Page 55 - Basic Structured Grid Generation

44 Basic Structured Grid Generation

If this matrix has rank 2 at a point P(x 0 ,y 0 ,z 0 ), at least one of the second-order
sub-determinants is non-zero there. Suppose that this is
∂x ∂y ∂y ∂x
− = 0.
∂u ∂v ∂u ∂v
This is the condition for the non-vanishing of the Jacobian of the transformation
(u, v) → (x, y), which implies that the inverse mapping u = u(x, y), v = v(x, y)
exists in some neighbourhood of the projected point (x 0 ,y 0 ) in the plane Oxy. Hence
it is possible to write

z = z(u, v) = z(u(x, y), v(x, y)) = f (x, y),
as in eqn (3.1), which is equivalent to

F(x, y, z) = f (x, y) − z = 0
It follows that F has non-zero gradient (∂f/∂x, ∂f/∂y, −1) and thus P is non-singular
according to our previous definition.
Singular points arise when the rank of M is 0 or 1. When rankM = 1 at all points,
the surface reduces to a curve. For example, the equations
2
x = u + v, y = (u + v) , z = (u + v) 3
represent a curve.
It is possible for one sub-determinant of M to be zero, as in the case of the circular
cylinder of radius a
x = a cos u, y = a sin u, z = v (3.9)
and for two, as in the case of the planar surface

x = u, y = v, z = 1,
but rankM = 2 at all points in both cases.
Singular points may arise out of the nature of the surface, but may now also appear
due to a chosen parametrization (3.3) in the situation where rankM < 2. For example,
for the spherical surface (3.4) we have

cos θ cos ϕ cos θ sin ϕ − sin θ
M = ,
− sin θ sin ϕ sin θ cos ϕ 0
with rankM = 2, except for the singular point at the ‘North Pole’ of the sphere
(Fig. 3.1), where θ = 0and rankM = 1.
A surface with an obvious singular point is the right circular cone (Fig. 3.2) with
semi-vertical angle α, which can be represented in terms of the parameters u, ϕ,by
the parametric equations
x = u sin α cos ϕ, y = u sin α sin ϕ, z = u cos α. (3.10)

Here

sin α cos ϕ sin α sin ϕ cos α
M = ,
−u sin α sin ϕ u sin α cos ϕ 0