Conformal Representation

Conformality and Uniformization


By an isogonal representation of two areas on one another we mean a one-one, continuous, and continuously differentiable representation of the areas, which is such that two curves of the first area which intersect at an angle \( \alpha \) are transformed into two curves intersecting at the same angle \( \alpha . \) If the sense of rotation of a tangent is preserved, an isogonal transformation is called conformal.

Disregarding as trivial the Euclidean magnification of the plane, we may say that the oldest known transformation of this kind is the stereographic projection of the sphere, which was devised by Ptolemy for the representation of the celestial sphere; it transforms the sphere conformally into a plane. A quite different conformal representation of the sphere on a plane area is given by Mercator's Projection; in this the spherical earth, cut along a meridian circle, is conformally represented on a plane strip.


Uniformisation Theorem:

The universal covering surface \( \overline{\boldsymbol{S}} \) of any open or closed Riemann surface S can always be represented conformally on
(1) a closed sphere
(2) an Euclidean plane
(3) the interior of a circle \( |t|<R \) these possibilities being mutually exclusive

In this way we obtain a complete conformal pattern in the \( t \) -plane of the triangulation of \( \overline{\boldsymbol{S}} \). Its edges are Jordan curves and do not necessarily have tangents at their end-points, so that in general the angles of the triangles are not defined. However, the given triangulation can always be replaced by an equivalent one for which the edges are analytic arcs or, even more specially, are geodesics in the particular metric involved.

-C. Caratheodory

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