Degree Reduction for NURBS Symbolic

Computation on Curves

Xianming ChenRichard F. Riesenfeld

Elaine Cohen

School of Computing, University of Utah

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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.

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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

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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

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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 …

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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

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Find Critical Curvature – squaring approach

Numerator of (2) : C-1 B-spline of deg 24.

(2)=2=0.Thus 2 extra zeros from =0.

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Critical curvature of a cubic -3

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Critical curvature of a rational quadratic-1

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Critical curvature of a rational quadratic-2

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Why the magic?

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Critical Curvature of Plane B-spline -1

Brute force squaring approach

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Critical Curvature of Plane B-spline -2

A better way for polynomial case

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Critical Curvature of Plane B-spline -3

An even better way for rational case

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Evolute of rational B-spline -1

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Evolute of rational B-spline -2

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Similar Result for Space Curve

Torsion Tangent developable Normal scroll (ruled surface) Binormal scroll

However, Focal curve is not even rational

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Point-Curve Bisector -1

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Point-Curve Bisector -2

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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

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Bisector Surface of Two Space Curves -1

As solution to a linear system

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Bisector Surface of Two Space Curves -2

Polynomialization for Rational Case

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Point/Curve Bisector Surface -1

Directrix Approach

An under-determined system

[Elber et al 1998a] Add a constraint to solve for the directrix

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Point/Curve Bisector Surface -2

Our Direct Approach

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Point/Ellipse Bisector Surface -1

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Point/Ellipse Bisector Surface -2

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Point/Ellipse Bisector Curve -1

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Point/Ellipse Bisector Curve -2

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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.

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Thanks!