Lecture

Harmonic Mean: Geometric Projective Geometry

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In course

MATH-124: Geometry for architects I

Ce cours entend exposer les fondements de la géométrie à un triple titre : 1/ de technique mathématique essentielle au processus de conception du projet, 2/ d'objet privilégié des logiciels de conception paramétrique 3/ de composante à part entière de l'h ...

Description

This lecture explores the harmonic mean as a fundamental component of visual arts, allowing the development of the two fundamental invariants of projective geometry: harmonic & anharmonic bi-ratios. It delves into the historical context, from Claude Perrault's architectural contributions to Desargues' perspective division. The lecture also discusses the essential role of harmonic bundles and the invariance properties under various transformations, such as homotheties and projections. The construction of the harmonic mean is demonstrated through ruler and compass constructions, emphasizing its significance in modern projective geometry.

Instructor

Bernard Cache

Official source

Related concepts (32)

Harmonic mean

In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations.

Straightedge and compass construction

In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances.

Geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers a1, a2, ..., an, the geometric mean is defined as or, equivalently, as the arithmetic mean in logscale: Most commonly the numbers are restricted to being non-negative, to avoid complications related to negative numbers not having real roots, and frequently they are restricted to being positive, to enable the use of logarithms.

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.

Mean

There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the arithmetic mean, also known as "arithmetic average", is a measure of central tendency of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead bar, .

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