The Morgan Kaufmann Series in Interactive 3D Technology: Visualizing Quaternions

Chapter 23. Quaternion Volumes

We have now treated in detail two fundamental families of frames: the 1D family of frames corresponding to points on a curve carrying a continuous frame, with one axis following the curve’s tangent vector and the 2D family of frames attached to a surface, with one axis following the normal vector perpendicular to the surface at each point. By transforming each of these families of frames to quaternion form, we achieved a map from each point in the manifold to a manifold of the same dimension in quaternion space. Because the metric properties of quaternion space permit quantitative evaluation of proximity among frames, the geometry of these maps reveals both local and global properties of the frame assignments to ...