splines, version 0.2.0
Polynomial Curves in Euclidean Space
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Splines in Euclidean Space and Beyond
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Polynomial Parametric Curves
Polynomial Curves in Euclidean Space
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Polynomial Parametric Curves
Lagrange Interpolation
One-dimensional Example
Neville’s Algorithm
Two-dimensional Example
Runge’s Phenomenon
Hermite Splines
Properties
Derivation (Uniform)
Basis Matrix
Basis Polynomials
Example Plot
Relation to Bézier Splines
Derivation (Non-Uniform)
Basis Matrix
Basis Polynomials
Example Plot
Utilizing the Uniform Basis Matrix
Natural Splines
Properties
Derivation (Uniform)
End Conditions
Natural
Clamped
Closed
Solving the System of Equations
Derivation (Non-Uniform)
End Conditions
Bézier Splines
Properties
De Casteljau’s Algorithm
Preparations
Degree 1 (Linear)
Degree 2 (Quadratic)
Quadratic Tangent Vectors
Degree 3 (Cubic)
Cubic Tangent Vectors
Cubic Bézier to Hermite Segments
Degree 4 (Quartic)
Quartic Tangent Vectors
Arbitrary Degree
Non-Uniform (Cubic) Bézier Splines
Tangent Vectors
Control Points From Tangent Vectors
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
Using Bézier Segments
Derivation (Non-Uniform)
Tangent Vectors
Using Non-Uniform Bézier Segments
Animation
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
Basis Matrix
Basis Polynomials
Derivation (Non-Uniform)
End Conditions
Natural End Conditions
Begin
End
Example
Bézier Control Points
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
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Splines in Euclidean Space and Beyond
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Polynomial Parametric Curves
Table of Contents
Polynomial Curves in Euclidean Space
Polynomial Parametric Curves
Lagrange Interpolation
Hermite Splines
Natural Splines
Bézier Splines
Catmull–Rom Splines
Kochanek–Bartels Splines
End Conditions
Piecewise Monotone Interpolation
Rotation Splines
Python module
References
General Index