[share_ebook] Stereographic Projection

Description

• Author: B. A. Rosenfeld, N. D. Sergeeva
• Publisher: Mir Publishers Moscow
• Publish Date: 1977
• Pages: 53

Little Mathematics Library

The present booklet is devoted to I'roofs of the properties of the stereographic projection and to the presentation of some of its applications. The booklet consists of eight sections.

Sec. I gives a definition of the stereographic projection and proofs of its basic properties. In Sec. 2 we establish the connection between the stereographic projection and a remarkable transformation of a plane onto itself in which the circles are also transformed into circles or straight lines and the angles between the lines are transformed into the angles equal to them - this transformation
is called the inversion with respect to a circle; in the same section we establish the relation of the stereographic projectionto the similar transformation of space - the inversion with respect to a sphere. In Sec. 3 the basic properties of the stereographic projection are proved in a different way, namely by means of
coordinates. Sec. 4 establishes the relation between the stereographic projection and the complex numbers: when the projection plane is considered to be a plane of a complex variable, mapping of complex numbers by the points on the sphere is realized by means of a stereographic projection. This mapping is frequently utilized in the theory of functions of complex variables since the so-called point at infinity of the plane of the complex variable, which cannot be mapped on the plane itself, is given on the sphere by the very projection centre. The same section discusses the so-called spherical metric on a plane when the distance between two points of the plane is assumed equal to a spherical distance between the corresponding points on the sphere; this distance is expressed in the simplest form by means of complex numbers. In Sec. 5 we show how the rotations of the sphere are mapped by the plane transformations in the stereographic projection; these transformations are also expressed most simply by means of complex numbers. Sec. 6 gives an account of the history of stereographic projection which was developed already in antiquity and was very popular in the Middle Ages. Sec. 7 describes how the stereographic projection applies to astronomy - medieval astrolabes were based on this projection - and to geography where this projection is used to draw nautical maps. Sec. 8 presents the definition of the Lobachevskian plane, demonstrates how a peculiar stereographic projection can yield a projection of the Lobachevskian plane onto an ordinary plane so that the circles and some other curves on the Lobachevskian plane are mapped as circles or straight lines while the angles between the lines of the Lobachevskian plane are mapped as the angles equal to them.

The booklet is aimed to be used in the senior grades of the high schools and by the first- and second-year students.

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