From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 4

Example 1.2 ( E

A 1 ,A 1 ,A 2 → S r , isoparametric mapping).

As an example of the mapping f : M

l

→ M

r and the commutative diagram ( f, f ,

Φ l , Φ r ), think

of an ellipsoid-of-revolution

:= X

3 +

X 2 + Y 2
A 1

+ Z 2

2
A 1 ,A 1 ,A 2

+

E ∈ R

A 2 =1 ,

R A 1 >A 2 ∈ R (1.9)

2 2

A 1 ,A 1 ,A 2

of semi-major axis A 1 and semi-minor axis A 2 as the left Riemann manifold M

l =

E ,and

think of a sphere

r := x

+ 2 2

x 2 + y 2 + z 2 = r 2 ,r S ∈ R | ∈ R (1.10)

of radius r as the right Riemann manifold M

r = S r , f being the pointwise mapping of E A 1 ,A 1 ,A 2

r one-to-one. f could be illustrated by a transformation of

to S { ellipsoidal longitude Λ , ellipsoidal

latitude Φ one-to-one. The mapping f =id

is called isoparametric if {Λ = λ, Φ = φ} or {U = u,V = v} in general coordinates of the left
Riemann manifold and the right Riemann manifold, respectively. Accordingly, in an isoparametric
mapping, { ellipsoidal longitude, ellipsoidal latitude } and { spherical longitude, spherical latitude }
are identical.

} onto { spherical longitude λ , spherical latitude φ }

End of Example.

E 2 A 1 ,A 1 ,A 2 toasphere S r ; f : E 2 A 1 ,A 1 ,A 2 → S r ;

Fig. 1.2. Bijective mapping of an ellipsoid-of-revolution Φ l :=

[ Λ,Φ ] , Φ r := [ λ, φ ]; isoparametric mapping f = id, namely { Λ,Φ } = { λ, φ }

An isoparametric mapping of this type is illustrated by the commutative diagram of Fig. 1.2 .We
take notice that the differential mappings, conventionally called f ∗ and f ∗ , respectively, between
the bell-shaped surface of revolution and the torus illustrated by Fig. 1.1 do not generate a diffeo-
morphism due to the different genus of the two surfaces. While Fig. 1.3 illustrates simply connected
regions in

3 , respectively, Fig. 1.4 demonstrates regions which are not simply connected.
Those regions are characterized by closed curves which can be laid around the inner holes and
which cannot be contracted to a point within the region. The holes are against contraction. The

R

2 and

R