Graphics Reference
In-Depth Information
We would like to find an orthonormal basis (
u
1
,
u
2
) of eigenvectors of D
n
(
p
). If
p
=
(
)
r cos
q
,
r
sin
q
03
,
,
0
then let
u
1
= (-sin q
0
,cos q
0
,0) and let
u
2
=
e
3
. Note that the curve
q
q
Ê
Ë
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
ˆ
¯
()
=
gq
r
cos
+
q
,
r
sin
+
q
,
p
0
0
3
r
r
has the property that g(0) =
p
and g¢(0) =
u
1
. It is now easy to check from the formula
for D
n
(
p
) above that
( ( )
=
()
( ( )
=
D
np u
1
0.
r
u
1
1
D
np u
2
This shows that K = 0 and H =-1/2r.
Although we do not have time to discuss mean curvature here, it certainly has
geometrical significance. Here are two facts:
(1) The mean curvature at a point is the average of the normal curvatures there.
See [MilP77] or [DoCa76].
(2) The mean curvature plays an important role in the study of surfaces that have
minimum area for a fixed boundary.
Plateau's Problem:
Given a closed curve
C
, to find the surface of minimum area that
has
C
for its boundary.
Plateau's problem is an old one and solving it has motivated a lot of research
over the years. It is well known that soap films spanning wireframe bound-
aries create minimum area surfaces. It can be shown that a necessary condi-
tion for a surface to minimizes area is that its mean curvature vanish.
Therefore another active research area is the study of minimal surfaces.
Definition.
A
minimal surface
is a surface for which the mean curvature van-
ishes everywhere.
Minimal surfaces arise in other contexts, not just where one is minimizing
area. See [Osse69] or [Gray98]. There are lots of questions here about their
existence, uniqueness, construction, and characterization.
Definition.
The
Dupin indicatrix
at a point
p
of the surface
S
is the subset
X
of T
p
(
S
)
defined by
=Œ
()
{
()
=±
}
Xu
T
S
u
1.
p
II
9.9.13. Theorem.
Let
p
be a point on a surface
S
with Gauss curvature K. Then
(1) If K > 0, then the Dupin indicatrix is an ellipse.
(2) If K < 0, then the Dupin indicatrix consists of two pairs of hyperbolas.
