Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation

PDF Version Also Available for Download.

Description

We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we ... continued below

Physical Description

vi, 93 leaves: ill.

Creation Information

Debrecht, Johanna M. August 1998.

Context

This thesis is part of the collection entitled: UNT Theses and Dissertations and was provided by UNT Libraries to Digital Library, a digital repository hosted by the UNT Libraries. It has been viewed 84 times , with 9 in the last month . More information about this thesis can be viewed below.

Who

People and organizations associated with either the creation of this thesis or its content.

Chair

Committee Members

Publisher

Rights Holder

For guidance see Citations, Rights, Re-Use.

  • Debrecht, Johanna M.

Provided By

UNT Libraries

With locations on the Denton campus of the University of North Texas and one in Dallas, UNT Libraries serves the school and the community by providing access to physical and online collections; The Portal to Texas History and UNT Digital Libraries; academic research, and much, much more.

Contact Us

What

Descriptive information to help identify this thesis. Follow the links below to find similar items on the Digital Library.

Degree Information

Description

We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.

Physical Description

vi, 93 leaves: ill.

Subjects

Language

Identifier

Unique identifying numbers for this thesis in the Digital Library or other systems.

Collections

This thesis is part of the following collection of related materials.

UNT Theses and Dissertations

Theses and dissertations represent a wealth of scholarly and artistic content created by masters and doctoral students in the degree-seeking process. Some ETDs in this collection are restricted to use by the UNT community.

What responsibilities do I have when using this thesis?

When

Dates and time periods associated with this thesis.

Creation Date

  • August 1998

Added to The UNT Digital Library

  • March 24, 2014, 8:07 p.m.

Description Last Updated

  • Oct. 2, 2014, 1:27 p.m.

Usage Statistics

When was this thesis last used?

Yesterday: 0
Past 30 days: 9
Total Uses: 84

Interact With This Thesis

Here are some suggestions for what to do next.

Start Reading

PDF Version Also Available for Download.

Citations, Rights, Re-Use

Debrecht, Johanna M. Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation, thesis, August 1998; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc278501/: accessed October 22, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .