Point (geometry) in the context of "Coordinate axis"

A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimensional spaces are often called planes, or, more generally, surfaces. These include analogs to physical spaces, like flat planes, and curved surfaces like spheres, cylinders, and cones, which can be infinite or finite. Some two-dimensional mathematical spaces are not used to represent physical positions, like an affine plane or complex plane.

In geometry, a three-dimensional space is a mathematical space in which three values (termed coordinates) are required to determine the position of a point. Alternatively, it can be referred to as 3D space, 3-space or, rarely, tri-dimensional space. Most commonly, it means the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may refer colloquially to a subset of space, a three-dimensional region (or 3D domain), a solid figure.

Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space. The set of these n-tuples is commonly denoted ${\displaystyle \mathbb {R} ^{n},}$ and can be identified to the pair formed by a n-dimensional Euclidean space and a Cartesian coordinate system.When n = 3, this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics, it serves as a model of the physical universe, in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that these directions do not lie in the same plane. Furthermore, if these directions are pairwise perpendicular, the three values are often labeled by the terms width/breadth, height/depth, and length.

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 branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.

A point source of pollution is a single identifiable source of air, water, thermal, noise or light pollution. A point source has negligible extent, distinguishing it from other pollution source geometrics (such as nonpoint source or area source). The sources are called point sources because in mathematical modeling, they can be approximated as a mathematical point to simplify analysis. Pollution point sources are identical to other physics, engineering, optics, and chemistry point sources and include:

• Air pollution from an industrial source (rather than an airport or a road, considered a line source, or a forest fire, which is considered an area source, or volume source)
• Water pollution from factories, power plants, municipal sewage treatment plants and some farms (see concentrated animal feeding operation). The U.S. Clean Water Act also defines municipal separate storm sewer systems and industrial stormwater discharges (such as construction sites) as point sources.
  • Man-made, natural, and groundwater reservoirs can all be contaminated by point source pollution which can threaten human health and safety.
• Noise pollution from a jet engine
• Disruptive seismic vibration from a localized seismic study
• Light pollution from an intrusive street light
• Radio emissions from an interference-producing electrical device

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.

A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians (180 degrees); and the other arc, the major arc, subtends an angle greater than π radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula or a mathematical expression. More formally, a mathematical symbol is any grapheme used in mathematical formulas and expressions. As formulas and expressions are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other types of mathematical object. As the number of these types has increased, the Greek alphabet and some Hebrew letters have also come to be used. For more symbols, other typefaces are also used, mainly boldface ⁠${\displaystyle \mathbf {a,A,b,B} ,\ldots }$⁠, script typeface ${\displaystyle {\mathcal {A,B}},\ldots }$ (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur ⁠${\displaystyle {\mathfrak {a,A,b,B}},\ldots }$⁠, and blackboard bold ⁠${\displaystyle \mathbb {N,Z,Q,R,C,H,F} _{q}}$⁠ (the other letters are rarely used in this face, or their use is unconventional). It is commonplace to use alphabets, fonts and typefaces to group symbols by type (for example, boldface is often used for vectors and uppercase for matrices).

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.

Most commonly, the ambient space is n-dimensional Euclidean space, in which case the hyperplanes are the (n − 1)-dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces.

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.

A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.

In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.