# Point (geometry)

﻿
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object. Because of their nature as one of the simplest geometric concepts, they are often used in one form or another as the fundamental constituents of geometry, physics, vector graphics, and many other fields.

Points in Euclidean geometry

Points are most often considered within the framework of Euclidean geometry, where they are one of the fundamental objects. Euclid originally defined the point vaguely, as "that which has no part". In two dimensional Euclidean space, a point is represented by an ordered pair, $, \left(x,y\right)$, of numbers, where the first number conventionally represents the horizontal and is often denoted by $, x$, and the second number conventionally represents the vertical and is often denoted by $, y$. This idea is easily generalized to three dimensional Euclidean space, where a point is represented by an ordered triplet, $, \left(x,y,z\right)$, with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms,$, \left(a_1,a_2,...,a_n\right)$ where n is the dimension of the space in which the point is located.

Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line is an infinite set of points of the form $, L = lbrace \left(a_1,a_2,...a_n\right)|a_1c_1 + a_2c_2 + ... a_nc_n = d brace$, where $, c_1$ through $, c_n$ and $, d$ are constants and n is the dimension of the space. Similar constructions exist that define the plane, line segment and other related concepts.

In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line, this is easily confirmed under modern expansions of Euclidean geometry, and had grave consequences at the time of its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's axiomatization of points was neither complete nor definitive, as he occasionally assumed facts that didn't follow directly from his axioms, such as the ordering of points on the line or the existence of specific points, but in spite of this, modern expansions of the system have since removed these assumptions.

Points in Branches of Mathematics

A point in point-set topology is defined as a member of the underlying set of a topological space.

ee also

* Accumulation point
* Affine space
* Boundary point
* Critical point
* Singular point
* Cusp

* [http://www.mathopenref.com/point.html Definition of Point] with interactive applet
* [http://www.mathopenref.com/tocs/pointstoc.html Points definition pages] With interactive animations that are also useful in a classroom setting. Math Open Reference
*planetmath reference|id=8173|title=Point

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Mass point geometry — Mass point geometry, colloquially known as mass points, is a geometry problem solving technique which applies the physical principle of the center of mass to geometry problems involving triangles and intersecting cevians.[1] All problems that can …   Wikipedia

• Point — may refer to the following: Contents 1 Business and finance 2 Engineering 3 Entertainment …   Wikipedia

• Geometry — (Greek γεωμετρία ; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of… …   Wikipedia

• Geometry Wars — est une série de jeux vidéo créée par Bizarre Creations et éditée par Microsoft Game Studios, débutée en novembre 2003. Chaque titre de la série est un shoot them up multidirectionnel de type manic shooter au style graphique géométrique.… …   Wikipédia en Français

• Geometry Wars: Galaxies — Geometry Wars Geometry Wars Éditeur Microsoft Game Studios Genre Shoot them up multidirectionnel Début de la série Novembre 2003 Plate forme Nintendo DS Wii …   Wikipédia en Français

• Geometry Wars: Retro Evolved — Geometry Wars Geometry Wars Éditeur Microsoft Game Studios Genre Shoot them up multidirectionnel Début de la série Novembre 2003 Plate forme Nintendo DS Wii …   Wikipédia en Français

• Geometry Wars: Retro Evolved 2 — Geometry Wars Geometry Wars Éditeur Microsoft Game Studios Genre Shoot them up multidirectionnel Début de la série Novembre 2003 Plate forme Nintendo DS Wii …   Wikipédia en Français

• point — [point] n. [OFr, dot, prick < L punctum, dot, neut. of punctus, pp. of pungere, to prick (< IE base * peuĝ , *peuk̑, to prick, jab > Ger fichte, spruce tree, L pugil, boxer, pugnus, fist); also < OFr pointe, sharp end < ML puncta… …   English World dictionary

• point — ► NOUN 1) the tapered, sharp end of a tool, weapon, or other object. 2) a particular spot, place, or moment. 3) an item, detail, or idea in a discussion, text, etc. 4) (the point) the most significant or relevant factor or element. 5) advantage… …   English terms dictionary

• Point plotting — is an elementary mathematical skill required in analytic geometry. Invented by René Descartes and originally used to locate positions on military maps, this skill is now assumed of everyone who wants to locate grid 7A on any map.Using point… …   Wikipedia