# Two-dimensional space

Euclidean planetwo-dimensional2Dtwo dimensions2-Dtwo dimensional2-dimensionalplanetwoaffine plane
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).wikipedia
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### Dimension

dimensionsdimensionalone-dimensional
For a generalization of the concept, see dimension.
A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere.

### Area

surface areaArea (geometry)area formula
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.
Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane.

### Cartesian coordinate system

Cartesian coordinatesCartesian coordinateCartesian
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.
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.

### Analytic geometry

analytical geometryCartesian geometrycoordinate geometry
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates.
Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions).

### Manifold

manifoldsboundarymanifold with boundary
The hypersphere in 2 dimensions is a circle, sometimes called a 1-sphere (S 1 ) because it is a one-dimensional manifold.
For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because (among other properties) it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts).

### Digon

{2}digons2
The regular henagon {1} and regular digon {2} can be considered degenerate regular polygons.
Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space.

### Conic section

conicconic sectionsconics
There are an infinitude of other curved shapes in two dimensions, notably including the conic sections: the ellipse, the parabola, and the hyperbola.
The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions.

### Angle

acute angleobtuse angleoblique
where θ is the angle between A and B.
Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane.

### Plane (geometry)

planeplanarplanes
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. Two-dimensional space can be seen as a projection of the physical universe onto a plane.
A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.

### Euclidean distance

Euclidean metricEuclideandistance
: the formula for the Euclidean length of the vector.
In the Euclidean plane, if p = (p 1, p 2 ) and q = (q 1, q 2 ) then the distance is given by

### Surface (topology)

surfaceclosed surfacesurfaces
In topology, the plane is characterized as being the unique contractible 2-manifold.
A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions (it has two degrees of freedom).

### Curve

closed curvespace curvesmooth curve
For some scalar field f : U ⊆ R 2 → R, the line integral along a piecewise smooth curve C ⊂ U is defined as
A plane curve is a curve for which X is the Euclidean plane—these are the examples first encountered—or in some cases the projective plane.

### Picture function

* Picture function
A picture function is a mathematical representation of a two-dimensional image as a function of two spatial variables.

### Parameter

parametersparametricargument
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).

### Universe

physical worldThe Universeuniverses
Two-dimensional space can be seen as a projection of the physical universe onto a plane.

### Euclidean space

EuclideanspaceEuclidean vector space
Usually, it is thought of as a Euclidean space and the two dimensions are called length and width.

### Euclid's Elements

ElementsEuclid's ''ElementsEuclid
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.

### Pythagorean theorem

Pythagoras' theoremPythagorasPythagoras's theorem
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.

### Point (geometry)

pointpointslocation
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. 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).

### Number

number systemnumericalnumbers
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.

### Negative number

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.

### Perpendicular

perpendicularlyPerpendicularitynormal
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.

### Unit vector

unit vectorsnormalizedunit
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.

### Origin (mathematics)

origincoordinate originzero point
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).

### Projection (linear algebra)

orthogonal projectionprojectionprojection operator
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.