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!
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