1. Shortest Geodesics on Polyhedral Surfaces Project Summary Efrat Barak

2. Introduction Project Objective Mathematical Review The New Theoretical Algorithms The Database Logical Algorithms Results Conclusions Suggestions for Future Projects

3. Project Objectives: 1.Implement two new algorithms for calculating geodesics on a polyhedral surface. 2. Confirm the equivalence of the algorithms.

4. 3. Find the straightest shortest line for cutting the cylindrical surface in order to span it to a rectangular. 4. Evaluate the relation between the sides of the rectangular in several methods. Project Objectives:

5. Mathematical Review Definition: Let M be a smooth two- dimensional surface. A smooth curve with is a geodesic if one of the equivalent properties holds: 1. is a locally shortest curve. 2. is parallel to the surface normal. 3. has vanishing geodesic curvature

6. Mathematical Review On polyhedral surfaces, the concepts of shortest and straightest geodesics are equivalent only locally. Straightest geodesics solve uniquely the initial value problem on polyhedral surfaces.

7. The New Algorithms Algorithm A: Projecting Neighboring Triangles on the Plane of the Current Triangle

8. The New Algorithms Algorithm B: Calculation of the Angles to the Neighboring Vertices

9. The Database CT Scans

10. The 3D-Slicer

11. MATLAB Representation of the raw data:

12. Logical Algorithms Triangulation Algorithm Finding Neighbor Triangles Algorithm Finding Edge Triangles and Vertices Algorithm Calculating the Cylinder Edges’ Lengths Algorithm

13. The Triangulation Algorithm 1. Cutting the cylinder of samples 2. Projecting each of the halves of the cylinder on the x-z plane

14. The Triangulation Algorithm 3. Triangulating the samples points on the plane 4. Reshaping the plane to it’s former form

15. Finding Edge Triangles and Vertices Algorithm

17. An Algorithm for Calculating the Cylinder Edges’ Lengths The Problem:

18. An Algorithm for Calculating the Cylinder Edges’ Lengths The Solution:

19. Results Results of Algorithm A:

20. Results Results of Algorithm A:

21. Results Results of Algorithm B:

22. Comparison of Algorithms Algorithm A: Algorithm B:

23. Comparison of Algorithms Algorithm A: Algorithm B:

24. Results An Edge Case:

25. The Straightest Shortest Curve

26. Definition of a new parameter that measures that straightness of a curve: L - The length of the curve D - The distance between the first and the last points of the curve q was the smallest for the shortest curve !

27. Calculation of the Module M(Q) of the Rectangular a – The length of the longer side of the rectangle b – The length of the shorter side of the rectangle Definitions: – The length of the straightest geodesic – The average length of the edges of the cylinder

28. Calculation of the Module M(Q) of the Rectangular Method A: Method B: Method C: Method D:

29. Conclusions The two new algorithms are highly suited for calculating straightest curves on polyhedral surfaces The two algorithms are equivalent

30. Conclusions The straightest curve that was found on the polyhedral cylinder was also the shortest. Methods A, B and D for calculating M(Q) are quite accurate

31. Suggestions for Future Projects Implement the two algorithms for calculating the straightest curve on a very large polyhedral surface. Implement a non-linear transformation that would span the cylinder into a rectangular

32. THE END