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 Two-dimensional space Two-dimensional space or 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 ℝ2. For a generalization of the concept, see dimension. Two-dimensional space 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.Contents1 History 2 In geometry2.1 Coordinate systems 2.2 Polytopes2.2.1 Convex 2.2.2 Degenerate (spherical) 2.2.3 Non-convex2.3 Circle 2.4 Other shapes3 In linear algebra3.1 Dot product, angle, and length4 In calculus4.1 Gradient 4.2 Line integrals and double integrals 4.3 Fundamental theorem of line integrals 4.4 Green's theorem5 In topology 6 In graph theory 7 References 8 See alsoHistory Books I through IV and VI of Euclid's Elements Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem 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 La Géométrie was translated into Latin in 1649 by Frans van Schooten 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 Norwegian-Danish 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 Main article: Plane (geometry) See also: Euclidean geometry Coordinate systems Further information: Coordinate system "Plane coordinates" redirects here. It is not to be confused with Coordinate plane. 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.Cartesian coordinate systemPolar coordinate systemPolytopes Main article: Polygon In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below: Convex The Schläfli symbol Schläfli symbol p represents a regular p-gon.Name Triangle (2-simplex) Square (2-orthoplex) (2-cube) Pentagon Hexagon Heptagon OctagonSchläfli 3 4 5 6 7 8 ImageName Nonagon Decagon Hendecagon Dodecagon Tridecagon TetradecagonSchläfli 9 10 11 12 13 14 ImageName Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon ...n-gonSchläfli 15 16 17 18 19 20 n ImageDegenerate (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 Henagon DigonSchläfli 1 2 ImageNon-convex There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli 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 Schläfli symbols n/m for all m such that m < n/2 (strictly speaking n/m = n/(n − m) ) and m and n are coprime.Name Pentagram Heptagrams Octagram Enneagrams Decagram ...n-agramsSchläfli 5/2 7/2 7/3 8/3 9/2 9/4 10/3 n/m Image Circle Main article: CircleThe 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 A = π r 2 displaystyle A=pi r^ 2 where r displaystyle r is the radius. Other shapes Main article: List of two-dimensional geometric 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 Main article: Dot product The dot product of two vectors A = [A1, A2] and B = [B1, B2] is defined as:[4] A ⋅ B = A 1 B 1 + A 2 B 2 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 ‖ A ‖ displaystyle mathbf A . In this viewpoint, the dot product of two Euclidean vectors A and B is defined by[5] A ⋅ B = ‖ A ‖ ‖ B ‖ cos ⁡ θ , 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 A ⋅ A = ‖ A ‖ 2 , displaystyle mathbf A cdot mathbf A =mathbf A ^ 2 , which gives ‖ A ‖ = A ⋅ A , displaystyle mathbf A = sqrt mathbf A cdot mathbf A , the formula for the Euclidean length Euclidean length of the vector. In calculus Gradient In a rectangular coordinate system, the gradient is given by ∇ f = ∂ f ∂ x i + ∂ f ∂ y j 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 : U ⊆ R2 → R, the line integral along a piecewise smooth curve C ⊂ U is defined as ∫ C f d s = ∫ a b f ( r ( t ) ) r ′ ( t ) d t . 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 a < b displaystyle a

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