Page 326 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 26: Lines of Curvature

The term asymptotic stems from the fact that those curve have their tangent   along the Notes

j
1
asymptotes of the Dupin indicatrix, the conic section k    in the tangent space. Since the
i
ij
Dupin indicatrix has no asymptotes when K > 0, we see that the Gauss curvature must be
non-positive along any asymptotic line.
The following Theorem can be proved by the same method as used above to obtain Theorem 1.
Theorem 2. Let X : U   be a parametric surface, and let u be a hyperbolic point. Then there
3
0

is neighborhood U of u and a diffeomorphism : U  0  U such that X  X  is parametrized

0
0
0
by asymptotic lines.
26.2 Examples
A surface of revolution is a parametric surface of the form:

X(u,v)  f(u)cos(v),f(u)sin(v),g(u) ,

where f(t), g(t) is a regular curve, called the generator, which satisfies f(t)  0. Without loss of
generality, we may assume that f(t) > 0. The curves


 v (t)  f(t)cos(v),f(t)sin(v),g(t) , v fixed.
are called meridians and the curves

 u (t)  f(u)cos(t),f(u)sin(t),g(u) , u fixed.
are called parallels. Note that every meridian is a planar curve congruent to the generator and
is furthermore also a normal section, and every parallel is a circle of radius f(u). It is not difficult
to see that parallels and meridians are lines of curvature. Indeed, let  be a meridian, then
v
choosing as in the paragraph following the correct orientation in the plane of  , its spherical
v
k .
image under the Gauss map is   N   e , and by the Frenet equations,    ke     v


v
2
v
v
1
Thus, using Proposition 1 and the comment immediately following it, we see that  is a line of
v
curvature with associated principal curvature k. Since the parallels  are perpendicular to the
u
meridians  , it follows immediately that they are also lines of curvature. We derive this also
v
follows from Proposition 1 and furthermore obtain the associated principal curvature. A
straightforward computation gives that the spherical image of  under the Gauss map is:
u
  N   c  B

u
u
u
3
where B  and c  are constants. Thus,   c  u and  is a line of curvature with associated

u
u
principal curvature c.
The plane, the sphere, the cylinder, and the hyperboloid are all surfaces of revolution. We
discuss one more example.
The catenoid is the parametric surface of revolution obtained from the generating curve
cosh(t),t :


X(u,v)  cosh(u),cos(v),cosh(u),sin(v),u .
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