Constrained optimal framings of curves and surfaces using quaternion Gauss maps

We propose a general paradigm for computing optimal coordinate frame fields that may be exploited to visualize 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 heuristic generalization of parallel transport. Our approach differs from minimal tangential acceleration approaches due to the addition of "sliding ring" constraints that fix one frame axis, but allow an axial rotational freedom whose value is varied in the optimization process. 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., length- or area-minimizing, quaternion frame fields in this constrained space.