B-Spline

B-Splines

Is it possible to achieve both $C^{2}$-continuity and local controllability?
- B-splines can do!
Uniform cubic B-spline basis functions

Uniform B-spline basis functions

Bell-shaped basis function for each control points

Overlapping basis functions
- Control points correspond to knot points

Uniform B-splines

We have $(n+1)+2$ unknowns.

B-spline Properties

Variation Diminishing
$C^{2}$-continuity
Local Controllability

Demo

https://www.ibiblio.org/e-notes/Splines/basis.html

B-spline basis functions

B-spline basis functions The equation for k-order B-spline with n+1 control points (P0 , P1 , ... , Pn ) is P(t) = ∑i=0,n Ni,k(t) Pi , tk-1 ≤ t ≤ tn+1 . In a B-spline each control point is associated with a basis function Ni,k which is gi

www.ibiblio.org

Summary

Polynomial Interpolation
- Lagrange Polynomial
- Oscillation Problem
Spline Interpolation
- Natural Cubic Spline
  - $C^{2}$-continuity
  - No Local Controllability
- Catmull-Rom Spline
  - $C^{1}$-continuity
  - Local Controllability
- B-Spline
  - $C^{2}$-continuity
  - Local Controllability

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