**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Projective geometry

Summary

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.
Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an e

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related units

No results

Related courses (15)

MATH-535: Topics in algebraic geometry

This course is aimed to give students an introduction to the theory of algebraic curves and surfaces. In particular, it aims to develop the students' geometric intuition and combined with the basic algebraic geometry courses to build a strong foundation for further study.

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.

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

Related publications (17)

Loading

Loading

Loading

Related concepts (85)

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 b

Conic section

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the c

Projective space

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extensi

Related lectures (32)

Related people

No results

Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH (0)(A) and HH (0)(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.

This work is dedicated to developing algebraic methods for channel coding. Its goal is to show that in different contexts, namely single-antenna Rayleigh fading channels, coherent and non-coherent MIMO channels, algebraic techniques can provide useful tools for building efficient coding schemes. Rotated lattice signal constellations have been proposed as an alternative for transmission over the single-antenna Rayleigh fading channel. It has been shown that the performance of such modulation schemes essentially depends on two design parameters: the modulation diversity and the minimum product distance. Algebraic lattices, i.e., lattices constructed by the canonical embedding of an algebraic number field, or more precisely ideal lattices, provide an efficient tool for designing such codes, since the design criteria are related to properties of the underlying number field: the maximal diversity is guaranteed when using totally real number fields and the minimum product distance is optimized by considering fields with small discriminant. Furthermore, both shaping and labelling constraints are taken care of by constructing Zn-lattices. We present here the construction of several families of such n-dimensional lattices for any n, and compute their performance. We then give an upper bound on their minimal product distance, and show that with respect to this bound, existing lattice codes are optimal in the sense that no further significant coding gain could be reached. Cyclic division algebras have been introduced recently in the context of coherent Space-Time coding. These are non-commutative algebras which naturally yield families of invertible matrices, or in other words, linear codes that fulfill the rank criterion. In this work, we further exploit the algebraic structures of cyclic algebras to build Space-Time Block codes (STBCs) that satisfy the following properties: they have full rate, full diversity, non-vanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We give algebraic constructions of such STBCs for 2, 3, 4 and 6 antennas and show that these are the only cases where they exist. We finally consider the problem of designing Space-Time codes in the noncoherent case. The goal is to construct maximal diversity Space-Time codewords, subject to a fixed constellation constraint. Using an interpretation of the noncoherent coding problem in terms of packing subspaces according to a given metric, we consider the construction of non-intersecting subspaces on finite alphabets. Techniques used here mainly derive from finite projective geometry.

While over fields of characteristic at least 5, a normal, projective and Gorenstein del Pezzo surface is geometrically normal, this does not hold for characteristic 2 and 3. There is no characterization of all such non-geometrically normal surfaces, but there is a complete characterization of all possible surfaces that can arise as the normalization of the base change to the algebraic closure of the base field. In four of these instances, for a given normalization, we describe the construction of such a non-geometrically normal surface.