic Splines

“…Yang [28] constructs a curvature continuous conic spline by first fitting a tangent continuous conic spline to a point set and fairing the resulting curve. Li et al [16] show how to divide the initial curve into simple segments which can be efficiently approximated with rational quadratic Bézier curves. These methods have many limitations, among which the dependence on the specific parameterization of the curve, the large number of conic segments produced or the lack of accuracy and absence of control of the error.…”

Approximate implicitization of parametric surfaces by using compactly supported radial basis functions

“…Chen et al [5] presented the concept of interval implicitization of rational curves and developed the corresponding effective algorithm. Li et al [6] considered approximate implicitization of plane parametric curves using the piecewise quadratic Bézier spline curves with G 1 continuity. …”