Lecture

Projective Geometry: Birth and Foundations

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In course

MATH-126: Geometry for architects II

Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective. ...

Description

This lecture introduces the birth of projective geometry, exploring the perspective of Desargues and the gnomonic foundation. It delves into the vanishing point invention, the meetings of cones and planes in stereotomy, and cylindrical conical tubes.

Instructor

Bernard Cache

Official source

Related concepts (31)

Pascal's theorem

In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem, Latin for mystical hexagram) states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after Blaise Pascal.

Brianchon's theorem

In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1783–1864). Let be a hexagon formed by six tangent lines of a conic section. Then lines (extended diagonals each connecting opposite vertices) intersect at a single point , the Brianchon point. The polar reciprocal and projective dual of this theorem give Pascal's theorem.

Girard Desargues

Girard Desargues (ʒiʁaʁ dezaʁɡ; 21 February 1591 - September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour. Born in Lyon, Desargues came from a family devoted to service to the French crown. His father was a royal notary, an investigating commissioner of the Seneschal's court in Lyon (1574), the collector of the tithes on ecclesiastical revenues for the city of Lyon (1583) and for the diocese of Lyon.

Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points.

Projective harmonic conjugate

In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction: Given three collinear points A, B, C, let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively. If AN and BM meet at K, and LK meets AB at D, then D is called the harmonic conjugate of C with respect to A and B. The point D does not depend on what point L is taken initially, nor upon what line through C is used to find M and N.

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