Little Mathematics Library – Stereographic Projection

We now come to another book in the Little Mathematics Library titled Stereographic Projection by B. A. Rosenfeld and N. D. Sergeeva. As the title suggests the book deals with projections on planes.

The present booklet is devoted to proofs of the aforesaid properties of the stereographic projection and to the presentation of some of its applications. The booklet consists of eight sections dealing with different properties of projections. …The booklet is aimed to be used in the senior grades of the high schools and by the first- and second-year students.

The book was translated from the Russian by Vitaly Kisin and was first published by Mir in 1977. You can get the book here. All credits to the original uploader.

Sections and description:

1. Definition and Basic Properties of the Stereographic Projection 11

Sec. I gives a definition of the stereographic projection and proofs of its basic properties.

2. Stereographic Projection and Inversion 20

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 projection to the similar transformation of space – the inversion with respect to a sphere.

3. Proof of the Properties of the Stereographic Projection by Means of Coordinates 25

In Sec. 3 the basic properties of the stereographic projection are proved in a different way, namely by means of coordinates.

4. Spherical Metric on a Plane. Application of Complex numbers 30

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.

5. Mapping of Sphere Rotations on a Plane 37

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.

6. History of the Stereographic Projection 40

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.

7. Application of the Stereographic Projection to Astronomy and Geography 42

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.

8. Application of the Stereographic Projection to the Lobachevskian Geometry 47

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.

Bibliography 54

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