Page 54 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 43

z

P
q
O
f

x y

Fig. 3.1 Surface of sphere with spherical polar co-ordinates.

Standard relations between the functions in (3.1) and (3.2), in the case where they
represent the same surface, may be obtained as follows. Suppose that at a point P
on the surface with co-ordinates (x 0 ,y 0 ,z 0 ) the partial derivative ∂F/∂z = 0. Then
the Implicit Function Theorem of calculus implies that in some neighbourhood of P
eqn (3.2) can be solved for z as a function of x and y, giving eqn (3.1). Thus in this
neighbourhood we have, for all x and y,
F(x, y, f (x, y)) = 0. (3.5)
Now differentiating partially with respect to x gives
∂F ∂F ∂f
+ = 0,
∂x ∂z ∂x
from which we derive
−1
∂f ∂F ∂F
=− , (3.6)
∂x ∂x ∂z
and similarly
∂f ∂F ∂F −1
=− . (3.7)
∂y ∂y ∂z
Of course, eqn (3.1) is just a special case of (3.3), in which x = u, y = v,and
z = f(u, v).
Given a representation (3.2) of a surface, the tangent plane to the surface at a point
P with co-ordinates (x 0 ,y 0 ,z 0 ) is the best linear approximation to the surface at P.
Consideration of the first-order increment formula δF = F x δx + F y δy + F z δz = 0
from P to a neighbouring point on the surface, where the partial derivatives of F at P
are denoted by F x ,F y ,F z , shows that the equation of the tangent plane is
F x (x − x 0 ) + F y (y − y 0 ) + F z (z − z 0 ) = 0.
It follows that the vector of partial derivatives (F x ,F y ,F z ), i.e. the gradient vector
∇F, is normal to the tangent plane, and hence normal to the surface, at P. A point P
on the surface is said to be non-singular if ∇F = 0 there; thus it is possible to specify
the normal to the surface at a non-singular point.
Another approach to non-singular points comes from considering the matrix M of
partial derivatives of the representation (3.3):

∂x/∂u ∂y/∂u ∂z/∂u
M = . (3.8)
∂x/∂v ∂y/∂v ∂z/∂v