Lecture Notes for 120 - UCLA Department of Mathematics

4.2. TANGENT SPACES AND MAPS 77
(10) Show that
x 2 + y 2 + z 2 + R 2 r 2 2 =4R 2 x 2 + y 2
defines a surface when R>r>0. Showthatitisrotationallysymmetric
and a torus.
(11) The helicoid is given by the equation
tan z h = y x
where h 6= 0is a fixed constant.
(a) Show that this defines a surface for suitable (x, y, z).
(b) Show that the surface can be parametrized by
q (r, ✓) =(r cos ✓, r sin ✓, h✓)
and determine for which r, ✓ this defines a parametrized surface. Note
that for fixed r we obtain helices.
(12) Enneper’s surface is defined by the parametrization
q (u, v) =
✓
u
1
3 u3 + uv 2 ,v
◆
1
3 v3 + vu 2 ,u 2 v 2
(a) For which u, v does this define a parametrization?
(b) Show that Enneper’s surface satisfies the equation
✓ y
2
x 2
+ 2z2
2z 9 + 2 ◆ 3
y
2
x
=6✓ 2 1
✓x 2 + y 2 + 8 ◆
3
4z 4
9 z2 + 2 9
4.2. Tangent Spaces and Maps
Definition 4.2.1. The tangent space at p 2 M of a (parametrized) surface is
defined as
⇢ @q
T p M = span
@u , @q ,
@v
and normal space
N p M =(T p M) ?
Remark 4.2.2. For a parametrized surface with self-intersections this is a bit
ambivalent as the tangent in that case depends on the parameter values (u, v) and
not just the point p = q (u, v). This is just as for curves where the tangent line
at a point really is the tangent line at a point with respect to a specific parameter
value.
Proposition 4.2.3. Both tangent and normal spaces are subspaces that do not
change under reparametrization.
Proof. This would seem intuitively clear, just as with curves, where the tangent
line does not depend on parametrizations. For curves it boils down to the
simple fact that velocities for different parametrizations are proportional and hence
define the same tangent lines. With surfaces something similar happens, but it is
abitmoreinvolved. Supposewehavetwodifferentparametrizationsofthesame
surface:
q (s, t) =q (u, v)
◆ 2