Efficiently Computing Geodesic loop for Interactive Segmentation of 3D Mesh

Introduction: Shape segmentation is a fundamental problem of computer graphics and geometric modeling. Although the existence segmentation algorithms of shapes have been widely studied in mathematics community, little progress has been made on how to compute them on polygonal surfaces interactively using geodesic loops. Method: We compute the geodesic distance fields with improved Fast March Method (FMM) proposed by Xin and Wang. We propose a new algorithm to compute geodesic loops over a triangulate surface and a new interactive shape segmentation manner on triangulate surface. Result: The average computation time on 50K vertices model is less than 0.08s. Discussion: In the future, we will use an accurate geodesic algorithm and parallel computing techniques to improve our algorithm to obtain better smooth geodesic loop. Conclusion: A large number of experimental results show that the algorithm proposed in this paper can effectively achieve high precision geodesic loop paths, and our method can also be used to interactive shape segmentation in real time.

ENGLERT-TYPE SOLUTIONS OF d = 11 SUPERGRAVITY

A family of geometries on S7 arise as solutions of the classical equations of motion in 11 dimensions. In addition to the conventional Riemannian geometry and the two exceptional Cartan–Schouten compact flat geometries with torsion, one can also obtain non-flat geometries with torsion. This torsion is given locally by the structure constants of a non-associative geodesic loop in the affinely connected space.

SOME NUMERICAL INVARIANTS OF FLAT VIRTUAL LINKS ARE ALWAYS REALIZED BY REDUCED DIAGRAMS

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.