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Derivative bounds of rational B ézier curves and surfaces. Hui-xia Xu Wednesday, Nov. 22, 2006. Research background. Bound of derivative direction can help in detecting intersections between two curves or surfaces

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Derivative bounds of rational b zier curves and surfaces
Derivative bounds of rational Bézier curves and surfaces

Hui-xia Xu

Wednesday,Nov. 22, 2006


Research background
Research background

  • Bound ofderivative direction can help in detecting intersections between two curves or surfaces

  • Bound of derivative magnitude can enhance the efficiency of various algorithms for curves and surfaces


Methods
Methods

  • Recursive Algorithms

  • Hodograph and Homogeneous Coordinate

  • Straightforward Computation


Related works 1
Related works(1)

  • Farin, G., 1983. Algorithms for rational Bézier curves. Computer-Aided Design 15(2), 73-77.

  • Floater, M.S., 1992. Derivatives of rational Bézier curves. Computer Aided Geometric Design 9(3), 161-174.

  • Selimovic, I., 2005. New bounds on the magnitude of the derivative of rational Bézier curves and surfaces. Computer Aided Geometric Design 22(4), 321-326.

  • Zhang, R.-J., Ma, W.-Y., 2006. Some improvements on the derivative bounds of rational Bézier curves and surfaces. Computer Aided Geometric Design 23(7), 563-572.


Related works 2
Related works(2)

  • Sederberg, T.W., Wang, X., 1987. Rational hodographs. Computer Aided Geometric Design 4(4), 333-335.

  • Hermann, T., 1992. On a tolerance problem of parametric curves and surfaces. Computer Aided Geometric Design 9(2), 109-117.

  • Satio, T., Wang, G.-J., Sederberg, T.W., 1995. Hodographs and normals of rational curves and surfaces. Computer Aided Geometric Design 12(4), 417-430.

  • Wang, G.-J., Sederberg, T.W., Satio, T., 1997. Partial derivatives of rational Bézier surfaces. Computer Aided Geometric Design 14(4), 377-381.


Related works 3
Related works(3)

  • Hermann, T., 1999. On the derivatives of second and third degree rational Bézier curves. Computer Aided Geometric Design 16(3), 157-163.

  • Zhang, R.-J., Wang, G.-J., 2004. The proof of Hermann’s conjecture. Applied Mathematics Letters 17(12), 1387-1390.

  • Wu, Z., Lin, F., Seah, H.S., Chan, K.Y., 2004. Evaluation of difference bounds for computing rational Bézier curves and surfaces. Computer & Graphics 28(4), 551-558.

  • Huang, Y.-D., Su, H.-M., 2006. The bound on derivatives of rational Bézier curves. Computer Aided Geometric Design 23(9), 698-702.


Derivatives of rational b zier curves
Derivatives of rational Bézier curves

M.S., Floater

CAGD 9(1992), 161-174


About m s floater
About M.S. Floater

  • Professor of University of Oslo

  • Research interests: Geometric modelling, numerical analysis, approximation theory


Outline
Outline

  • What to do

  • The key and innovation points

  • Main results


What to do
What to do

Recursive

Algorithm

Two formulas about derivative P'(t)

Rational Bézier curve P(t)

Two bounds on the derivative magnitude

Higher derivatives, curvature and torsion


The key and innovation points
The key andinnovation points


Definition
Definition

  • The rational Béziercurve P of degree n as

    where


Recursive algorithm
Recursive algorithm

  • Defining the intermediate weights and the intermediate points respectively as


Recursive algorithm1
Recursive algorithm

  • Computing using the de Casteljau algorithm

    The former two identities represent the recursive algorithm!



Derivative formula 1
Derivative formula(1)

  • The expression of the derivative formula


Derivative formula 11
Derivative formula(1)

  • Rewrite P(t) as

    where


Derivative formula 12
Derivative formula(1)

  • Rewrite a’(t) and b’(t) as

    with the principle “accordance with degree”, then after some computation, finally get the derivative formula (1).


Derivative formula 2
Derivative formula(2)

  • The expression of the derivative formula

    where

    or




Derivative formula 21
Derivative formula(2)

  • Rewrite P(t) as

  • Method of undetermined coefficient





Some improvements on the derivative bounds of rational b zier curves and surfaces
Some improvements on the derivative bounds of rational Bézier curves and surfaces

Ren-Jiang Zhang and Weiyin Ma

CAGD23(2006), 563-572


About weiyin ma
About Weiyin Ma

  • Associate professor of city university of HongKong

  • Research interests:

    Computer Aided Geometric Design, CAD/CAM, Virtual Reality for Product Design, Reverse Engineering, Rapid Prototyping and Manufacturing.


Outline1
Outline

  • What to do

  • Main results

  • Innovative points and techniques


What to do1
What to do

Hodograph

Derivative bound of rational Bézier curves of degree n=2,3 and n=4,5,6

Degree elevation

Extension to surfaces

Derivative bound of rational Bézier curves of degree n≥2

Recursive algorithm


Definition1
Definition

  • A rational Bézier curve of degree n is given by

  • A rational Bézier surface of degree mxn is given by



Main results for curves 1
Main results for curves(1)

  • For every Bézier curve of degree n=2,3

    where


Main results for curves 2
Main results for curves(2)

  • For every Bézier curve of degree n=4,5,6

    where


Main results for curves 3
Main results for curves(3)

  • For every Bézier curve of degree n≥2

    where


Main results for surfaces 1
Main results for surfaces(1)

  • For every Bézier surface of degree m=2,3


Main results for surfaces 2
Main results for surfaces(2)

  • For every Bézier surface of degree m=4,5,6


Main results for surfaces 3
Main results for surfaces(3)

  • For every Bézier surface of degree m≥2

    where



Innovative points and techniques1
Innovative points and techniques1

  • Represent P’(t) as

    where


Innovative points and techniques11
Innovative points and techniques1

  • Then P’(t) satisfies

    where


Innovative points and techniques12
Innovative points and techniques1

  • Let and are positive numbers, then

  • and are the same as above, then


Innovative points and techniques13
Innovative points and techniques1

  • Let m>0 and then

    where


Proof method
Proof method

  • Applying the corresponding innovative points and techniques

  • In the simplification process based on the principle :


Innovative points and techniques2
Innovative points and techniques2

  • Derivative formula(1)

  • Recursive algorithm


About results for curves 3
About results for curves (3)

  • Proof the results for curves n≥2

  • Point out the result is always stronger than the inequality


Results for curves of degree n 7
Results for curves of degree n=7

  • The bound for a rational Bézier curve of degree n=7:


The bound on derivatives of rational b zier curves
The bound on derivatives of rational Bézier curves

Huang Youdu and Su Huaming

CAGD 23(2006), 698-702


About authors
About authors

  • Huang Youdu: Professor of Hefei University of Technology , and computation mathematics and computer graphics are his research interests.

  • Su Huaming: Professor of Hefei University of Technology, and his research interest is computation mathematics.


Outline2
Outline

  • What to do

  • The key and techniques

  • Main results


What to do2
What to do

Property of Bernstein

New bounds on the curve

Rational Bézier curve P(t)

Degree elevation

On condition some weights are zero

Modifying the results



Definition2
Definition

  • A rational Bézier curve of degree n is given by


The key and techniques1
The key and techniques

  • Represent P’(t) as

  • Two identities:


The key and techniques2
The key and techniques

  • If ai and bi are positive real numbers, then


Main results 1
Main results(1)

  • New bound on the rational Bézier curve is


Superiority
superiority

  • Suppose vector then

  • Applying the results above, main results (1) can be proved that it is superior than the following:


Proof techniques
Proof techniques

  • Elevating and to degree n, then applying the inequality:


Main results 2
Main results (2)

  • The other new bounds on the curve:

  • where


The case some weights are zero
The case some weights are zero

  • Let , and about the denominator of P’(t) on [0,1], then

  • And with the property:


Main results 3
Main results(3)

  • On the case , the bound on it is



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