B-Splines [toc]

Demos

These demos illustrate some uses of the splinekit.bsplines module.

The demos rely on the Jupyter Lab deployment of Python. They run in-browser and do not necessitate anything to be installed beforehand. Edits to the notebooks will persist for a time in the cache of your browser. This duration is ephemeral and uncontrolled; dot not rely on it for persistent storage.

Hint

• To discard your modifications and revert to the notebooks as distributed, choose the entry <   ⃠ Clear Browser Data> from the menu <Help>.

• To run code, click on a notebook cell and hit the <SHIFT-CR> keys ⇧⏎.

Contents:

• Hello B-Spline Family!
  • A Family of B-Splines
  • Gradient
  • Integral
• B-Spline Members
  • Polynomial B-spline
  • Polynomial B-Spline Derivatives
  • Polynomial B-Spline Integral
• Numeric Stability and Speed of B-Splines
  • Splines
  • B-Splines
    • Classic Formula
    • De Boor
    • splinekit
  • Experiments
    • Critical Regime
      • Results
      • Discussion for the Critical Regime
    • Global Regime, Low Degrees
      • Results
      • Discussion for the Global Regime, Low Degrees
    • Global Regime
      • Results
      • Discussion for the Global Regime
• M-Scale Relation of B-Splines
• B-Spline vs \(\pi\)
  • Value of a B-Spline at the Origin
  • Alternating Sum of Samples
• Splines, B-Splines, and Polynomials
  • Splines
  • Polynomials
    • Polynomial of Degree \(\left(-1\right)\)
    • Polynomials of Degree \(0\) and Partition of Unity
    • Partition of Monomials
    • Spline Coefficients Made of Discrete Monomials
  • Monomial Convolved with a B-Spline
• Splines and Engineering Functions
  • Sine
  • Cosine
  • Interlude—Trigonometric Exercise
  • Exponential
  • Gauss Error Function
  • Interlude—Gaussian Exercise
• Spline Bases
  • Roadmap
  • Representation of Numbers
    • Uniform Splines
    • Bases
  • B-Spline
  • Cardinal B-Spline
    • Relation to the Cardinal Sine Function
  • Dual B-Spline
    • Defining Property of Dual B-Splines
  • Orthonormal B-Spline
    • Defining Property of Orthonormal B-Splines
• B-Spline Poles
  • Poles
  • Imbrication of Poles
  • Annihilating Sequences
  • Pole Values
    • Explicit Expressions
      • Quadratic Pole
      • Cubic Pole
      • Quartic Poles
      • Quintic Poles
      • Sextic Poles
      • Septimic Poles
      • Octic Poles
      • Nonic Poles
• Inverse of B-Spline Sequences
  • Usefulness
  • Non-Periodic Case
  • Periodic Case