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Covering Rotations in R 3 using Quaternions Abhijit Guria Advisor: Herbert Edelsbrunner

Covering Rotations in R 3 using Quaternions Abhijit Guria Advisor: Herbert Edelsbrunner. Motivation. Protein-Protein docking Rotate and fit. Outline of the Talk. Representing rotations Distance between rotations Measures of evenness Starting configurations

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Covering Rotations in R 3 using Quaternions Abhijit Guria Advisor: Herbert Edelsbrunner

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  1. Covering Rotations in R3 using Quaternions Abhijit Guria Advisor: Herbert Edelsbrunner

  2. Motivation • Protein-Protein docking • Rotate and fit

  3. Outline of the Talk • Representing rotations • Distance between rotations • Measures of evenness • Starting configurations • Samples refinement heuristics • Local improvement heuristics • Summary of results • Open questions

  4. Representing Rotations • Unit quaternion, q = (q1,q2,q3,q4) • Antipodal pair in S3

  5. Distance Between Rotations • Shorter geodesic • Approximation by Euclidean Distance

  6. Measures of “Evenness” • Covering radius • RMS error • 3D-Area of Convex hull • 4D-Volume of Convex hull • Packing radius • Triangulation properties

  7. Starting Configurations • Parameterization • Random • 120-tope

  8. Samples Refinement Heuristics Step 1: Start with a configuration Iterate N times Step 2: Compute the Delaunay triangulation on S3 (the convex hull). Step 3: Add new samples based on this triangulation. End iteration

  9. Refining with Edgewise Subdivision

  10. Refining with Voronoi Sequence

  11. Local Improvement Heuristics Step 1: Start with a configuration Iterate N times Step 2: Compute the Voronoi diagram on S3 (the dual of convex hull). Step 3: Update each sample locally End iteration

  12. Improving Covering Radius Locally

  13. Improving Volume Locally

  14. Improving RMS Error Locally

  15. Summary of Results • After refinement and local improvement covering density is around 2. • The variation in improved covering density across different starting configuration is small. • We are able to generate good samples as large as nearly 50,000 rotations.

  16. Questions • Is covering density comparable to B.C.C. achievable at all for small samples? • Given a tetrahedron, can it be cut into similar (need not be of same size) tetrahedrons? • How to tile S3using tetrahedrons of given shape(s)?

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