Inversive Geometry

Frank Morley, F.V. Morley

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486783062
• Topic: Matemáticas
• Subtopic: Geometría algebraica

This introduction to algebraic geometry makes particular reference to the operation of inversion and is suitable for advanced undergraduates and graduate students of mathematics. One of the major contributions to the relatively small literature on inversive geometry, the text illustrates the field's applications to comparatively elementary and practical questions and offers a solid foundation for more advanced courses.
The two-part treatment begins with the applications of numbers to Euclid's planar geometry, covering inversions; quadratics; the inversive group of the plane; finite inversive groups; parabolic, hyperbolic, and elliptic geometries; the celestial sphere; flow; and differential geometry. The second part addresses the line and the circle; regular polygons; motions; the triangle; invariants under homologies; rational curves; conics; the cardioid and the deltoid; Cremona transformations; and the n-line.

PART I

CHAPTER I

OPERATIONS OF ELEMENTARY GEOMETRY

§ 1. Instruments – Euclid’s plane geometry deals with figures, such as triangles, parallelograms, circles, in a plane. These are not pre-existent; they have to be either drawn or printed. In the phrase of Enriques, * they are werdende, not fertige. The drawing is done by instruments such as a straight rod or rule, or a combination of rods such as a triangle or a pair of compasses.

Let the instrument be a card lying on a table. We can then (thanks to the axioms) move the card freely over the table.

The card has a boundary; we take as the boundary a triangle, with three straight edges 1, 2, 3, and with the upper face marked. What we understand by straight is that the card can be turned over, keeping, say, the edge 1 fixed. The card will again fit the table, if both card and table are flat, and if the edge is straight. The marked face is now down.

We take this overturn as one of the fundamental operations of Euclidean geometry. If at first a point marked b of the card is over a point marked a of the table, then after the overturn the point marked b is over another point, say, a of the table. There is then set up in the table a correspondence of points with a line of fixed points, which we call a reflexion.

In the table is the record of the operation. By a usual trick of language, the same word reflexion will serve both for the operation and the correspondence which it sets up. Thus, instead of overturn we write reflexion.

Let us now make a second reflexion or overturn of the card, say about the edge marked 2. The marked face is now up. It seems that the result of the two reflexions could be obtained by moving the card along the table. Thus we get the idea that a planar displacement can be analysed into two reflexions. And this we take as a definition: a planar displacement is the product of two...