Page 74 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 63

z

a u
O b x

Fig. 3.5 Circle of radius a to be rotated around Oz to form torus.

elliptic points where −π/2 <u< π/2, hyperbolic points where π/2 <u< 3π/2,
and a curve of parabolic points where u =±π/2.
If a surface curve C is a normal section, obtained from the intersection of S with a
plane at P containing N,then κ g = 0 and κ = κ N , given by eqns (3.91) and (3.16) as
the ratio of quadratic forms

2
α
b αβ du du β L(du) + 2M du dv + N(dv) 2
κ N = = . (3.101)
γ
2
a γδ du du δ E(du) + 2F du dv + G(dv) 2
If C is not a normal section, with principal normal n making an angle φ with the
surface normal N, then taking the scalar product of eqn (3.90) with N gives Meusnier’s
Theorem
κ cos φ = κ N , (3.102)
which, expressed in terms of the radius of curvature ρ of C and the radius of curvature
ρ N of the normal section with the same surface tangent at P, is equivalent to

ρ = ρ N cos φ. (3.103)
If C is a geodesic, we know from eqn (3.78) that κ has the direction of N. Hence
for a geodesic, as for a normal section, κ g = 0; a geodesic through a point in a certain
direction has the same curvature as a normal section through that point in the same
direction. In the case of a spherical surface, normal sections at a point on the surface
are the same as geodesics (great circles), but this is not generally the case.
In general, it can be shown that the magnitude κ g of κ g is the geodesic curvature
defined in the last section. It gives a measure of how much the curvature of a surface
curve differs at a point from that of a geodesic curve in the same direction passing
through that point. By eqn (3.90) we have
2
2
2
κ = κ + κ . (3.104)
N g
3.7 Principal curvatures and lines of curvature

From eqn (3.101) the curvature κ N of a normal section at a point P satisfies
α β
(b αβ − κ N a αβ ) du du = 0,