Quaternion Gauss Maps and Optimal Framings of Curves and Surfaces

We propose a general paradigm for generating optimal coordinate frame fields that may be exploited to annotate and display curves and surfaces. Parallel-transport framings, which work well for open curves, generally fail to have desirable properties for cyclic curves and for surfaces. We suggest that minimal quaternion measure provides an appropriate generalization of parallel transport. Our fundamental tool is the “quaternion Gauss map,” a generalization to quaternion space of the tangent map for curves and of the Gauss map for surfaces. The quaternion Gauss map takes 3D coordinate frame fields for curves and surfaces into corresponding curves and surfaces constrained to the space of possible orientations in quaternion space. Standard optimization tools provide application-specific means of choosing optimal, e.g., lengthor area-minimizing, quaternion frame fields in this constrained space. We observe that some structures may have distinct classes of minimal quaternion framings, e.g, one disconnected from its quaternion reflection, and another that continuously includes its own quaternion reflection. We suggest an effective method for visualizing the geometry of quaternion maps that is used throughout. Quaternion derivations of the general moving-frame equations for both curves and surfaces are given; these equations are the quaternion analogs of the Frenet and Weingarten equations, respectively. We present examples of results of the suggested optimization procedures and the corresponding tubings of space curves and sets of frames for surfaces and surface patches.