A survey on the study of Motion of Curves in Euclidean Space and the Motion Shrinking Space Curves to Plane Curves

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ABSTRACT

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the IL2 metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses.

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Table of Contents

ABSTRACT3

CHAPTER 1: INTRODUCTION5

History5

Triangles5

Points and Lines6

OBJECTIVE8

CHAPTER 2: LITERATURE SURVEY9

Geometric Development in Ancient Civilizations9

The Elements and Euclidean Geometry9

The Development of Analytic Geometry10

Non-Euclidean Geometry11

More Considerable Contributions to Geometry11

Hyperbolic Geometry12

Example:13

Solution:13

Taxicab Geometry13

Example 1:14

Solution:14

CHAPTER 3: DESIGN16

CHAPTER 4: ANTICIPATED OUTCOME18

REFERENCES19

CHAPTER 1: INTRODUCTION

History

Euclidean Geometry is based on five assumptions, or postulates.

A line can be drawn between any two points.

Any line segment can be extended to infinity in either direction.

A circle can be drawn with any given point as the center and with any given radius.

All right angles are equal (Bishop, 2007).

For any given line l, and any given point A, there is only one line through A that does not intersect l. For many years mathematicians tried to prove the fifth postulate, commonly known as the Parallel Postulate, and a new system of geometry was developed.

Therefore, Non-Euclidean Geometry was derived from Euclidean Geometry.

Triangles

k = the curvature of the figure

Euclidean Geometry: The sum of the angles in a triangle will always be equal to 180°.Spherical Geometry: The sum of the angles in a triangle will always be greater than 180°.Hyperbolic Geometry: The sum of the angles in a triangle will always be less than 180°.

Example:

Based on the information above, what is the maximum number of right angles ...