Page 53 - Basic Structured Grid Generation

3

Differential geometry of surfaces

in E 3

3.1 Equations of surfaces

The classical differential geometry of curves and surfaces is presented in many text-
books, of which we quote several in the bibliography. Here we are concerned with
surfaces embedded in three-dimensional Euclidean space, which may be represented
by a variety of mathematical expressions in terms of rectangular cartesian co-ordinates
x, y, z.
A function of two variables f (x, y) gives a surface

z = f (x, y). (3.1)
A function of three variables F(x, y, z) gives a surface

F(x, y, z) = 0. (3.2)

Two real parameters u and v maybeused togiveasurface r = r(u, v), or, in terms
of three functions of two variables,

x = x(u, v), y = y(u, v), z = z(u, v). (3.3)
The parameters u and v here may be referred to as surface (or Gaussian) co-ordinates.
We assume that all functions are continuously differentiable unless otherwise stated.
In each of the above cases the functions may be defined only for a restricted range of
the independent variables. For example, the hemispherical surface obtained by slicing
2
2
2
the unit sphere x + y + z = 1 in half through the plane Oxy and discarding the
part with z< 0 may be expressed as

2
2
2
z = f (x, y) =+ 1 − x − y 2 with x + y 1
or
2
2
2
2
2
F(x, y, z) = x + y + z − 1 = 0 with 0 z 1and x + y 1
or, in terms of spherical polar co-ordinates θ, ϕ as parameters (Fig. 3.1),
x = sin θ cos ϕ, y = sin θ sin ϕ, z = cos θ, with 0 θ π/2, 0 ϕ 2π.
(3.4)