By A. I. Markushevich
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Additional resources for Complex Numbers and Conformal Mappings
If we put a light source (a bulb) emitting the projecting rays not on the South but on the North Pole of the globe, we can use the same procedure to make a map of the Southern hemisphere with all the angles remaining full-sized. Each map obtained this way constitutes a plane figure which, when subjected to conformal mapping, will pass into a new figure which can also be considered as a geographic map. Since conformal mapping preserves the angles the new map will show the full-sized angles between the directions 34 on the earth's surface.
We wish to note that the method we used to prove the conformity of mapping z' = Z2 is also applicable to other functions, for instance to the linear-fractional function z' = =: =. ~ or to Zhukovsky's function z' = ~ (z + +). But in this case we obtain some other expressions for the angle of rotation of the tangent. Thus, we shall have for a linear-fractional function that the tangents to the curves passing through the point Zo a-b rotate through an angle equal to Arg (zo _ b)2 , and in the case of the Arg (1 - Zhukovsky z~).
If we fix a: small but very bright electric bulb on the South pole of the globe and put a screen in front of the globe at right angles to its axis, then in a dark room we shall see on the screen a contour map of the Northern 33 hemisphere. It can be proved in terms of geometry that on such a map (it is called a map in a stereoqraphic projection) all the angles between any lines on a globe in the Southern hemisphere are represented full-sized. If we leave unpainted the (curvilinear) sides of some angle PQR with vertex at an arbitrary point of the Northern hemisphere then in the stereographic projection the angle will be represented full-sized (Fig.