Lecture 8 - 10.03.2016
• Last time: knot insertion
• Today: more knot insertion: 4.2-4.4
A=
Computing discrete B-splines
Proof:
Reccurence for discrete B-splines
Discrete B-splines
B-splines
The Oslo-algorithms
The Oslo-algorithms
Knot insertion example: p=2
Observation
The B-spline coefficients are functions of the knots!
In particular:
Duality
Multi-affine functions
Affine functions in one variable
Characterized by
Affine functions in two variables
Affine functions in three variables
In general 2p terms in affine functions of p variables
Multi-affine functions
Symmetric affine functions
In general p+1 terms
The Blossom
Example: g(x)=x
Blossoms of monomials
Example: g(x)=x2
) = x1x2
(x1x2 + x1x3 + x2x3)/3
Proof:
(4.23) Show that the RHS is the blossom for k=p. Differentiate p-k times wrt y
(4.24) Show that the RHS is the blossom
Blossoms of B-splines
Proof:
1.
2.
3.
Each element of Rk(xi) is affine in xi
Symmetry by (3.7)
Diagonal property holds
Proof: Fix j. Show that one can pick any k s.t. k=j,...,j+p and then
and so one can compute cj from fk