Degree Reduction for NURBS Symbolic Computation on Curves Xianming Chen Richard F. Riesenfeld Elaine Cohen School of Computing, University of Utah

2 Motivation  NURBS Symbolic Computation closed algebraic operations on NURBS  One Big Problem Fast raising degree when rational B- splines involved differentiation doubles degree,  contrasting to polynomial case, when degree is reduced by 1.

3 Related Work Many pioneering research work on Bezier and NURBS symbolic computation; however, When coming to rational case,  Quotient rule is used indiscriminately, resulting unnecessary high or huge degrees in many situations  A common practice, in CAD systems, is to approximate rationals with polynomials  Differentiation typically amplifies error

4 Related Work – cont.  [Chen et al. 2005] Extended forward difference operator on Bezier control polygon to rational case.  Higher order derivatives The order of the denominator effectively stays at 2

5 Contribution Develop several strategies to get around of the quotient rule for many typical NURBS symbolic computation on curves, incl. Zero curvature enquiry Critical curvature enquiry Evolutes Bisector curves/surfaces …

6 Critical curvature of a cubic -1  Cubic polynomial B-spline of 6 segments.   vanishes at 6 pts 1. evolute has 2 extra cusps at break pts

7 Find Critical Curvature – squaring approach  Numerator of ( 2 ) : C -1 B-spline of deg 24.  ( 2 )=2=0. Thus 2 extra zeros from =0.

8 Critical curvature of a cubic -3

9 Critical curvature of a rational quadratic-1

10 Critical curvature of a rational quadratic-2

11 Why the magic?

12 Critical Curvature of Plane B-spline -1 Brute force squaring approach

13 Critical Curvature of Plane B-spline -2 A better way for polynomial case

14 Critical Curvature of Plane B-spline -3 An even better way for rational case

15 Evolute of rational B-spline -1 

16 Evolute of rational B-spline -2 

17 Similar Result for Space Curve  Torsion  Tangent developable  Normal scroll (ruled surface)  Binormal scroll  However, Focal curve is not even rational

18 Point-Curve Bisector -1 

19 Point-Curve Bisector -2 

20 Optimal Degree for Bisectors  [Elber&Kim 1998 a&b] computed various bisector surfaces  [Farouki et al. 1994] proved point/plane-curve bisector curve has degree 3d-1 (resp. 4d-1) for polynomial (resp. rational) case.  We show (3d-1/4d-1)-result applies to bisector surfaces as well

21 Bisector Surface of Two Space Curves -1  As solution to a linear system

22 Bisector Surface of Two Space Curves -2 Polynomialization for Rational Case

23 Point/Curve Bisector Surface -1 Directrix Approach  An under-determined system  [Elber et al 1998a] Add a constraint to solve for the directrix

24 Point/Curve Bisector Surface -2 Our Direct Approach

25 Point/Ellipse Bisector Surface -1

26 Point/Ellipse Bisector Surface -2 

27 Point/Ellipse Bisector Curve -1

28 Point/Ellipse Bisector Curve -2 

29 Conclusion  presented several degree reduction strategies for NURBS symbolic computation on curves, incl. eliminating higher degree terms resulting from irrelevant lower order derivatives canceling common scalar factors polynomialization  Degree reduction is significant.

30 Thanks!