Derivative bounds of rational Bézier curves and surfaces

Hui-xia XuWednesday, Nov. 22, 2006

Research background

Bound of derivative 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.

Derivatives of rational Bézier curves

M.S., Floater CAGD 9(1992), 161-174

About M.S. Floater

Professor of University of Oslo

Research interests: Geometric modelling, numerical analysis, approximation theory

OutlineWhat to do

The key and innovation points

Main results

What to do

Rational BRational Bézier ézier curve P(t)curve P(t)

Two formulas Two formulas about derivative about derivative

P'(t)P'(t)

RecursiveRecursiveAlgorithmAlgorithm

Two bounds on the Two bounds on the derivative derivative magnitudemagnitude

Higher derivatives, Higher derivatives, curvature and curvature and

torsiontorsion

The key and innovation points

Definition The rational Bézier curve P of degree n as

where

,0

,0

( )( ) ,

( )

n

i n i iin

i n ii

B t PP t

B t

,0, ( ) (1 ) .i n ii i n

nB t t t

i

Recursive algorithm Defining the intermediate weights and

the intermediate points respectively as

, ( )i kP t

,0, , ,

0 ,0

( )( ) ( ) , ( ) .

( )

kk

j k i j i jji k j k i j i k k

j j k i jj

B t Pt B t P t

B t

, ( )i k t

,0 0,

,0 0,

( ) , ( ) ( )

( ) , ( ) ( )i i n

i i n

t t t

P t P P t P t

Recursive algorithm Computing using the de Casteljau algori

thm

The former two identities represent the recursive algorithm!

, ,( ), ( )i k i kt P t

, , 1 1, 1(1 ) ,i k i k i kt t

, , , 1 , 1 1, 1 1, 1(1 ) .i k i k i k i k i k i kP t P t P

Property

Derivative formula(1) The expression of the derivative formula

0, 1 1, 1'1, 1 0, 12

0,

( ) ( )( ) ( ( ) ( )).

( )n n

n n

n

t tP t n P t P t

t

Derivative formula(1) Rewrite P(t) as

where

( )( )

( )

a tP t

b t

' '' ( ) ( ) ( )( )

( )

a t b t P tP t

b t

1'

, 1 1 10

1'

, 1 10

( ) ( )( )

( ) ( )( )

n

i n i i i ii

n

i n i ii

a t n B t P P

b t n B t

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).

'1, 1 1, 1 0, 1 0, 1

'1, 1 0, 1

( ) ( )

( ) ( )

n n n n

n n

a t n P P

b t n

Derivative formula(2) The expression of the derivative formula

where

or

1'

10

( ) ( )( )n

i i ii

P t t P P

0,

, 1 1, 1 1, 1 , 120 1

( ) ( ( ) ( ) ( ) ( ))( )

n

i n

i j n k n j n k n j kj k i

nt B t B t B t B t

t

0,

, ,20 1

1( ) ( ) ( ) ( )

(1 ) ( )n

i n

i j n k n j kj k i

t k j B t B tt t t

Hodograph property

Two identities

, ,

' ', , 1, 1 , 1 , 1 1, 1( )

i n j nj n i n i n j n i n j nB B B B n B B B B

, ,

, ,' ', , ( )

(1 )i n j n

i n j nj n i n

B BB B B B i j

t t

Derivative formula(2) Rewrite P(t) as

Method of undetermined coefficient

, ,

0 0,,0

( ) ( )( ) ( ) , ( )

( )( )

ni n i n

i i i i ni nk n kk

B t B tP t t P t

tB t

Main results

Upper bounds(1)

'

, 0, ,( ) max ,i j

i j n

WP t n P P

max , mini i i iW

where

Upper bounds(2)

2'

12 0, , 1( ) max ,i i

i n

WP t n P P

where

max , mini i i iW

Some improvements on the derivative bounds of rational Bézier cu

rves and surfaces

Ren-Jiang Zhang and Weiyin MaCAGD23(2006), 563-572

About Weiyin Ma Associate professor of city university of HongKong

Research interests: Computer Aided Geometric Design, CAD/CAM, Virtu

al Reality for Product Design, Reverse Engineering, Rapid Prototyping and Manufacturing.

OutlineWhat to do

Main results

Innovative points and techniques

What to do

Hodograph

Degree elevation

Recursive algorithm

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

=4,5,6 Extension to

surfacesDerivative bound of rational Bézier curves

of degree n≥2

Definition A rational Bézier curve of degree n is given by

A rational Bézier surface of degree mxn is given by

0

0

( )( ) , 0 1

( )

n ni ii

n nii

i

i

B t PP t t

B t

, ,0 0

,0 0

( ) ( )( , ) , 0 , 1

( ) ( )

m n m ni j i ji j

m n m ni ji j

i j

i j

B u B v PF u v u v

B u B v

Main results

Main results for curves(1)

For every Bézier curve of degree n=2,3

where

'1

1( ) max( , ) max ,i i

iP t n P P

1

: max .ii

i

Main results for curves(2) For every Bézier curve of degree n=4,5,6

where

2'1

1( ) max( , ) max ,i i

iP t n P P

1

: max .ii

i

Main results for curves(3) For every Bézier curve of degree n≥2

where

'

,( ) max ,i j

i jP t n P P

1 11 1max , , 0.i i

i ni i

i n i n i i

n n n n

Main results for surfaces(1)

For every Bézier surface of degree m=2,3

,1, ,

, ,1, ,

1,

( , ) 1max max , max .

max

i ji h i k

i j i h ki j i j

ji j

F u vm P P

u

Main results for surfaces(2) For every Bézier surface of degree m=4,5,6

,1, ,

, ,1, ,

1,

2

( , ) 1max max , max .

max

i ji h i k

i j i h ki j i j

ji j

F u vm P P

u

Main results for surfaces(3) For every Bézier surface of degree m≥2

where

, ,, , ,

( , )max ,m i h j ki j h k

F u vm P P

u

1, 1,,

, ,

max , .i j i jm i j

i j i j

i m i m i i

m m m m

Innovative points and techniques

Innovative points and techniques1

Represent P’(t) as

where

2 2 2 2

' 02

0

( )( ) ,

( )

n nii

n nii

i

i

B t DP t

B t

2

1 1max 0, 1

12 1 .

2 2 1

i

i j i j i j jj i n

n nD i j P P

n j i j

i

Innovative points and techniques1

Then P’(t) satisfies

where 2 2 2 2

' 012

0

(1 )( ) max ,

( )

n n i iii

i iin n

ii i

t t dP t n P P

B t

22

1max 0, 1

12 1 .

1

i

i j i jj i n

n nd i j

j i jn

Innovative points and techniques1

Let and are positive numbers, then

and are the same as above, then

0

0

max .

n

ii in i

iii

i i

0

0

(1 )max , .

(1 )

n n i iii i

n n i i iiii

t tt

t t

i i

Innovative points and techniques1

Let m>0 and then

where

0( ) (1 ) , ( ) ( ),

nn n i i ni n i ii

H t t t p t a H t

0

( ) ( ),m n m n

n i iip t b H t

0

1 00 0

1 01 1

1

1 1

, .

m

m m

m m

m

mm

mm

m n nmm m n n

im

Cb a

C Cb a

C Cm

Ci

C

Cb a

C

c

Proof method Applying the corresponding innovative

points and techniques

In the simplification process based on the principle :

i

i

1( )

Innovative points and techniques2

Derivative formula(1)

Recursive algorithm

0, 1 1, 1'1, 1 0, 12

0,

( ) ( )( ) ( ( ) ( )).

( )n n

n n

n

t tP t n P t P t

t

, , 1 1, 1(1 ) ,i k i k i kt t

, , , 1 , 1 1, 1 1, 1(1 ) .i k i k i k i k i k i kP t P t P

About results for curves (3)

Proof the results for curves n≥2

Point out the result is always stronger than the inequality

'

,

1( ) max( , ) max i j

i jP t n P P

Results for curves of degree n=7

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

' 31

1( ) max( , ) max .i i

iP t n P P

The bound on derivatives of rational Bézier curvesHuang 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.

OutlineWhat to do

The key and techniques

Main results

What to do

Rational Bézier Rational Bézier curve P(t)curve P(t)

New New bounds on bounds on the curvethe curve

Property Property of of

BernsteinBernstein

Modifying Modifying the resultsthe results

Degree Degree elevatioelevatio

nnOn condition On condition

some weights are some weights are zerozero

The key and techniques

Definition A rational Bézier curve of degree n is given by

0

0

( )( ) 0 1)

( )

n ni ii

in nii

i

i

B t PP t t

B t

The key and techniques Represent P’(t) as

Two identities:

' ( )( ) .

( )

tP t

t

'1, 1 , 1,

( ) ( ( ) ( )).i n i ni nB t n B t B t

, 1, 1 , 1( ) ( ) (1 ) ( ),i n i n i nB t tB t t B t

The key and techniques If ai and bi are positive real numbers, then

1

1

max .

n

ii in i

iii

a a

bb

Main results(1) New bound on the rational Bézier curve is

1 1'

,1

( ) ( )( ) max .

min ,i j i i j i

i ji i

P P P PP t n

superiority Suppose vector then

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

1 2(1 ) , 0 1,r a r ar a

1 2max , .r r r

' 1

,1

( ) max max ,max max .i ii j

i i i ji i

P t n P P

Proof techniques Elevating and to degree n, then

applying the inequality:( )t ( )t

1

1

max .

n

ii in i

iii

a a

bb

Main results (2) The other new bounds on the curve:

where

(0),'

,( ) max .

i j

i ji

QP t n

(0) (0) (0)0 0, , 1, ,, , .j j nj n j ij i j i j

i n iQ Q Q Q Q Q Q

n n

, 1 1( ) ( ).i j i i j i i jQ P P P P

The case some weights are zero

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

And with the property:

0min , nc

2 2

, 2 20

( ) ( ) .2

n

i i n ni

ct B t

1

, 1 ,0 0( ) ( ) 1

n n

i n j ni jB t B t

Main results(3) On the case , the bound

on it is2 2

'2,

2 2' 0

2,

2( ) max ,

2( ) max .

n

j iji j

n

ji j ij

P t n Qc

P t n Qc

0 n i

Thank you!Thank you!