Splines in Euclidean Space and Beyond

… with focus on univariate, non-uniform piecewise cubic polynomial curves in one, two and three spatial dimensions, as well as rotation splines.

• Installation:

  python3 -m pip install splines
• Online documentation:

  https://splines.readthedocs.io/
• Documentation notebooks on Binder:

  https://mybinder.org/v2/gh/AudioSceneDescriptionFormat/splines/master?filepath=doc/index.ipynb
• Source code repository (and issue tracker):

  https://github.com/AudioSceneDescriptionFormat/splines
• License:

  MIT – see the file LICENSE for details.

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• Polynomial Curves in Euclidean Space
  • Polynomial Parametric Curves
  • Lagrange Interpolation
    • One-dimensional Example
    • Neville’s Algorithm
    • Two-dimensional Example
    • Runge’s Phenomenon
  • Hermite Splines
    • Properties
    • Derivation (Uniform)
      • Relation to Bézier Splines
    • Derivation (Non-Uniform)
  • Natural Splines
    • Properties
    • Derivation (Uniform)
      • End Conditions
        • Natural
        • Clamped
        • Closed
      • Solving the System of Equations
    • Derivation (Non-Uniform)
      • End Conditions
        • Natural
        • Other End Conditions
  • Bézier Splines
    • Properties
    • De Casteljau's Algorithm
      • Preparations
      • Degree 1, a.k.a. linear
      • Degree 2, a.k.a. quadratic
      • Degree 3, a.k.a. cubic
      • Degree 4, a.k.a. quartic
      • Arbitrary Degree
  • Catmull--Rom Splines
    • Properties
      • Tangent Vectors
      • Cusps and Self-Intersections
      • Chordal Parameterization
      • Centripetal Parameterization
      • Parameterized Parameterization
    • Derivation (Uniform)
      • Blending Functions
      • Cardinal Functions
      • Example Plot
      • Basis Polynomials
      • Basis Matrix
      • Tangent Vectors
    • Derivation (Non-Uniform)
      • Tangent Vectors
      • Basis Matrix
    • Barry--Goldman Algorithm
      • Triangular Schemes
      • Neville’s Algorithm
      • De Boor’s Algorithm
      • Combining Both Algorithms
      • Step by Step
        • First Stage
        • Second Stage
        • Third Stage
      • Tangent Vectors
      • Animation
  • Kochanek--Bartels Splines
    • Properties
      • Tension
      • Continuity
      • Bias
      • Combinations
    • Derivation (Uniform)
      • Parameters
        • Tension
        • Continuity
        • Bias
        • All Three Combined
      • Calculation
    • Derivation (Non-Uniform)
      • Parameters
      • Calculation
  • End Conditions
    • Natural End Conditions
      • Begin
      • End
      • Example
  • Piecewise Monotone Interpolation
    • Examples
      • Providing Slopes
    • Generating and Modifying the Slopes at Segment Boundaries
    • PCHIP/PCHIM
    • More Examples
    • Monotone Interpolation
    • End Conditions
    • Even More Examples
• Rotation Splines
  • Quaternions
    • Quaternion Representations
      • Algebraic
      • Scalar and Vector
      • 4D Space
      • More Representations
    • Quaternion Operations
    • Unit Quaternions as Rotations
    • Unit Quaternion Operations
      • Relative Rotation (Global Frame of Reference)
      • Relative Rotation (Local Frame of Reference)
  • Spherical Linear Interpolation (Slerp)
    • Derivation
    • Visualization
    • Piecewise Slerp
    • Slerp vs. Nlerp
  • De Casteljau's Algorithm
    • “Cubic”
    • Arbitrary “Degree”
    • Constant Angular Speed
    • Joining Curves
    • Joining Curves, Another Approach
  • Uniform Catmull--Rom-Like QuaternionSplines
  • Non-Uniform Catmull--Rom-Like QuaternionSplines
    • Parameterization
  • Kochanek--Bartels-like Rotation Splines
    • Uniform Catmull–Rom
    • Non-Uniform Catmull–Rom
    • TCB
    • Edge Cases
  • "Natural" End Conditions
    • Begin
    • End
    • (Non-)Symmetries
  • Barry--Goldman Algorithm
    • Constant Angular Speed
  • Cumulative Form
    • Piecewise Slerp
    • Cumulative Bézier/Bernstein Curve
    • Comparison with De Casteljau’s Algorithm
  • Naive 4D Quaternion Interpolation
  • Naive Interpolation of EulerAngles
• Python module
  • splines
  • splines.quaternion
• External Resources
• References