Graphics Reference
In-Depth Information
Proof.
At a nonumbilical point the principal directions are orthogonal and by defi-
nition F = 0 if and only if F
u
and F
v
are orthogonal. If F = 0, then M = 0 if and only
if the matrix in (9.50) is diagonal, which is equivalent to F
u
and F
v
being principal
directions and Rodrigues's formula holding.
From this we get immediately
9.9.24. Corollary.
Given a regular parameterization F(u,v) of a surface with no
umbilical points, then the u- and v-parameter curves on the patch are lines of curva-
ture if and only if F = M = 0.
9.9.25. Theorem.
If the vectors F
u
and F
v
of a regular parameterization F(u,v) of
a surface are the principal directions at a point, then the principal normal curvatures
at that point are given by
L
E
N
G
.
k
=
and
k
=
1
2
Proof.
Given the hypothesis, Theorem 9.9.23 implies that F = M = 0. Substituting
into the formulas for the Gauss curvature and mean curvature given in Theorems
9.9.15 and 9.9.16 and solving Equation (9.55) for this special case gives the desired
two roots.
9.9.26. Corollary.
If the u- and v-parameter curves of a regular parameterization
F(u,v) of a surface are lines of curvature, then the principal curvatures at that point
are given by
L
E
N
G
.
k
=
and
k
=
1
2
We are almost done with our foray into the geometry of surfaces, but it
seems appropriate to take it a little further and finish with the so-called fundamental
theorem of surfaces (Theorem 9.9.28 below) that is the analog of Theorem 9.4.7
for curves. Even though most of the results in the rest of this section are stated
without proofs (they are actually not that hard), the reader will at least get an
overview of what needs to be done. For details see [DoCa76], [Lips69], [Spiv70b], or
[Spiv75].
The main missing ingredient in what we have done so far is to express the second
derivatives F
uu
, F
uv
, and F
vv
in terms of the first derivatives and
n
. This can be done
because the three vectors form a basis for
R
3
. Suppose we write
1
1
1
2
FGFGF
FGFGF
FGFGF
FGFGF
=
+
+
a
a
a
a
n
n
n
n
uu
u
v
11
1
1
1
2
=
+
+
uv
u
v
12
2
1
2
2
=
+
+
vu
u
v
21
1
2
=
+
+
(9.58)
vv
u
v
22
22
22
