On the degree elevation of B-spline curves and corner cutting 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 Degree elevation of B-spline curves. CAGD 1( ) Prautzsch,H.,Piper,B A fast algorithm to raise the degree of B-spline curves. CAGD 8( ) Barry P.J.,Goldman R.N A recursive proof of a B-spline identity for degree elevation. CAGD 5( ) Liu,W A simple,efficient degree raising algorithm for B- spline curve. CAGD 14( ) Huang,Q.,Hu S.,Martin,R Fast degree elevation and knot insertion for B-spline curves. CAGD 22( ) Sederberg T.W., Zheng J., Song X., Knot intervals and multi-degree splines. CAGD 20( )

Problem The traditional method The new method The advantages

Problem and Approach Problem: Given a B-spline curve of degree k: To elevate the degree to k+1

Problem and Approach The traditional method: Step1 Update to Step2 are represented by (k+1)-degree B- spline basis functions. Step3 The control points are computed according the transforming formulas between and

Problem and Approach The new method: In each step we only increase the multiplicity of 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,and elevate the degree of the basis functions only in knot interval.

Problem and Approach Advantage of the method: 1) Obtain more simply formulas 2) The degree elevation algorithm can be interpreted as corner cutting algorithm.

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 ( ) if and otherwise

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

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

The bi-degree B-spline Transforming formulas Noting that We have

The bi-degree B-spline The initial basis functions and

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

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

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:

The bi-degree B-spline 4)Positivity: for ( and ) or ( and ) 5)Linear independence: are linearly independence on

The bi-degree B-spline The bi-degree B-spline curve Property: in it is (k+1)-degre curve, and in it is k-degree curve. Other : convex hull, geometric invariance, local control, variation diminishing.

Corner cutting Theorem 3 If, are bi-degree B- spline curves defined on,, and they are the same curves, then their control points, satisfy

Corner cutting Theorem 4 The degree elevation of B-spline curves is corner cutting.

Corner cutting

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

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.