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