Graphics Reference
In-Depth Information
Note.
Before we start we need to make something clear. Unless explicitly stated
otherwise, we shall assume throughout this chapter that our curves, surfaces, or man-
ifolds are submanifolds of
R
n
for some n and that tangent vectors and tangent spaces
are defined as in Section 8.4. However, and this is very important to keep in mind,
the reader should think of the manifolds as spaces that exist on their own and inde-
pendent of the engulfing space
R
n
. The only reason for bringing
R
n
into the picture
is to simplify the definition of tangent vectors and certain formulas. Everything else
will rely on parameterizations, so that the theory would translate to abstract mani-
folds as defined in Section 8.8 with very little effort.
9.2
Curve Length
Chapter 8 already discussed a few facts about curves. Of interest in this chapter are
their metric properties and these will be studied by means of parameterizations. We
begin with curves in
R
n
. This will be a warm up for studying curves in manifolds.
Defining the length of a curve is the first order of business.
To understand the notion of length, one builds on the everyday meaning of this
term in simple cases. To begin with, everyone agrees that the length of a line segment
should be the distance between its end points. Next, one wants length to be additive.
That leads to defining the length of a polygonal curve as the the sum of the lengths
of its segments, which in turn suggests the definition for a general curve.
Let
[
]
Æ
R
n
Fab
:
,
be an arbitrary parametric curve. The standard approach to defining its length is
to approximate the curve with a piecewise linear one and then use the length of the
piecewise linear version as an approximation for the length of the curve we are
after. More precisely, let P = (t
0
,t
1
,...,t
k
) be a partition of [a,b] and let
p
i
= F(t
i
). See
Figure 9.1.
The
length
of the parametric curve F, denoted by l
b
F, is defined by
Definition.
p
k
= F(t
k
)
p
k-1
F
a = t
0
t
1
···
t
k
= b
p
1
= F(t
1
)
Figure 9.1.
Curve length is computed
from polygonal approxi-
mations.
p
0
= F(t
0
)
