# Two-dimensional space

Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). In Mathematics, it is commonly represented by the symbol 2. For a generalization of the concept, see dimension.

Two-dimensional space can be seen as a projection of the physical universe onto a plane. Usually, it is thought of as a Euclidean space and the two dimensions are called length and width.

## History

Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics.

Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions, and did not publish the discovery.[1] Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.[2]

Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematician Caspar Wessel (1745–1818).[3] Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

## In geometry

### Coordinate systems

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin. They are usually labeled x and y. Relative to these axes, the position of any point in two-dimensional space is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the other axis.

Another widely used coordinate system is the polar coordinate system, which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray.

### Polytopes

In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below:

#### Convex

The Schläfli symbol {p} represents a regular p-gon.

Name Triangle
(2-simplex)
Square
(2-orthoplex)
(2-cube)
Pentagon Hexagon Heptagon Octagon
Schläfli {3} {4} {5} {6} {7} {8}
Image
Name Nonagon Decagon Hendecagon Dodecagon Tridecagon Tetradecagon
Schläfli {9} {10} {11} {12} {13} {14}
Image
Schläfli {15} {16} {17} {18} {19} {20} {n}
Image

#### Degenerate (spherical)

The regular henagon {1} and regular digon {2} can be considered degenerate regular polygons. They can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus.

Name Schläfli Henagon Digon {1} {2}

#### Non-convex

There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.

In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m} = {n/(nm)}) and m and n are coprime.

 Name Schläfli Image Pentagram Heptagrams Octagram Enneagrams Decagram ...n-agrams {5/2} {7/2} {7/3} {8/3} {9/2} {9/4} {10/3} {n/m}

### Circle

The hypersphere in 2 dimensions is a circle, sometimes called a 1-sphere (S1) because it is a one-dimensional manifold. In a Euclidean plane, it has the length 2πr and the area of its interior is

${\displaystyle A=\pi r^{2}}$

where ${\displaystyle r}$ is the radius.

### Other shapes

There are an infinitude of other curved shapes in two dimensions, notably including the conic sections: the ellipse, the parabola, and the hyperbola.

## In linear algebra

Another mathematical way of viewing two-dimensional space is found in linear algebra, where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors.

### Dot product, angle, and length

The dot product of two vectors A = [A1, A2] and B = [B1, B2] is defined as:[4]

${\displaystyle \mathbf {A} \cdot \mathbf {B} =A_{1}B_{1}+A_{2}B_{2}}$

A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by ${\displaystyle \|\mathbf {A} \|}$. In this viewpoint, the dot product of two Euclidean vectors A and B is defined by[5]

${\displaystyle \mathbf {A} \cdot \mathbf {B} =\|\mathbf {A} \|\,\|\mathbf {B} \|\cos \theta ,}$

where θ is the angle between A and B.

The dot product of a vector A by itself is

${\displaystyle \mathbf {A} \cdot \mathbf {A} =\|\mathbf {A} \|^{2},}$

which gives

${\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }},}$

the formula for the Euclidean length of the vector.

## In calculus

In a rectangular coordinate system, the gradient is given by

${\displaystyle \nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} }$

### Line integrals and double integrals

For some scalar field f : UR2R, the line integral along a piecewise smooth curve CU is defined as

${\displaystyle \int \limits _{C}f\,ds=\int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt.}$

where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and ${\displaystyle a.

For a vector field F : UR2R2, the line integral along a piecewise smooth curve CU, in the direction of r, is defined as

${\displaystyle \int \limits _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt.}$

where · is the dot product and r: [a, b] → C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.

A double integral refers to an integral within a region D in R2 of a function ${\displaystyle f(x,y),}$ and is usually written as:

${\displaystyle \iint \limits _{D}f(x,y)\,dx\,dy.}$

### Fundamental theorem of line integrals

The fundamental theorem of line integrals says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

Let ${\displaystyle \varphi :U\subseteq \mathbb {R} ^{2}\to \mathbb {R} }$. Then

${\displaystyle \varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)=\int _{\gamma [\mathbf {p} ,\,\mathbf {q} ]}\nabla \varphi (\mathbf {r} )\cdot d\mathbf {r} .}$

### Green's theorem

Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then[6][7]

${\displaystyle \oint _{C}(L\,dx+M\,dy)=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dx\,dy}$

where the path of integration along C is counterclockwise.

## In topology

In topology, the plane is characterized as being the unique contractible 2-manifold.

Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not simply connected.

## In graph theory

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other.[8] Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

## References

1. ^ "Analytic geometry". Encyclopædia Britannica (Encyclopædia Britannica Online ed.). 2008.
2. ^ Burton 2011, p. 374
3. ^ Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806. (Whittaker & Watson, 1927, p. 9)
4. ^ S. Lipschutz; M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1.
5. ^ M.R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7.
6. ^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
7. ^ Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7
8. ^ Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 64. ISBN 978-0-486-67870-2. Retrieved 8 August 2012. Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.

2D

2D or 2-D may refer to:

Two-dimensional space

2D geometric model

2D computer graphics

2-D (character), a member of the virtual band Gorillaz

Index finger, the second digit (abbreviated 2D) of the hand

Oflag II-D

Stalag II-D

Transcription factor II D

Two Dickinson Street Co-op, a student dining cooperative at Princeton University

Two dimensional correlation analysis

Nintendo 2DS

2degrees, New Zealand telecommunications provider

Twopence (British pre-decimal coin), routinely abbreviated 2d.

ACG (subculture)

The ACG is an abbreviation of "Anime, Comic and Games", used in some subcultures of Greater China . Because a strong economic and cultural connection exists between anime, manga and games in the Japanese market, ACG is used to describe this phenomenon in relative fields. The term refers in particular to Japanese anime, manga and video games, with the video games usually referring to galgames. The term is not normally translated into Chinese; if the meaning needs to be translated, it is usually "動漫遊戲" (dòngmànyóuxì, animation, comics and games), "two-dimensional space" (二次元, Èr cìyuán; Japanese: 2次元, translit. nijigen) or "動漫遊" (dòngmànyóu, animation, comics and games).

Apeirogonal tiling

In geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this type include:

Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces

Order-3 apeirogonal tiling, hyperbolic tiling with 3 apeirogons around a vertex

Order-4 apeirogonal tiling, hyperbolic tiling with 4 apeirogons around a vertex

Order-5 apeirogonal tiling, hyperbolic tiling with 5 apeirogons around a vertex

Infinite-order apeirogonal tiling, hyperbolic tiling with an infinite number of apeirogons around a vertex

Cauchy–Schwarz inequality

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics.The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first proved by

Viktor Bunyakovsky (1859). The modern proof of the integral inequality was given by Hermann Amandus Schwarz (1888).

Circular segment

In geometry, a circular segment (symbol: ⌓) is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc (of less than 180°) of a circle and by the chord connecting the endpoints of the arc.

Cohen–Sutherland algorithm

The Cohen–Sutherland algorithm is a computer-graphics algorithm used for line clipping. The algorithm divides a two-dimensional space into 9 regions and then efficiently determines the lines and portions of lines that are visible in the central region of interest (the viewport).

The algorithm was developed in 1967 during flight-simulator work by Danny Cohen and Ivan Sutherland.

Covariance matrix

In probability theory and statistics, a covariance matrix, also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix, is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of observed empirical values or a finite or infinite number of potential values. The potential values are specified by a theoretical joint probability distribution.

Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the ${\displaystyle x}$ and ${\displaystyle y}$ directions contain all of the necessary information; a ${\displaystyle 2\times 2}$ matrix would be necessary to fully characterize the two-dimensional variation.

Because the covariance of the i-th random variable with itself is simply that random variable's variance, each element on the principal diagonal of the covariance matrix is the variance of one of the random variables. Because the covariance of the i-th random variable with the j-th one is the same thing as the covariance of the j-th random variable with the i-th random variable, every covariance matrix is symmetric. Also, every covariance matrix is positive semi-definite.

The auto-covariance matrix of a random vector ${\displaystyle \mathbf {X} }$ is typically denoted by ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}$ or ${\displaystyle \Sigma }$.

Cross section (geometry)

In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross sections. The boundary of a cross section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation.

In technical drawing a cross-section, being a projection of an object onto a plane that intersects it, is a common tool used to depict the internal arrangement of a 3-dimensional object in two dimensions. It is traditionally crosshatched with the style of crosshatching often indicating the types of materials being used.

With computed axial tomography, computers construct cross-sections from x-ray data.

Degrees of freedom (mechanics)

In physics, the degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration. It is the number of parameters that determine the state of a physical system and is important to the analysis of systems of bodies in mechanical engineering, aeronautical engineering, robotics, and structural engineering.

The position of a single railcar (engine) moving along a track has one degree of freedom because the position of the car is defined by the distance along the track. A train of rigid cars connected by hinges to an engine still has only one degree of freedom because the positions of the cars behind the engine are constrained by the shape of the track.

An automobile with highly stiff suspension can be considered to be a rigid body traveling on a plane (a flat, two-dimensional space). This body has three independent degrees of freedom consisting of two components of translation and one angle of rotation. Skidding or drifting is a good example of an automobile's three independent degrees of freedom.

The position and orientation of a rigid body in space is defined by three components of translation and three components of rotation, which means that it has six degrees of freedom.

The exact constraint mechanical design method manages the degrees of freedom to neither underconstrain nor overconstrain a device.

Flat (geometry)

In geometry, a flat is a subset of a Euclidean space that is congruent to a Euclidean space of lower dimension. The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes.

In a n-dimensional space, there are flats of every dimension from 0 to n − 1. Flats of dimension n − 1 are called hyperplanes.

Flats are the affine subspaces of Euclidean spaces, which means that they are similar to linear subspaces, except that they need not pass through the origin. Flats occurs in linear algebra, as geometric realizations of solution sets of systems of linear equations.

A flat is manifold and an algebraic variety, and is sometimes called linear manifold or linear variety for distinguishing it from other manifolds or varieties.

Frustrated triangular lattice

Geometrical frustration occurs when a set of degrees of freedom is incompatible with the space it occupies.

A purely geometric example of this is the impossibility of close-packing pentagons in two dimensions. Another is atomic magnetic moments with antiferromagnetic interactions. These moments lower their interaction energy by pointing antiparallel to their neighbors.

In the case of two dimensional space, the triangular lattice is the simplest example of such frustration. With a triangular lattice, the two spins can easily accommodate two sides, but the third spin is frustrated. If this third spin is up, then two arrangements out of the three are compatible, but one is incompatible. This leads to a huge degeneracy in the ground state with non-zero entropy. This frustration leads to breaking symmetry, which leads to ferroelectricity.

Molecule editor

A molecule editor is a computer program for creating and modifying representations of chemical structures.

Molecule editors can manipulate chemical structure representations in either a simulated two-dimensional space or three-dimensional space, via 2D computer graphics or 3D computer graphics, respectively. Two-dimensional output is used as illustrations or to query chemical databases. Three-dimensional output is used to build molecular models, usually as part of molecular modelling software packages.

Database molecular editors such as Leatherface, RECAP, and Molecule Slicer allow large numbers of molecules to be modified automatically according to rules such as 'deprotonate carboxylic acids' or 'break exocyclic bonds' that can be specified by a user.

Molecule editors typically support reading and writing at least one file format or line notation. Examples of each include Molfile and simplified molecular input line entry specification (SMILES), respectively.

Files generated by molecule editors can be displayed by molecular graphics tools.

Overlapping circles grid

An overlapping circles grid is a geometric pattern of repeating, overlapping circles of equal radii in two-dimensional space. Commonly, designs are based on circles centered on triangles (with the simple, two circle form named vesica piscis) or on the square lattice pattern of points.

Patterns of seven overlapping circles appear in historical artefacts from the 7th century BC onwards; they become a frequently used ornament in the Roman Empire period, and survive into medieval artistic traditions both in Islamic art (girih decorations) and in Gothic art. The name "Flower of Life" is given to the overlapping circles pattern in New Age publications.

Of special interest is the six petal rosette derived from the "seven overlapping circles" pattern, also known as "Sun of the Alps" from its frequent use in alpine folk art in the 17th and 18th century.

Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

When working exclusively in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, or, in other words, in the plane.

In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes (integrating over lines is known as the X-ray transform). It was later generalized to higher-dimensional Euclidean spaces, and more broadly in the context of integral geometry. The complex analog of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.

Solid torus

In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product ${\displaystyle S^{1}\times D^{2}}$ of the disk and the circle, endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus.

Two-dimensional graph

A two-dimensional graph is a set of points in two-dimensional space. If the points are real and if Cartesian coordinates are used, each axis depicts the potential values of a particular real variable. Often the variable on the horizontal axis is called x and the one on the vertical axis is called y, in which case the horizontal and vertical axes are sometimes called the x axis and y axis respectively. With real variables on the axes, each point in the graph depicts the values of two real variables.

Alternatively, each point in a graph may depict the value of a single complex variable. This two-dimensional graph is called Argand diagram. In the Argand diagram, the horizontal axis is called the real axis and depicts the potential values of the real part of the complex number, while the vertical axis is called the imaginary axis and depicts the potential values of the imaginary part of the complex number.

Two-dimensional liquid

A two-dimensional liquid (2D liquid) is a collection of objects constrained to move in a planar or other two-dimensional space in a liquid state.

Uncinate process of ethmoid bone

In the ethmoid bone, a sickle shaped projection, the uncinate process, projects posteroinferiorly from the ethmoid labyrinth.

Between the posterior edge of this process and the anterior surface of the ethmoid bulla, there is a two-dimensional space, resembling a crescent shape. This space continues laterally as a three-dimensional slit-like space - the ethmoidal infundibulum. This is bounded by the uncinate process, medially, the orbital lamina of ethmoid bone (lamina papyracea), laterally, and the ethmoidal bulla, posterosuperiorly. This concept is easier to understand if one imagine the infundibulum as a prism so that its medial face is the hiatus semilunaris. The "lateral face" of this infundibulum contains the ostium of the maxillary sinus, which, therefore, opens into the infundibulum.

Dimensional spaces
Other dimensions
Polytopes and shapes
Dimensions by number

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