Abstract:
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., length- or area-minimizing, quaternion frame fields in this constrained space. We observe that some structures may have distinct classes of minimal quaternion framings, e.g,...
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