Projects for Differential Geometry
Exercises for projects are all taken from the book Differential
Geometry and its Applications by John Oprea (Prentice-Hall 1997).
- Project 1
- Involutes and Evolutes.
This project will explore
ways of making new types of curves from old ones. For example, an
involute of the circle is given by the tip of thread which is
unwound tautly from a circular spool. Involutes are important in gear
tooth design. Relevant exercises include 2.4, 2.5, 4.6, 4.7, 4.8,
4.9, 4.10, 5.9, 6.6, 6.7 all in Chapter 1.
Project 2
- Spherical Curves.
This project explores the special qualities of curves which lie on a sphere.
What is special about the curvature and torsion of such curves? How does the
geometry of the curve interact with the geometry of the surface? Relevant
exercises include 1.3.17, 1.3.18, 1.3.19, 1.3.20, 1.3.21, 1.6.13, 2.4.1 as well
as basic geometric information about the sphere.
Project 3
- Curves of Constant Precession.
This project explores
a special class of curves determined by sine and cosine curvature
and torsion functions. These curves are not only beautiful, but they
lead to interesting questions about their tangent curves as well.
Relevant exercises include 3.9, 3.10, 3.11, 3.12, 6.12, 6.13 in
Chapter 1 as well as information about hyperboloids of one sheet
and showing that the curves of constant procession always lie on
these hyperboloids.
Project 4
- Cones and Cylinders.
This project explores familiar
geometric objects, the cone and cylinder, from the viewpoint of
differential geometry. In particular, you will see that the cone
and cylinder deserve the name `flat'. You will also learn to
unroll these ruled surfaces, understand their geodesics and
determine when the cone may be lassoed. relevant exercises include
5.5.2, 5.5.3, 5.5.4, 5.6.4, 6.3.9, 6.3.10.
Project 5
- The Gauss Map.
The Gauss map has input a point on a
surface in $\Bbb R^3$ and output the unit normal to the surface at
that point. So, it is a mapping from the surface to the unit sphere.
This mapping describes virtually everything about the geometry of the
surface and provides an alternative view to our usual calculational
approach to curvature. Relevant exercises include 2.3.4, 2.3.5, 2.3.6,
2.3.7, 5.5.9, 7.3.21, 7.3.22 as well as material on p. 76-78, 90 and
Proposition 7.3.6.
Project 6
- Soap Films and Minimal Surfaces
. This project
explores the boundary between mathematics and physics by understanding
soap films (and bubbles) as surfaces whose mean curvature is zero
(or constant). Relevant exercises include 2.1.9, 4.2.1,
4.2.2, 4.2.5, 7.5.2 and Chapter 4: p. 125-127, Chapter 7: p.
251-253.
Project 7
- The Brachistochrone.
This project will explore the
shape of the wire on which a bead may slide under gravity from
one point to another in the least time. The general principle
of minimization as in Chapter 8 may then be used to understand
curves such as the catenary from several viewpoints. Relevant
exercises include 1.1.5, 1.1.6, 2.1.6, 4.3.4, 8.2.1, 8.2.3,
8.4.5, Chapter 8: p. 261 as well as Maple procedures from Chapter 8.
Project 8
- Student Suggestions.
You may suggest your own
project. For instance, in the past, some students have done
projects on general relativity. There is some material for this in
Chapter 9 as well as in several other texts which are available. For
this project or other ones (not listed in 1-7 above) you may wish to
do, you must first get the instructor's approval.
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