Page 62 - Basic Structured Grid Generation

Differential geometry of surfaces in E 3 51

3
properties of the surface are induced by the Euclidean metric of E . However, once
formulas for length, angle, and area have been established in terms of a αβ ,measurement
may be carried out on the surface without regard to the embedding space.
When two surfaces (or parts of two surfaces) are such that they admit surface co-
ordinate systems which give identical first fundamental forms, then the intrinsic geom-
etry of the surfaces is the same. The existence of these surface co-ordinates implies the
existence of a mapping between points on the surfaces such that corresponding curves
have the same length or intersect at the same angle. This means that there is no differ-
ence locally between the surfaces as far as measurement of lengths, angles, and areas
is concerned, no matter how different the surfaces may appear from the standpoint of
the enveloping three-dimensional space. Such surfaces are called isometric.
A simple example is the isometry between the circular cylinder (3.9) and a portion
of a plane. The first fundamental form of the cylinder is easily seen to be
2
2
a (dϕ) + (dz) 2
2
2
which is equivalent to (dx) + (dy) for the plane with cartesian co-ordinates under
the transformation aϕ → x and z → y.

3.3 Surface covariant differentiation

Surface Christoffel symbols of first and second kinds can be defined. It is important to
note at the outset, however, that there is no immediate surface equivalent of eqn (1.97),
β
since in general ∂a α /∂u is not a surface vector and cannot be expressed in terms of a 1
and a 2 . But we can effectively use the equivalents of eqns (1.98) and (1.99) to define
the surface Christoffel symbols, and then the same procedure that led to eqn (1.108)
gives
2
∂a α ∂ r ∂r 1 ∂a αγ ∂a βγ ∂a αβ
[αβ, γ ]= · a γ = · = + − (3.49)
α
∂u β ∂u ∂u β ∂u γ 2 ∂u β ∂u α ∂u γ
and
γ ∂a α γ γδ
αβ = · a = a [αβ, δ], (3.50)
∂u β
where the Greek indices can take only the values 1 and 2.
2
1
Explicitly, using eqns (3.30), we can write, putting u = u , v = u ,
1 ∂a 11 ∂a 12 1 ∂a 11 1 ∂a 11
[11, 1]= , [11, 2]= − , [12, 1]=[21, 1]= ,
2 ∂u ∂u 2 ∂v 2 ∂v
1 ∂a 22 ∂a 12 1 ∂a 22 1 ∂a 22
[12, 2]= [21, 2]= , [22, 1]= − , [22, 2]=
2 ∂u ∂v 2 ∂u 2 ∂v
(3.51)
and
1 ∂a 11 ∂a 11 ∂a 12
1
11 = a 22 + a 12 − 2
2a ∂u ∂v ∂u
1 ∂a 11 ∂a 22
1
1
12 = 21 = a 22 − a 12
2a ∂v ∂u