Chapter 20. Three-Dimensional Curves
The methods of 3D frames are taught as a basic area of study in differential geometry. However, classical methods do not take advantage of the quaternion techniques that can be used to clarify the nature of moving frames. In this chapter we review the classical differential geometry of moving frames on 3D space curves, and then introduce the equivalent quaternion formulation along with the quaternion tangent map, which allows us to see global properties of a framing choice at a single glance.
The fundamental difference between 2D space curves and 3D space curves is that although the tangent direction is still determinable directly from the space curve, there is an additional degree of rotational freedom in the ...
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