Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space. In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as 'that which has no part'. In two-dimensional Euclidean space, a point is represented by an ordered pair (x, y) 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 (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a1, a2, … , an) 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 = { ( a 1 , a 2 , . . . a n ) | a 1 c 1 + a 2 c 2 + . . . a n c n = d } {displaystyle scriptstyle {L=lbrace (a_{1},a_{2},...a_{n})|a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d brace }} , where c1 through cn 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. A line segment consisting of only a single point is called a degenerate line segment.