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Derivative bounds of rational B ézier curves and surfacesPowerPoint Presentation

Derivative bounds of rational B ézier curves and surfaces

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Derivative bounds of rational B ézier curves and surfaces

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Derivative bounds of rational B ézier curves and surfaces

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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

- Recursive Algorithms
- Hodograph and Homogeneous Coordinate
- Straightforward Computation

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)

- 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)

- 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.

About M.S. Floater

- Professor of University of Oslo
- Research interests: Geometric modelling, numerical analysis, approximation theory

Outline

- What to do
- The key and innovation points
- Main results

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 andinnovation points

Definition

- The rational Béziercurve P of degree n as
where

Recursive algorithm

- Defining the intermediate weights and the intermediate points respectively as

Recursive algorithm

- Computing using the de Casteljau algorithm
The former two identities represent the recursive algorithm!

Derivative formula(1)

- The expression of the derivative formula

Derivative formula(1)

- Rewrite P(t) as
where

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)

- The expression of the derivative formula
where

or

Derivative formula(2)

- Rewrite P(t) as
- Method of undetermined coefficient

Upper bounds(1)

where

Upper bounds(2)

where

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

- 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.

Outline

- What to do
- Main results
- Innovative points and techniques

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

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)

- For every Bézier curve of degree n=2,3
where

Main results for curves(2)

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

Main results for curves(3)

- For every Bézier curve of degree n≥2
where

Main results for surfaces(1)

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

Main results for surfaces(2)

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

Main results for surfaces(3)

- For every Bézier surface of degree m≥2
where

Innovative points and techniques1

- Represent P’(t) as
where

Innovative points and techniques1

- Then P’(t) satisfies
where

Innovative points and techniques1

- Let and are positive numbers, then
- and are the same as above, then

Innovative points and techniques1

- Let m>0 and then
where

Proof method

- Applying the corresponding innovative points and techniques
- In the simplification process based on the principle :

Innovative points and techniques2

- Derivative formula(1)
- Recursive algorithm

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

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

The bound on derivatives of rational Bézier curves

Huang Youdu and Su Huaming

CAGD 23(2006), 698-702

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.

Outline

- What to do
- The key and techniques
- Main results

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

Definition

- A rational Bézier curve of degree n is given by

The key and techniques

- Represent P’(t) as
- Two identities:

The key and techniques

- If ai and bi are positive real numbers, then

Main results(1)

- New bound on the rational Bézier curve is

superiority

- Suppose vector then
- Applying the results above, main results (1) can be proved that it is superior than the following:

Proof techniques

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

Main results (2)

- The other new bounds on the curve:
- where

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)

- On the case , the bound on it is