Page 56 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 45

z

P

a
O
f
y
x N

Fig. 3.2 Right circular cone with semi-angle α and vertex O.

Curves of
constant u

r v
P
r u
Curves of
constant v

Fig. 3.3 Surface with curvilinear co-ordinates u, v, and base vectors r u , r v .

with rankM = 2, except at the vertex of the cone, where u = 0and rankM = 1. The
vertex will clearly be a singular point whatever the parametrization of the surface.
In general, when a surface is parametrized as in eqn (3.3), we may assume that the
surface can be covered by a grid consisting of two families of co-ordinate curves; on
the members of one family u varies but v is constant, while on the other v varies and
u is constant (Fig. 3.3). Through a point P of the surface there will generally pass one
member of each family, and tangent vectors in the directions of these curves will be
given by ∂r/∂u and ∂r/∂v, which we shall write here as r u and r v respectively.
Now the rows of M as defined in eqn (3.8) are just the cartesian components of r u
and r v , and the condition rankM = 2 is equivalent to the condition that r u and r v
should be non-zero and independent, i.e.
r u × r v = 0. (3.11)
This condition holds good under a change of parametrization from (u, v) to (u, v),
given some relation u = u(u, v), v = v(u, v), provided that the Jacobian of the
transformation is non-zero, that is,
∂u ∂v ∂u ∂v
− = 0.
∂u ∂v ∂v ∂u
For, using Chain Rules, we have

∂u ∂v ∂u ∂v ∂u ∂v ∂u ∂v
r u × r v = r u + r v × r u + r v = − (r u × r v ).
∂u ∂u ∂v ∂v ∂u ∂v ∂v ∂u