Reading Notes on B-splines

Reading Notes on
B-splines

Wenjia Bai

November 28, 2008

Title: http://en.wikipedia.org/wiki/B-spline
Title: Applied Numerical Analysis
Author: Curtis F. Gerald, Patrick O. Wheatley
Publisher: Addison Wesley
ISBN: 020187072X 978-0201870725

1 B-spline

A spline is a special function defined piecewise by polynomials.
A B-spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition.
- Given m + 1 knots t_i [0, 1] with
  $t0 ≤ t1 ≤ ...≤ tm$
  
  a B-spline of degree n with m + 1 knots is a parametric curve,
  $2 B : [t0,tm] → ℝ$
  
  composed of basis B-splines of degree n,
  $m∑-n-1 B (t) = pibi,n(t),t ∈ [tn,tm-n] i=0$ (1)
  
  where p_i denotes the control point.
  The basis B-splines of degree n are defined using the Cox-de Boor recursion formula,
  ${ bi,0(t) := 1 if ti ≤ t < ti+1 (2) 0 otherwise -t --ti-- --ti+n+1---t-- bi,n(t) := t - t bi,n-1(t) + t - t bi+1,n-1(t) (3) i+n i i+n+1 i+1$
  
  Examples of basis B-splines are illustrated later.
- When the knots are equidistant we say the B-spline is uniform. Otherwise, we call it non-uniform. When the B-spline is uniform, the basis B-splines are just shifted copies of each other.
- When the number of knots is the same as the degree, the B-spline degenerates into a Bezier curve.
- Note that B-spilne are not really interpolating splines, because the curves do not normally pass throught all of the control points. They are an approximation of curves.
- Given the points p_i = (x_i,y_i), i = 0, 1,…,n, the uniform cubic B-spline for the interval (p_i,p_i+1), i = 1, 2,…,n - 1, is
  $⌊ - 1 3 - 3 1 ⌋ ⌊ p ⌋ 1[ ]| | | i- 1 | Bi (u) = --u3,u2, u,1 | 3 - 6 3 0 | | pi | 6 ⌈ - 3 0 3 0 ⌉ ⌈ pi+1 ⌉ 1 4 1 0 pi+2 uTM p = ----b-, 0 ≤ u ≤ 1 (4) 6$

2 Properties

A fundamental theorem states that every spline function of a given degree, smoothness and domain partition, can be represented as a linear combination of B-splines of that same degree and smoothness, and over the same partition.
B-splines are pieced together. They agree at their joints in three ways,
1. B_i(1) = B_i+1(0)
2. B_i^′(1) = B_i+1^′(0)
3. B_i^′′(1) = B_i+1^′′(0)
The portion of the curve determined by each group of four points is within the convex hull of these points.

3 Polynomial Approximation of Surfaces

A B-spline surface patch depends on 16 points,
$⌊ v ⌋ [ ] | 3 | xij(u,v ) = 1--u3, u2,u,1 MbXi,jM bT| v2 | 36 ⌈ v ⌉ 1$ (5)

where X_i,j is the 4×4 matrix,
$⌊ ⌋ | xi-1,j-1 xi-1,j xi-1,j+1 xi- 1,j+2 | | xi,j-1 xi,j xi,j+1 xi,j+2 | ⌈ xi+1,j-1 xi+1,j xi+1,j+1 xi+1,j+2 ⌉ xi+2,j-1 xi+2,j xi+2,j+1 xi+2,j+2$

The y and z equations are then obtained merely by substituting the corresponding matrices Y _i,j and Z_i,j.
$⌊ 3⌋ v 1--[ 3 2 ] T || v2 || yij(u,v) = 36 u ,u ,u,1 MbYi,jM b ⌈ v ⌉ (6) 1 ⌊ ⌋ v3 1--[ 3 2 ] T || v2 || zij(u,v) = 36 u ,u ,u,1 MbZi,jM b ⌈ v ⌉ (7) 1$

Because each of these equations is cubic in u and v, they are referred to as bicubic equations.