1

MAT4170/9170 - Lecture 5 - 14/2-2018

Last time (2.3-2.5, 3.1)• Spline control polygon• Spline functions and curves• Matrix notation• Evaluation• B-spline matrix properties• Dual polynomials

Today (3.1-3.2)• Oblig 1• Marsdens identity• Representation of polynomials• Local linear independence• Global linear independence• Differentiation and smoothness

2

Dual polynomials and B-spline matrices

3

Marsdens identity

If we set y=0, we get

Therefore

So, we can express the polynomial xd as a linear combination of B-splines of degree d

Can differentiate (3.10) wrt y and set y=0 to obtain xd-1 as a linear combination of B-splines of degree d

Can be re-arranged to get

For certain B-spline coefficients

4

Representation of polynomials by B-splines

5

The four active B-splines

span the 4-dimensional

space of polynomials

on the interval

6

Local linear independence

7

Global linear independence

8

Piecewise continuity and smoothness (3.2)

9

Derivatives of B-splines

10

Derivatives of B-splinesUse the previous result to differentiate

where

11

Use this to simplify the derivative

Proof: Differentiate the following wrt z:

which can be differentiated again. In general

So, the r'th derivative is a spline of degree d-r

12