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B-Spline

B-Splines

  • Is it possible to achieve both C2-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
  • C2-continuity
  • Local Controllability

 

Demo

 

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
      • C2-continuity
      • No Local Controllability
    • Catmull-Rom Spline
      • C1-continuity
      • Local Controllability
    • B-Spline
      • C2-continuity
      • Local Controllability
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📖 Contents 📖
B-SplineB-SplinesUniform B-spline basis functionsUniform B-splinesB-spline PropertiesDemoSummary