Page 73 - Basic Structured Grid Generation

62 Basic Structured Grid Generation

which gives
2
2
2
1 ∂ r 2 ∂ r ∂ r 2
N · δr = N · (δu) + 2 δuδv + (δv)
2 ∂u 2 ∂u∂v ∂v 2
1 2 2 1 α β
= L(δu) + 2Mδuδv + N(δv) = b αβ δu δu , (3.100)
2 2
since N is perpendicular to the surface tangent vectors a 1 and a 2 . The sign of the
resulting expression will be different for points on S lying on different sides of the
tangent plane. So if the second fundamental form is positive definite or negative defi-
nite, which is the case if det(b αβ )> 0 at P, all points in the immediate neighbourhood
of P will lie on the same side of the tangent plane. Such points P may be called elliptic.
But if det(b αβ )< 0, there will be points in a neighbourhood of P lying on different
sides of the tangent plane, and P may be called hyperbolic.The third possibility is that
the second fundamental form is positive or negative semi-definite, which occurs when
det(b αβ ) = 0. This is the case, for example, for a circular cylinder, all points on the
surface of which may be called parabolic.
Example: A surface which exhibits all three types of points is the torus, the surface
(Fig. 3.4) formed by revolving the circle
2 2 2
(x − b) + z = a
in the Oxz plane, where b> a, about the z-axis (Fig. 3.5). This is a surface of
revolution of the form eqn (3.32), with f(u) = b + a cos u,and g(u) = a sin u.
Substituting into eqns (3.99), we obtain

L = a, M = 0, N = (b + a cos u) cos u.
2
Hence det(b αβ ) = LN − M = a cos u(b + a cos u).Since (b + a cos u) > 0for all
u, it follows that the sign of det(b αβ ) is the same as the sign of cos u. Hence there are

z

Hyperbolic points

O
x
Parabolic points
Elliptic points y

Fig. 3.4 Torus.