Polynomial Curves in Euclidean Space

This section is mostly about different types of univariate non-rational polynomial splines in one-, two- and three-dimensional Euclidean space – for an application in a four-dimensional space, see the section about 4D quaternion interpolation.

But before diving into splines – and before even defining what they are – we will discuss a few basics about polynomial curves and a spline-less interpolation method called Lagrange interpolation.

Many of the approaches shown in this section will later be adapted to the context of rotation splines.

• Parametric Polynomial Curves
• Lagrange Interpolation
  • One-dimensional Example
  • Neville’s Algorithm
  • Two-Dimensional Example
  • Runge’s Phenomenon
• Splines
  • Definition
  • Properties
  • Types
• 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
• Quadrangle Interpolation
  • Basis Polynomials
  • Basis Matrix
  • Tangent Vectors
  • Quadrangle to Hermite Control Values
  • Quadrangle to Bézier Control Points
  • Non-Uniform Parameterization
• Catmull–Rom Splines
  • Properties
    • Tangent Vectors
    • Wrong 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
    • Using Quadrangle Interpolation
  • Derivation (Non-Uniform)
    • Tangent Vectors
    • Using Non-Uniform Bézier Segments
    • Using Non-Uniform Quadrangle Interpolation
    • 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
• Re-Parameterization
  • Arc-Length Parameterization
  • Spline-Based Re-Parameterization