Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
, an ancient Greek mathematician. It is the name of:
Geometry
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Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the two-dimensional plane and three-dimensional space of Euclidean geometry as well as their higher dimensional generalizations
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Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, the study of the properties of Euclidean spaces
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Non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
, systems of points, lines, and planes analogous to Euclidean geometry but without uniquely determined parallel lines
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Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
, the distance between pairs of points in Euclidean spaces
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Euclidean ball, the set of points within some fixed distance from a center point
Number theory
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Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
, the division which produces a quotient and a remainder
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Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an e ...
, a method for finding greatest common divisors
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Extended Euclidean algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's id ...
, a method for solving the Diophantine equation ''ax'' + ''by'' = ''d'' where ''d'' is the greatest common divisor of ''a'' and ''b''
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Euclid's lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely:
For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as we ...
: if a prime number divides a product of two numbers, then it divides at least one of those two numbers
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Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers ...
, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and the extended Euclidean algorithm to work
Other
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Euclidean relation In mathematics, Euclidean relations are a class of binary relations that formalize " Axiom 1" in Euclid's ''Elements'': "Magnitudes which are equal to the same are equal to each other."
Definition
A binary relation ''R'' on a set ''X'' is Euclide ...
, a property of binary relations related to transitivity
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Euclidean distance map
A distance transform, also known as distance map or distance field, is a derived representation of a digital image. The choice of the term depends on the point of view on the object in question: whether the initial image is transformed into anoth ...
, a digital image in which each pixel value represents the Euclidean distance to an obstacle
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Euclidean zoning
Zoning in the United States includes various land use laws falling under the police power rights of state governments and local governments to exercise authority over privately owned real property. Zoning laws in major cities originated with ...
, a system of land use management modeled after the zoning code of Euclid, Ohio
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Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
of the Intermediate Math League of Eastern Massachusetts
See also
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Euclid (disambiguation) Euclid, Euclides, or Eucleides generally refers to the ancient Greek mathematician Euclid of Alexandria (3rd century BC), who wrote a work on geometry called the ''Elements''.
It may also refer to:
People
* Euclid of Megara (c. 435 BC–c. 365 B ...
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Euclid's ''Elements'', a 13-book mathematical treatise written by Euclid, that includes both geometry and number theory
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Euclideon
Euclideon Pty Ltd is an Australian computer software company best known for a middleware 3D computer graphics, 3D Game engine, graphics engine, called Unlimited Detail. Euclideon is also the parent company and operator of Holoverse, a 'holograph ...
, an Australian computer graphics company
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Mathematics disambiguation pages