# Affine plane (incidence geometry)

In geometry, an affine plane is a system of points and lines that satisfy the following axioms: * Any two distinct points lie on a unique line. * Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. (Playfair's axiom) * There exist three non-collinear points (points not on a single line). In an affine plane, two lines are called parallel if they are equal or disjoint. Using this definition, Playfair's axiom above can be replaced by: * Given a point and a line, there is a unique line which contains the point and is parallel to the line. Parallelism is an equivalence relation on the lines of an affine plane. Since no concepts other than those involving the relationship between points and lines are involved in the axioms, an affine plane is an object of study belonging to incidence geometry. They are non-degenerate linear spaces satisfying Playfair's axiom. The familiar Euclidean plane is an affine plane. There are many finite and infinite affine planes. As well as affine planes over fields (and division rings), there are also many non-Desarguesian planes, not derived from coordinates in a division ring, satisfying these axioms. The Moulton plane is an example of one of these. (Wikipedia).

algebraic geometry 17 Affine and projective varieties

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the relation between affine and projective varieties, with some examples such as a cubic curve and the twisted cubic.

From playlist Algebraic geometry I: Varieties

Introduction to Projective Geometry (Part 2)

The second video in a series about projective geometry. We list the axioms for projective planes, give an examle of a projective plane with finitely many points, and define the real projective plane.

From playlist Introduction to Projective Geometry

algebraic geometry 5 Affine space and the Zariski topology

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the definition of affine space and its Zariski topology.

From playlist Algebraic geometry I: Varieties

algebraic geometry 15 Projective space

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It introduces projective space and describes the synthetic and analytic approaches to projective geometry

From playlist Algebraic geometry I: Varieties

Calculus and affine geometry of the magical parabola | Algebraic Calc and dCB curves 3 | Wild Egg

Algebraic Calculus naturally lives in affine geometry, not Euclidean geometry. Affine geometry is the geometry of parallelism, or (almost the same thing) --- the geometry of pure linear algebra. The parabola is characterized projectively in this geometry as the unique conic which is tangen

From playlist Algebraic Calculus One Info

algebraic geometry 24 Regular functions

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers regular functions on affine and quasiprojective varieties.

From playlist Algebraic geometry I: Varieties

Introduction to Signed Area b) | Algebraic Calculus One | Wild Egg

This is a lecture in the Algebraic Calculus One course, which will present an exciting new approach to calculus, sticking with rational numbers and high school algebra, and avoiding all "infinite processes", "real numbers" and other modern fantasies. The course will be carefully framed on

From playlist Algebraic Calculus One from Wild Egg

Points and Lines in the Affine Plane | Algebraic Calculus One | Wild Egg

This is the first lecture in the Algebraic Calculus One course, which will present an exciting new approach to calculus, sticking with rational numbers and high school algebra. The course will be carefully framed on careful definitions, explicit examples and concrete computations. In thi

From playlist Algebraic Calculus One

Filiz Dogru: Outer Billiards: A Comparison Between Affine, Hyperbolic, and Symplectic Geometry

Filiz Dogru, Grand Valley State University Title: Outer Billiards: A Comparison Between Affine Geometry, Hyperbolic Geometry, and Symplectic Geometry Outer billiards appeared first as an entertainment question. Its popularity increased after J. Moser’s description as a crude model of the p

Spectrahedral lifts of convex sets – Rekha Thomas – ICM2018

Control Theory and Optimization Invited Lecture 16.6 Spectrahedral lifts of convex sets Rekha Thomas Abstract: Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expr

From playlist Control Theory and Optimization

Lecture 2A: What is a "Mesh?" (Discrete Differential Geometry)

From playlist Discrete Differential Geometry - CMU 15-458/858

algebraic geometry 26 Affine algebraic sets and commutative rings

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the relation between morphisms of affine algebraic sets and homomorphisms of commutative rings. As examples it describes some homomorphisms of commutative rings

From playlist Algebraic geometry I: Varieties

A brief history of geometry II: The European epoch | Sociology and Pure Mathematics | N J Wildberger

Let's have a quick overview of some of the developments in the European story of geometry -- at least up to the 19th century. We'll discuss Cartesian geometry, Projective geometry, Descriptive geometry, Algebraic geometry and Differential geometry. This is meant for people from outside m

From playlist Sociology and Pure Mathematics

Laura Starkston: Unexpected symplectic fillings of links of rational surface singularities

HYBRID EVENT Recorded during the meeting "Milnor Fibrations, Degenerations and Deformations from Modern Perspectives" the September 09, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given

From playlist Virtual Conference

The topology of proper toric maps - Mark Andrea de Cataldo

Mark Andrea de Cataldo Stony Brook University; Member, School of Mathematics October 1, 2014 I will discuss some of the topology of the fibers of proper toric maps and a combinatorial invariant that comes out of this picture. Joint with Luca Migliorini and Mircea Mustata. More videos on

From playlist Mathematics

What are opposite rays

👉 Learn essential definitions of points, lines, and planes. A point defines a position in space. A line is a set of points. A line can be created by a minimum of two points. A plane is a flat surface made up of at least three points. A plane contains infinite number of lines. A ray is a li

From playlist Points Lines and Planes

Felix Klein Lectures 2020: Quiver moduli and applications, Markus Reineke (Bochum), Lecture 4

Quiver moduli spaces are algebraic varieties encoding the continuous parameters of linear algebra type classification problems. In recent years their topological and geometric properties have been explored, and applications to, among others, Donaldson-Thomas and Gromov-Witten theory have

What are opposite rays

👉 Learn essential definitions of points, lines, and planes. A point defines a position in space. A line is a set of points. A line can be created by a minimum of two points. A plane is a flat surface made up of at least three points. A plane contains infinite number of lines. A ray is a li

From playlist Points Lines and Planes

What are opposite Rays

👉 Learn essential definitions of points, lines, and planes. A point defines a position in space. A line is a set of points. A line can be created by a minimum of two points. A plane is a flat surface made up of at least three points. A plane contains infinite number of lines. A ray is a li

From playlist Points Lines and Planes