Derivative bounds of rational b zier curves and surfaces
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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!


Property

Property


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


Hodograph property

Hodograph property


Two identities

Two identities


Derivative formula 21

Derivative formula(2)

  • Rewrite P(t) as

  • Method of undetermined coefficient


Main results

Main results


Upper bounds 1

Upper bounds(1)

where


Upper bounds 2

Upper bounds(2)

where


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 results1

Main results


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 techniques

Innovative points and techniques


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


The key and techniques

The key and techniques


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


Derivative bounds of rational b zier curves and surfaces

Thank you!


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