Lecture Notes for 120 - UCLA Department of Mathematics

5.1. CURVES ON SURFACES 101
Taking inner products with the normal will eliminate the second term as it is a
tangent vector so we obtain
¨q · N = ⇥ ˙u ˙v ⇤ " #
@ 2 q
@u
· N @2 q apple 2 @v@u · N
@ 2 q
@u@v · N
@2 q
@v
· N
˙u˙v
2
= ⇥ ˙u ˙v ⇤ apple L uu L vu
L uv L vv
apple ˙u˙v
= II(˙q, ˙q)
Definition 5.1.2. We can now redefine NII (Z, Z) by selecting a curve on the
surface with ˙q = Z and then using that NII ( ˙q, ˙q) is the normal component of ¨q.
To compute NII (X, Y ) we can use polarization:
NII (X, Y )= 1 (NII (X + Y,X + Y ) NII (X, X) NII (Y,Y ))
2
As with space curves we define the unit tangent T for a regular curve q (t)
on a surface q (u, v). However, we use the normal N to the surface instead of
the principal normal component of the acceleration in relation to the unit tangent.
From these two vectors we can define the normal to the curve tangent to the surface
as
S = N ⇥ T
In this way curve theory on surfaces is closer to the theory of planar curves as we
can think of S as the signed normal to the curve in the surface. Using an arclength
parameter s we define the normal curvature
the geodesic curvature
and the geodesic torsion
apple n =II(T, T)
apple g = S · dT
ds
⌧ g = N · dS
ds
Example 5.1.3. Aplanealwayshasvanishingsecondfundamentalformasits
normal is constant. This means that any curve in this plane has vanishing normal
curvature and geodesic torsion. The geodesic curvature is the signed curvature apple ± .
Example 5.1.4. AsphereofradiusR centered at c is given by the equation
F (x, y, z) =|q c| 2 = R 2 > 0
The gradient is
rF =2(q c) =2(x a, y b, z c)
which cannot vanish unless q = c. This shows that the sphere is a surface and also
computes the two normals
N = ± 1 (q c)
R
as |q c| = R. The + gives us an outward pointing normal. If we have a
parametrization q (u, v), then this relationship between q and N implies that
@q
@w = ± 1 @N
R @w . This in turn yields II = ± 1 R I.
⇤