Point (geometry)

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The point in geometry is one of the fundamental entities of geometry, together with the line and the plane, since they are considered primary concepts, that is, it is only possible to describe them in relation to other similar or similar items. The point lacks length, thickness or thickness. They are usually described based on the characteristic postulates, which determine the relationships between the fundamental geometric entities. The point is the simplest, irreducibly minimal unit of visual communication; it is a dimensionless geometric figure, it also has no length, area, volume, or any other dimensional angle. It is not a physical object. Describes a position in the plane, determined with respect to a pre-established coordinate system.

History

The concept of point as a geometric entity arises from the ancient Greek conception of geometry compiled in Alexandria by Euclid in his treatise The Elements, giving an exclusive definition of point: «what has no part". The point, in classical geometry, is based on the idea that it was an intuitive concept, the geometric entity "without dimensions" and it was only necessary to assume the notion of point.

Graphic representation

Examples of eight points located in the Cartesian plane through their coordinate pairs.

In some geometry texts, a small cross (+), circle (o), square, or triangle is often used. In relation to other figures, it is usually represented with a small perpendicular segment when it belongs to a line, ray or segment.

Points are often named with a capital letter: A, B, C, etc. (to the lines with lowercase letters, and to the angles with Greek letters).

The way to represent a point by two intersecting segments (a small “cross” +) presupposes that the point is the intersection. When represented by a small circle, circumference, or other geometric figure, it assumes that the point is its center.

Geometric determination

A point can be determined with various reference systems:

In the Cartesian coordinate system, it is determined by the orthogonal distances to the principal axes, which are indicated by two letters or numbers: (x, y) in the plane; and with three in the space (x, y, z).

In polar coordinates, through its distance from the center and the angular measure with respect to the reference axis: (r, θ).

In spherical coordinates, through its distance from the center and the angular measure with respect to the reference axes: (r, θ, φ).

In cylindrical coordinates, using radial, azimuthal and height coordinates: (u, φ, z).

Elliptic, parabolic, spheroidal, toridal, etc. coordinate systems can also be used.

Dimension of a point

There are several non-equivalent definitions of dimension in mathematics. In all common definitions, a point is of dimension 0.

Dimension of vector space

The dimension of a vector space is the maximum size of a linearly independent subset. In a vector space consisting of a single point (which should be the zero vector 0), there is no linearly independent subset. The zero vector is not in itself linearly independent, because there is a non-trivial linear combination that makes it zero: ${displaystyle 1cdot mathbf {0} =mathbf {0} }$ .

Topological dimension

The topological dimension of a topological space $X$ defined as the minimum value of nsuch that all finite open coverage ${mathcal {A}}$ of $X$ supports a thin open deck ${mathcal {B}}$ of $X$ to refine ${mathcal {A}}$ where no point is included in more than n+1 elements. If there is no such minimum n, it is said that space is of infinite coverage dimension.

A point is zero dimensional with respect to the dimension that covers it because every open cover of space has a consistent refinement of an open set.

Hausdorff dimension

Sea X a metric space. Yeah. S X and d certified [0, ∞), the Hausdorff content- dimensional. d of S is the infimum number set δ ≥ 0 such that there is some collection (indexed) of balls ${displaystyle {B(x_{i},r_{i}):iin I}}$ covering S with r_i  0 for each i 한 I which satisfies $<math alttext="{displaystyle sum _{iin I}r_{i}^{d}␡ ␡ i한 한 Irid.δ δ {displaystyle sum _{iin I}r_{i}^{d}{d}<img alt="{displaystyle sum _{iin I}r_{i}^{d}$

The Hausdorff dimension of X is defined by

{displaystyle operatorname {dim} _{operatorname {H} }(X):=inf{dgeq 0:C_{H}^{d}(X)=0}.}

A point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius.

Points, lines and planes: relative positions

Given three or more points in the plane or in space (as appropriate), they can be divided into sets that do or do not satisfy the following conditions. Collinear: the so-called collinears are those contained in a straight line. Coplanar: Coplanar points are those that are contained in the same plane.

Some postulates and theorems related to the point

Postulates in Euclidean geometry

At one point they pass infinite straights and planes.
Two points determine one straight and only one.
A straight line contains infinite points.
A plane contains infinite points and infinite straight lines.
The space contains infinite points, straight and planes.

These postulates can be generalized for n-dimensional spaces.

Theorems in Euclidean geometry

Three unaligned points determine a plane and only one.

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