Degree reduction of NURBS curves

Abstract

Higher degree curves are used in applications because they are easier to manipulate interactively but require heavy computation. Most of the equations for curves used popularly in CAD software are of degree 2 and 3, because two curves of degree 3 can guarantee 2nd derivative continuity at the connection point. This study proposes a different but simpler method than any put forward before to deal with degree reduction of free-form curves. The reduced curves use the simplest knot vector type, i.e., the open uniform knot vector. Unlike other methods, this study does not modify or refine the knot vectors but perturb the control points globally. After obtaining an initial condition, a radiating web-like search algorithm is applied to detect the optimum positions. These NURBS curve formats reach basic industrial standards for CAD/CAM/CNC applications. By defining a global bound error function, this algorithm can achieve an optimum solution not only for NURBS curves but also Bezier/B-spline curves

A survey on shape preserving interpolation

“…In view of the algorithms of degree reduction approximation, there are mainly the geometric method based on control points approximation [39] and the algebraic way based on base transition [38,40]. In view of the curve/ surface type of degree reduction, curve mainly focus on Bézier curve [40], B spline [41], NURBS [42], C-Bézier [43]. Surface mainly focus on tensor product Bézier surface [38,44] and triangular Bézier surface [38,45].…”

Tool-path generation of planar NURBS curves

“…In order to achieve the goal of generating tool-path, an algorithm is presented to offset NURBS curves by an optimum procedure. Recently, Lai et al [2] proposed an optimum process to reduce the degree of NURBS curves. Following the optimum process, this paper describes a radiatingweb like searching path and a normalizing process to find the offsets of planar NURBS curves.…”

“…This method cannot inherit the original free-form curve's properties. (2) Offset the control polygon that consists of control points with the same degree and weights as the original one. This method is fast but the accuracy problem exists in most of free-form cases, and it works only on convex patterns.…”

“…The main targets of this paper are: (1) To keep a constant distance d between offset curve C d (t) and progenitor curve C(t) on the normal direction of C(t). (2) To alternate the order k of the basis function in offset curve C d (t). Generally, reducing of the order is better for calculating the initial conditions.…”