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Lecture

Projective Geometry: Historical Overview

Description

This lecture provides an overview of the historical development of projective geometry, focusing on key figures such as Alberti and Piero della Francesca. It explores how these mathematicians contributed to the understanding of perspective and distance points, shedding light on their innovative approaches and the impact of their work.

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Instructor

Bernard Cache

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Related concepts (24)

Duality (projective geometry)

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration.

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa.

Piero della Francesca

Piero della Francesca (ˌpjɛəroʊ_ˌdɛlə_frænˈtʃɛskə, USalso-_frɑːnˈ-, ˈpjɛːro della franˈtʃeska; ; 1415 – 12 October 1492) was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca is chiefly appreciated for his art. His painting is characterized by its serene humanism, its use of geometric forms and perspective. His most famous work is the cycle of frescoes The History of the True Cross in the church of San Francesco in the Tuscan town of Arezzo.

Distance

Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). Since spatial cognition is a rich source of conceptual metaphors in human thought, the term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as statistical distance between probability distributions or edit distance between strings of text) or a degree of separation (as exemplified by distance between people in a social network).

Great-circle distance

The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics.

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