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Derivative bounds of rational B ézier curves and surfaces

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

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

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 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)

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

Upper bounds(1)

where

Upper bounds(2)

where

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 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

Innovative points and techniques1

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

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:

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

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