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Shortest Geodesics on Polyhedral Surfaces
Project Summary
Efrat Barak
Introduction
• Project Objective• Mathematical Review• The New Theoretical Algorithms• The Database• Logical Algorithms • Results• Conclusions• Suggestions for Future Projects
Project Objectives:
1. Implement two new algorithms for calculating geodesics on a polyhedral surface.
2. Confirm the equivalence of the algorithms.
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:
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
MI : 1'
''
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.
The New AlgorithmsAlgorithm A: Projecting Neighboring Triangles on the Plane of the Current
Triangle
The New Algorithms
Algorithm B: Calculation of the Angles to the Neighboring Vertices
The Database
CT Scans
The 3D-Slicer
MATLAB
Representation of the raw data:
Logical Algorithms
• Triangulation Algorithm
• Finding Neighbor Triangles Algorithm
• Finding Edge Triangles and Vertices Algorithm
• Calculating the Cylinder Edges’ Lengths Algorithm
The Triangulation Algorithm
1. Cutting the cylinder of samples 2. Projecting each of the
halves of the cylinder on the x-z plane
The Triangulation Algorithm
3. Triangulating the samples points on the plane
4. Reshaping the plane to it’s former form
Finding Edge Triangles and Vertices Algorithm
Finding Edge Triangles and Vertices Algorithm
An Algorithm for Calculating the Cylinder
Edges’ Lengths
The Problem:
An Algorithm for Calculating the Cylinder Edges’ Lengths
The Solution:
Results
Results of Algorithm A:
Results
Results of Algorithm A:
Results
Results of Algorithm B:
Comparison of Algorithms
Algorithm A: Algorithm B:
Comparison of Algorithms
Algorithm A: Algorithm B:
ResultsAn Edge Case:
The Straightest Shortest Curve
The Straightest Shortest Curve
Definition of a new parameter that measures that straightness of a curve:
L - The length of the curveD - The distance between the first and the last
points of the curve
q was the smallest for the shortest curve !
D
Lq
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
( )a
M Qb
1L
2L
1 2
1 2
min ,
max ,
s
l
l L L
l L L
Calculation of the Module M(Q) of the
Rectangular
Method A:
Method B:
Method C:
Method D:
2
2
ln 1 2 1 2ln 1 2
( )
1 2ln 1 2 ln 1 2
l l
s s
l s
s l
l l
l lM Q
l ll l
2( ) ( ) 21.26
s
AreaM Q M Q
l
30.52 ( ) 2.7 10M Q
( ) ( ) 19.28l
s
lM Q M Q
l
2
2 )()(
)( l
s
l
QAreaQM
QArea
l 26.21)(497.17 QM
Conclusions
• The two new algorithms are highly suited for calculating straightest curves on polyhedral surfaces
• The two algorithms are equivalent
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
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
THE END