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Exact geodesics and shortest paths on polyhedral surfaces.
Mukund Balasubramanian1, Jonathan R Polimeni, Eric L Schwartz
1Department of Cognitive and Neural Systems, Boston University, 677 Beacon Street, Boston, MA 02215, USA. mukundb@cns.bu.edu
Summary
We developed algorithms for calculating exact geodesic and shortest path distances on polyhedral surfaces. These methods are accurate, efficient, and suitable for large-scale surface analysis and flattening.
Area of Science:
- Computational geometry
- Computer graphics
- Geometric modeling
Background:
- Calculating distances on complex 3D surfaces is crucial for various applications.
- Existing methods may lack accuracy or efficiency for large-scale polyhedral models.
Purpose of the Study:
- To present novel algorithms for computing exact geodesic and shortest path distances on polyhedral surfaces.
- To enable accurate surface flattening and analysis for complex geometric models.
Main Methods:
- Developed two algorithms: one for minimal-geodesic distances, another for shortest-path distances.
- Extended algorithms to compute exact paths, not just distances.
- Implemented and validated algorithms on known surfaces for accuracy and performance.
Main Results:
- Algorithms compute exact minimal-geodesic and shortest-path distances and paths.
- Run-time performance is cubic or less.
- Feasible for large-scale surfaces with tens of thousands of vertices.
Conclusions:
- The presented algorithms provide accurate and efficient solutions for distance computations on polyhedral surfaces.
- These methods are essential for advanced surface flattening techniques.
- Enables precise transformation of curved manifolds into flat representations.
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