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Lecture

Inversion and Circle Orthogonality

Description

This lecture covers the concept of inversion in geometry, focusing on the relationship between points on a disk and their inverses. The instructor explains how to find the inverse of a point with respect to a circle, demonstrating the construction process and the properties of this geometric transformation. The lecture also delves into the power of a point with respect to a circle and how it can be used to construct orthogonal circles, leading to a non-Euclidean geometry model with unique properties.

Instructor

Bernard Cache

Official source

Related concepts (32)

Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries.

Euclidean geometry

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.

Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point.

Elliptic geometry

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

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