Coordinate system in the context of Principal component analysis

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.

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.

Geodesy or geodetics is the science of measuring and representing the geometry, gravity, and spatial orientation of the Earth in temporally varying 3D. It is called planetary geodesy when studying other astronomical bodies, such as planets or circumplanetary systems.

Geodynamical phenomena, including crustal motion, tides, and polar motion, can be studied by designing global and national control networks, applying space geodesy and terrestrial geodetic techniques, and relying on datums and coordinate systems.

In modern mapping, a topographic map or topographic sheet is a type of map characterized by large-scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and artificial features. A topographic survey is typically based upon a systematic observation and published as a map series, made up of two or more map sheets that combine to form the whole map. A topographic map series uses a common specification that includes the range of cartographic symbols employed, as well as a standard geodetic framework that defines the map projection, coordinate system, ellipsoid and geodetic datum. Official topographic maps also adopt a national grid referencing system.

Natural Resources Canada provides this description of topographic maps:

In geometry, a Cartesian coordinate system (UK: /kɑːrˈtiːzjən/, US: /kɑːrˈtiːʒən/) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system. The point where the axes meet is called the origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame.

Similarly, the position of any point in three-dimensional space can be specified by three Cartesian coordinates, which are the signed distances from the point to three mutually perpendicular planes. More generally, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are the signed distances from the point to n mutually perpendicular fixed hyperplanes.

A nomogram (from Greek νόμος (nomos) 'law' and γράμμα (gramma) 'that which is drawn'), also called a nomograph, alignment chart, or abac, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. The field of nomography was invented in 1884 by the French engineer Philbert Maurice d'Ocagne (1862–1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision. Nomograms use a parallel coordinate system invented by d'Ocagne rather than standard Cartesian coordinates.

A nomogram consists of a set of n scales, one for each variable in an equation. Knowing the values of n-1 variables, the value of the unknown variable can be found, or by fixing the values of some variables, the relationship between the unfixed ones can be studied. The result is obtained by laying a straightedge across the known values on the scales and reading the unknown value from where it crosses the scale for that variable. The virtual or drawn line, created by the straightedge, is called an index line or isopleth.

In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin, orientation, and scale have been specified in physical space. It is based on a set of reference points, defined as geometric points whose position is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers).An important special case is that of an inertial reference frame, a stationary or uniformly moving frame.

For n dimensions, n + 1 reference points are sufficient to fully define a reference frame. Using rectangular Cartesian coordinates, a reference frame may be defined with a reference point at the origin and a reference point at one unit distance from the origin along each of the n coordinate axes.

A graticule or grid (from Latin crāticula 'grill/grating'), on a map, is a graphical depiction of a coordinate system as a grid of coordinate curves or "lines", each curve/line representing a constant coordinate value. It is thus a form of isoline, and is commonly found on maps of many kinds, at scales from local to global.

The term graticule is almost always used to specifically refer to the parallels and meridians of latitude and longitude, respectively. In modern usage, graticules are contrasted with grids, which display the eastings and northings of a projected coordinate reference system, such as Universal Transverse Mercator – usually the coordinate system in which the map is drawn.

In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are

• the point's distance from a reference point called the pole, and
• the point's direction from the pole relative to the direction of the polar axis, a ray drawn from the pole.

The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The pole is analogous to the origin in a Cartesian coordinate system.

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.

In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are

• the radial distance r along the line connecting the point to a fixed point called the origin;
• the polar angle θ between this radial line and a given polar axis; and
• the azimuthal angle φ, which is the angle of rotation of the radial line around the polar axis.

(See graphic regarding the "physics convention".)

A spatial reference system (SRS) or coordinate reference system (CRS) is a framework used to precisely measure locations on, or relative to, the surface of Earth as coordinates. It is thus the application of the abstract mathematics of coordinate systems and analytic geometry to geographic space. A particular SRS specification (for example, "Universal Transverse Mercator WGS 84 Zone 16N") comprises a choice of Earth ellipsoid, horizontal datum, map projection (except in the geographic coordinate system), origin point, and unit of measure. Thousands of coordinate systems have been specified for use around the world or in specific regions and for various purposes, necessitating transformations between different SRS.

Although they date to the Hellenistic period, spatial reference systems are now a crucial basis for the sciences and technologies of Geoinformatics, including cartography, geographic information systems, surveying, remote sensing, and civil engineering. This has led to their standardization in international specifications such as the EPSG codes and ISO 19111:2019 Geographic information—Spatial referencing by coordinates, prepared by ISO/TC 211, also published by the Open Geospatial Consortium as Abstract Specification, Topic 2: Spatial referencing by coordinate.

In solar observation and imaging, coordinate systems are used to identify and communicate locations on and around the Sun. The Sun is made of plasma, so there are no permanent demarcated points that can be referenced.

Streptococcus, from Ancient Greek στρεπτός (streptós), meaning "twisted", and κόκκος (kókkos), meaning "grain", is a genus of gram-positive spherical bacteria that belongs to the family Streptococcaceae, within the order Lactobacillales (lactic acid bacteria), in the phylum Bacillota. Cell division in streptococci occurs along a single axis, thus when growing they tend to form pairs or chains, which may appear bent or twisted. This differs from staphylococci, which divide along multiple axes, thereby generating irregular, grape-like clusters of cells. Most streptococci are oxidase-negative and catalase-negative, and many are facultative anaerobes (capable of growth both aerobically and anaerobically).

The term was coined in 1877 by Viennese surgeon Albert Theodor Billroth (1829–1894), from Ancient Greek στρεπτός (streptós), meaning "twisted", and κόκκος (kókkos), meaning "grain". In 1984, many bacteria formerly grouped in the genus Streptococcus were separated out into the genera Enterococcus and Lactococcus. Currently, over 50 species are recognised in this genus. This genus has been found to be part of the salivary microbiome.