Rotation Splines

There are many ways to implement rotation splines. Here we use unit quaternions to represent rotations. First, we’ll show what quaternions are, and how their subset of unit quaternions can be used to handle rotations. Based on a special form of linear interpolation called Slerp, we then use several algorithms that we have seen in the section about Euclidean splines – which all utilize linear interpolations (and extrapolations) – to implement rotation splines. In the end of this section, we present a few methods which are not based on Slerp, but it will turn out that they all have severe limitations.

• Quaternions
  • Quaternion Representations
  • Unit Quaternions
  • Unit Quaternions as Rotations
  • Axes Conventions
  • Quaternion Multiplication
  • Inverse
  • Relative Rotation (Global Frame of Reference)
  • Relative Rotation (Local Frame of Reference)
  • Exponentiation
  • Negation
  • Canonicalization
• Spherical Linear Interpolation (Slerp)
  • Derivation
  • Visualization
  • Piecewise Slerp
  • Slerp vs. Nlerp
• De Casteljau’s Algorithm With Slerp
  • “Cubic”
  • Arbitrary “Degree”
  • Constant Angular Speed
  • Joining Curves
• Uniform Catmull–Rom-Like Quaternion Splines
  • Relative Rotations
  • Tangent Space
  • Example
  • Shoemake’s Approach
• Non-Uniform Catmull–Rom-Like Quaternion Splines
  • Parameterization
• Kochanek–Bartels-like Rotation Splines
  • Examples
• “Natural” End Conditions
  • Examples
• Barry–Goldman Algorithm With Slerp
  • Constant Angular Speed
• Spherical Quadrangle Interpolation (Squad)
  • Non-Uniform Parameterization
• Cumulative Form
  • Piecewise Slerp
  • Cumulative Bézier/Bernstein Curve
  • Comparison with De Casteljau’s Algorithm
• Naive 4D Quaternion Interpolation
• Naive Interpolation of Euler Angles