On the degree elevation ofB-spline curves and cornercutting

Guozhao Wang,Chongyang Deng

Reporter ： Jingjing Yu

Outline

Introduction Previous works Problem and Approach The bi-degree B-spline Corner cutting Conclusions and future work

Previous works Prautzsch,H. 1984 Degree elevation of B-spline curves. CAG

D 1(193-198) Prautzsch,H.,Piper,B. 1991 A fast algorithm to raise the degre

e of B-spline curves. CAGD 8(253-266) Barry P.J.,Goldman R.N. 1988 A recursive proof of a B-spline i

dentity for degree elevation. CAGD 5(173-175) Liu,W. 1997 A simple,efficient degree raising algorithm for B-

spline curve. CAGD 14(693-698) Huang,Q.,Hu S.,Martin,R. 2005 Fast degree elevation and kno

t insertion for B-spline curves. CAGD 22(183-197) Sederberg T.W., Zheng J., Song X., 2003. Knot intervals and m

ulti-degree splines. CAGD 20(455-468)

Problem

The traditional method

The new method

The advantages

Problem and Approach

Problem:Given a B-spline curve of degree k:

,0

( ) ( )n

i k ii

P t N t P

1

1 1, , , , , ,

n

n n

z z

T t t t t

To elevate the degree to k+1

Problem and Approach The traditional method:Step1 Update

1

1 1, , , , , ,

n

n n

z z

T t t t t

to

1

1 1

1 1

, , , , , ,

n

n n

z z

T t t t t

Step2 are represented by (k+1)-degree B- spline basis functions .

,i kN t , 1i kN t

Step3 The control points are computed according the transforming formulas between and

,i kN t , 1i kN t

Problem and Approach The new method: In each step we only increase the multiplicityof one interior knot and elevate the degree of only in one knot interval. Denote for each ,we update to by increasing the multiplicity of by one,andelevate the degree of the basis functions only in knot interval .

,i kN t 0T T1,2, , 1j n 1jT jT

jt

1,j jt t

Problem and Approach

Advantage of the method: 1) Obtain more simply formulas 2) The degree elevation algorithmcan be interpreted as corner cuttingalgorithm.

The definition of bi-degree B-spline basis

Transforming formulas

Properties of the bi-degree B-spline basis

The bi-degree B-spline curve

The bi-degree B-spline The definition of bi-degree B-spline basis Initial functions over ( ) jT

1

2,0

2 1

1

0

ji

j ji i

jj ii j j

i i

t t

t t

t tN t

t t

if and 1j ji it t t 1 ji l

if and 1 2j ji it t t 1 1ji l

if and 1j ji it t t ji l

otherwise

1 2j jl z z z j

The bi-degree B-splineFor

1k

, , 1 , 1 1, 1 1, 1

tj j j j ji k i k i k i k i kN t N s N s ds

where

1

, ,j ji k i kN t dt

By the definition we know that are bi-degree B-spline basis functions: in they are -degree, and in they are k-degree.

,ji kN t

1 1, jt t

1k 1 ,j nt t

The bi-degree B-spline Theorem 1 Assume that , are the usualB-spline basis functions defined on and and , are basis functionsdefined on and , then and .

,i kN t , 1i kN t

T T 0

,i kN t 1,ni kN t

0T T 1nT T 0, ,i k i kN t N t

1, 1 ,

ni k i kN t N t

The bi-degree B-spline Transforming formulas Noting that 1

11

jij

i ji

tt

t

We have

,0

1,0 ,0 1

1,0

ji

j j ji i i

ji

N t

N t N t N t

N t

1ji l

ji l

1ji l

1ji l

1ji l

The bi-degree B-spline The initial basis functions

and 1

,0jiN t

,0jiN t

The bi-degree B-spline Theorem 2 For the bi-degree B-spline basis functions and , we have 1

,ji kN t ,

ji kN t

,

1, , , 1, 1,

1,

1

ji k

j j j j ji k i k i k i k i k

ji k

N t

N t a N t a N t

N t

1ji l k

1 1j jl k i l 1ji l

where ,0 0 1 1ji j ja l k i l

1, 1j

jl ka

1, ,, 1

, 1, 1, ,

1 11

j ji k i kj

i k j jj j j ji k i k i k i k

aa l k i l

a a

The bi-degree B-spline Proof: When k=0, it is obvious. Assume it holds for , then

0k

1, , , 1, 1,1j j j j ji k i k i k i k i kN t dt a N t a N t dt

So

, 1,1

,

, 1, 1, ,1

j ji k i kj

i k j j j ji k i k i k i ka a

We have

The bi-degree B-spline

, 1,

, , 1, 1,

, 1, 1, ,

2, 1,1, 1, 2, 2,

1, 2, 2, 1,

1, ,

, 1, 1

1

1

1

11

j jti k i k j j j j

i k i k i k i kj j j ji k i k i k i k

j jti k i k j j j j

i k i k i k i kj j j ji k i k i k i k

j ji k i k

j ji k i k i

aN s N s ds

a a

aN s N s ds

a a

a

a a

2, 1,, 1 1, 1

, , 1, 2, 2, 1,

, 1 , 1 1, 1 1, 1

1

1

j ji k i kj j

i k i kj j j j j jk i k i k i k i k i k

j j j ji k i k i k i k

aN t N t

a a

a N t a N t

1 1 1 1 1, 1 , , 1, 1,

tj j j j ji k i k i k i k i kN t N s N s ds

The bi-degree B-spline Properties of the bi-degree basis 1)Differential: is time

continuously differential at the knot with denotes the multiplicity of the knot .

2) Partition of unity: 3)Derivative:

,ji kN t jlk r

jlt

jltj

lr

'

, , 1 , 1 1, 1 1, 1j j j j ji k i k i k i k i kN t N t N t

, 1ji kiN t

The bi-degree B-spline4)Positivity: for ( and ) or ( and )

, 0ji kN t 1,j j

i i kt t t 2ji l k 1j ji k it t

2,j ji i kt t t 2ji l k 2

j ji k it t

5)Linear independence: are linearly independence on

,ji kN t

jT

The bi-degree B-spline The bi-degree B-spline curve , 1

0

( ) ( )n

j j ji k i n

i

P t N t P t t t

Property: in it is (k+1)-degre curve,

and in it is k-degree curve.

Other : convex hull, geometric invariance ,

local control, variation diminishing.

1 1, jt t

1 ,j nt t

Corner cutting Theorem 3 If , are bi-degree B-spline

curves defined on , , and they are the same curves, then their control points

, satisfy

1

1 1, , 1

11

1

ji

j j j j ji i k i i k i

ji

P

P a P a P

P

1ji l k

1 1j jl k i l

1ji l

1jP t jP t1jT jT

1jiP jiP

Corner cutting

Theorem 4 The degree elevation of B-splinecurves is corner cutting.

Corner cutting

Corner cutting

An example of degree elevation.A cubic B-spline which is defined by

and knot vector 0,1, ,4iP i

0,0,0,0,0.4,1,1,1,1

Conclusions and future work Conclusions: In this paper we have presented

the theory of bi-degree B-spline.Using it we prove that degree elevation of B-spline curve can be interpreted as corner cutting.

Future work: 1)computing the explicit coefficients of the

corner cutting. 2)to investigate more properties and applications of the bi-degree B-spline.