Graphics Reference
In-Depth Information
()
=
()
+
2
()
()
2
LLuMuv Nv
MLuuMuv
F
F
F
y
2
y
+
y
x
x
x
x
()
=
()
()
(
)
+
()
y
+
y
+
v uNv v
y
xy
xy
xy
xy
()
=
()
+
2
()
()
2
NLuMuv
y
2
y
+
Nv
y
y
y
y
y
Proof.
Use the appropriate chain rule.
We have not paid much attention to the local parameterizations that we used. By
choosing these more carefully we can make formulas simpler (as in the case of curves
where arc-length parameterization was superior to other ones).
Definition.
A
line of curvature
on a surface is a curve whose tangent at every point
is parallel to a principal direction.
9.9.20. Theorem.
(Rodrigues) Let F(u,v) be any regular parameterization of a
surface
S
. Then a curve g(t) in
S
defines a line of curvature if and only if
n
t
¢
()
=-
()
¢
()
k
t
g
t
,
(9.57)
where
n
(t) =
n
(g(t)) and k(t) is the principal curvature of the curve at g(t).
Proof.
If g(t) is a line of curvature, then g¢(t) must be an eigenvector of D
n
.
Equation (9.57) is called the
Rodrigues formula
.
9.9.21. Theorem.
Every nonumbilical point on a C
3
surface has a neighborhood on
which there exist two orthogonal families of lines of curvature.
Proof.
Proving this theorem amounts to solving the differential equations defined
by (9.57). Standard results about differential equations imply the existence and
uniqueness of local solutions.
Often the following weaker form of Theorem 9.9.21 is sufficient. It is much easier
to prove.
9.9.22. Theorem.
Given a point
p
on a C
2
surface we can always find a regular
parameterization F(u,v) of a neighborhood of
p
so that F
u
and F
v
at
p
are principal
directions.
Proof.
Choose any regular parameterization F(u,v) for a neighborhood of
p
. Assume
that F(0,0) =
p
. If F
u
and F
v
are not already principal directions, we shall simply
“rotate” the coordinate patch to achieve this. Let
u
1
and
u
2
be linearly independent
vectors in
R
2
with the property that DF(
u
i
) are principal directions and let T be
the linear transformation of
R
2
so that T(
e
i
) =
u
i
. Then j=F
-1
T is the desired
parameterization.
9.9.23. Theorem.
Given a regular parameterization F(u,v) of a surface, the vectors
F
u
and F
v
at a nonumbilical point are in the direction of the principal directions if
and only if F = M = 0 at that point.
