Graphics Reference
In-Depth Information
(1) Straight lines are not “curved”, that is, they have zero curvature.
(2) Straight lines define paths of shortest lengths between two points.
(3) Between any two points in
R
n
there is a unique straight line from one to the other.
(4) The tangent vectors along the straight line are all parallel.
The different ways of looking at a straight line have a close connection to principles
in physics, specifically, the mechanics of particles. It follows from basic principles of
classical mechanics that a particle that is not under the influence of any external forces
and that is moving in a surface will move along a geodesic in the surface. For example,
the shortest path property reflects Jacobi's form of the principle of least action. The
straightness of lines reflects the Hertz principle of least curvature. The shortest path
property is probably one of the most common ways of thinking of a straight line. A
nice elementary topic that investigates shortest paths (and other variational problems)
at great length is [Lyus64].
Before we get started generalizing the just-mentioned properties of straight lines,
we want to clarify a point that might otherwise cause some confusion in the reader's
mind later on. Basically, we again run into the problem that although we are really
interested in characterizing certain
sets
(reading the just-mentioned properties cer-
tainly suggests that we are talking about paths or sets), the analysis will proceed by
studying functions, namely, parameterizations of those sets. The term “geodesic” is
applied to both paths, where there will be no confusion, and functions, where the
confusion may arise with respect to the multiple definitions since there is no unique
parameterization of a path. Of course, we could restrict ourselves to arc-length para-
meterizations. This is a good approach from a theoretical point of view, and will be
our first one, because it would not only give us a unique parameterization for each
path but also greatly simplify the analysis. The problem with
only
using arc-length
parameterizations when dealing with geodesics is that one rarely sees these parame-
terizations in practice. Defining arc-length parameterizations is usually extremely
complicated. For practical reasons therefore one want to study geodesics in the
context of a class of functions that is at least broad enough to cover the kind of para-
meterizations one finds in the real world. The potential confusion to which we are
referring here, and which is what we want to bring out into the open, arises from the
fact that the classes of parameterizations, to which the different definitions for when
a parameterization is a geodesic apply, are often different. This difference alone would
therefore make the definitions different. For example, some definitions in the litera-
ture apply only to arc-length parameterizations, some to arbitrary regular parame-
terizations, and others to constant speed regular parameterizations. When using one
of these definitions one must then of course be careful to apply it to a parameteriza-
tion of the correct type. As long as the reader does that and understands what is going
on, a question such as “is this definition the same as that one?” may be interesting
from a technical point of view but is unimportant practically speaking. All definitions
agree on arc-length parameterizations. They all lead to the
same
geodesic
paths
.
We start the discussion of geodesics by introducing a new aspect of the curvature
of a space curve if it lies in a surface. Let
F :
US
Æ
be a regular one-to-one parameterization of a surface
S
in
R
3
and let
n
(
p
) be the
unit normal vector to the surface
S
at the point
p
=F(u,v). Let g : [a,b] Æ
S
be a curve
