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Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads
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Efficiently computing exact geodesic loops within finite steps.

Shi-Qing Xin1, Ying He, Chi-Wing Fu

  • 1Nanyang Technological University, Singapore. sqxin@ntu.edu.sg

IEEE Transactions on Visualization and Computer Graphics
|June 22, 2011
PubMed
Summary
This summary is machine-generated.

This study introduces the first algorithm to find exact geodesic loops on polygonal surfaces, offering an efficient method for computing these crucial geometric structures. The new approach significantly outperforms existing techniques in speed and accuracy.

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Area of Science:

  • Differential Topology
  • Differential Geometry
  • Computational Geometry

Background:

  • Closed geodesics are fundamental in topology and geometry but challenging to compute on polygonal surfaces.
  • Existing methods often approximate geodesics or restrict them to mesh edges, failing to capture true geodesic behavior.

Purpose of the Study:

  • To prove the existence and uniqueness of geodesic loops on closed face sequences.
  • To develop an efficient algorithm for computing exact geodesic loops on polygonal meshes.

Main Methods:

  • An iterative algorithm that evolves an initial closed path into an exact geodesic loop.
  • The method avoids numerical solvers and differential equations, directly applicable to triangular meshes.

Main Results:

  • The first proof of existence and uniqueness for geodesic loops on face sequences.
  • An algorithm with O(k) space and O(mk) experimental time complexity.
  • Interactive speed performance, achieving results in milliseconds for large meshes.

Conclusions:

  • The proposed algorithm provides an exact and efficient solution for computing geodesic loops on polygonal surfaces.
  • This method significantly advances the state-of-the-art in computational geometry for geodesic computations.
  • Potential applications include interactive shape segmentation and other geometric modeling tasks.