8.5: Normal Sections
Consider a curve on a surface with tangent vector
T(t0)
at the point
X(t0). We first observe that the normal curvature in this direction only depends
on the direction and not on the particular curve that has this direction
as its unit tangent vector. This follows because
κN(t)s'(t)2
= X''(t)·N(t) = -X'(t)·N'(t)
and this quantity can be calculated once we know
X'(t)
in any coordinate system. Specifically if
X'(
t) = a1X1(t)+a2X
2(t), then
N'(
t) = a1N1(t)+a2N
2(t)
so the normal curvature depends only on the direction.
One very important curve on the surface which has a tangent vector
T(t0) at the point X(t0) is the normal section
, the plane curve which is the intersection of the surface with
the plane determined by the vectors T(t0) and N(t0).
For such a curve, the vector P(t0) lies along the normal line, and
κ
NN(t0)
= Pκ(t0).
Hence, for any normal section,
κN(t0) = κ(t0)
Demonstration
The default surface is the graph of a saddle surface
f(u,v) = u2 - v2.
In this demonstration, we choose a point (u(t0),v(t0)) in the domain, which appears as X(u(t0),v(t0) on the surface. The domain window also allows us to choose a direction T(t0) by rotating the orange vector. Given a point and a direction, the demo draws a plane spanned by the vectors T(t0) and N(t0). The intersection of this plane and the surface produces the red curve, which we defined as the normal section. In addition, the osculating circle for the normal section at the point X(u(t0),v(t0)) is also shown.
Exercises
1. Using the demonstration, look at the normal section for the graph of a quadratic surface f(u,v) = u2 + buv + cv2
2. How do the size and orientation of the osculating circles for the normal section relate to the principal and asymptotic directions?
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