Euclidean space

EuclideanspaceEuclidean vector space
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane (dimension two). It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean is used to distinguish it from other spaces that were later discovered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical universe.

Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space.

Dot product

scalar productdotinner product
The name "dot product" is derived from the centered dot " · " that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector, as is the case for the vector product in three-dimensional space. The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.


Two parallel cubes can be connected to form a tesseract. ]] 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 because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. 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.


orthogonalorthogonallyorthogonal vector
The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace. In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa. Note that the geometric concept of two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle. In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.

Line (geometry)

linestraight linelines
The vector equation of the line through points A and B is given by (where λ is a scalar). If a is vector OA and b is vector OB, then the equation of the line can be written:. A ray starting at point A is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0. In three-dimensional space, a first degree equation in the variables x, y, and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in n-dimensional space n-1 first-degree equations in the n coordinate variables define a line under suitable conditions.

Linear algebra

linearlinear algebraiclinear-algebraic
There is a strong relationship between linear algebra and geometry, which started with the introduction by René Descartes, in 1637, of Cartesian coordinates. In this new (at that time) geometry, now called Cartesian geometry, points are represented by Cartesian coordinates, which are sequences of three real numbers (in the case of the usual three-dimensional space). The basic objects of geometry, which are lines and planes are represented by linear equations. Thus, computing intersections of lines and planes amounts to solving systems of linear equations. This was one of the main motivations for developing linear algebra.

Euclidean vector

vectorvectorsvector addition
Vector-valued functions can be differentiated and integrated by differentiating or integrating the components of the vector, and many of the familiar rules from calculus continue to hold for the derivative and integral of vector-valued functions. The position of a point x = (x 1, x 2, x 3 ) in three-dimensional space can be represented as a position vector whose base point is the origin :The position vector has dimensions of length. Given two points x = (x 1, x 2, x 3 ), y = (y 1, y 2, y 3 ) their displacement is a vector :which specifies the position of y relative to x. The length of this vector gives the straight-line distance from x to y. Displacement has the dimensions of length.

Cross product

vector cross productvector productcross-product
In mathematics, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space and is denoted by the symbol \times. Given two linearly independent vectors \mathbf{a} and \mathbf{b}, the cross product, (read "a cross b"), is a vector that is perpendicular to both \mathbf{a} and \mathbf{b} and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).

Real number

In French mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets are noted respectively ℝ + and ℝ −. In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted ℝ + * and ℝ − *. The notation R n refers to the Cartesian product of n copies of R, which is an n-dimensional vector space over the field of the real numbers; this vector space may be identified to the n-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter.


curvednegative curvatureextrinsic curvature
For a curve drawn on a surface (embedded in three-dimensional Euclidean space), several curvatures are defined, which relates the direction of curvature to the surface's unit normal vector. These are the normal curvature, geodesic curvature and geodesic torsion. Any non-singular curve on a smooth surface has its tangent vector T contained in the tangent plane of the surface.

Mathematical analysis

analysisclassical analysisanalytic
The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).

Differential geometry

differentialdifferential geometerdifferential geometry and topology
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations.


differentiationdifferentiablefirst derivative
Integral. Infinitesimal. Linearization. Mathematical analysis. Multiplicative inverse. Non-Newtonian calculus. Numerical differentiation. Rate (mathematics). Radon–Nikodym theorem. Symmetric derivative. Schwarzian derivative. Khan Academy: "Newton, Leibniz, and Usain Bolt". Online Derivative Calculator from Wolfram Alpha. Online Derivative Calculator from Wolfram Alpha.

Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
Kiran Kedlaya, Geometry Unbound (a treatment using analytic geometry; PDF format, GFDL licensed). Kiran Kedlaya, Geometry Unbound (a treatment using analytic geometry; PDF format, GFDL licensed).

Two-dimensional space

Euclidean planetwo-dimensional2D
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). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates.


A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points.

Elementary mathematics

Basic Mathematicselementarypure mathematics
Analytic geometry is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations.

Glossary of areas of mathematics

Spherical geometry: a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere. Spherical trigonometry: a branch of spherical geometry that studies polygons on the surface of a sphere. Usually the polygons are triangles. Statistics: although the term may refer to the more general study of statistics, the term is used in mathematics to refer to the mathematical study of statistics and related fields. This includes probability theory. Stochastic calculus. Stochastic calculus of variations. Stochastic geometry: the study of random patterns of points. Stratified Morse theory. Super category theory. Super linear algebra.


acute angleobtuse angleoblique
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Linear equation

linearlinear equationsslope-intercept form
In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1 ) in the Euclidean space of dimension n.


closed curvespace curvesmooth curve
Gallery of Bishop Curves and Other Spherical Curves, includes animations by Peter Moses. The Encyclopedia of Mathematics article on lines. The Manifold Atlas page on 1-manifolds.

Normal (geometry)

normalnormal vectorsurface normal
For a plane given by the equation, the vector is a normal. For a plane whose equation is given in parametric form :,where r 0 is a point on the plane and p, q are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both p and q, which can be found as the cross product.

Discrete geometry

combinatorial geometrydiscretecombinatorial
Topics in this area include: An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces). In comparison, an ordinary (i.e., non-oriented) matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered. A geometric graph is a graph in which the vertices or edges are associated with geometric objects.


A cylinder (from Greek κύλινδρος – kulindros, "roller", "tumbler" ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom. This traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology. The shift in the basic meaning (solid versus surface) has created some ambiguity with terminology. It is generally hoped that context makes the meaning clear.