This choice gives us two alternative forms of geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean and hyperbolic geometries.

Angles are usually presumed to be in a Euclidean plane, but are also defined in non-Euclidean geometry.

This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth.

This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).

These definitions are designed to be consistent with the underlying Euclidean geometry.

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

Richardson had believed, based on Euclidean geometry, that a coastline would approach a fixed length, as do similar estimations of regular geometric figures.

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

A circle is a simple shape of Euclidean geometry that is the set of points in the plane that are equidistant from a given point, the

The invention of Cartesian coordinates in the 17th century by René Descartes ( Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point " infinitesimally ", i. e. in the first order of approximation.

These are the closest analogues to the " ordinary " plane and space considered in Euclidean and non-Euclidean geometry.

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points ( the foci ) is constant.

In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms.

Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.

* Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

For over two thousand years, the adjective " Euclidean " was unnecessary because no other sort of geometry had been conceived.

Euclidean geometry is an axiomatic system, in which all theorems (" true statements ") are derived from a small number of axioms.

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula

The result of applying a Euclidean transformation to a point is given by the formula

A point in Euclidean space may be identified by a tuple of real numbers, and distances are defined using the Euclidean distance formula.

For example, it follows that any closed oriented Riemannian surface can be C < sup > 1 </ sup > isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space ( there is no such C < sup > 2 </ sup >- embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε < sup >- 2 </ sup >).

In mathematics, the Euclidean distance or Euclidean metric is the " ordinary " distance between two points that one would measure with a ruler, and is given by the Pythagorean formula.

By using this formula as distance, Euclidean space ( or even any inner product space ) becomes a metric space.

He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space.

This last formula is also valid for the curvature of curves in a Euclidean space of any dimension.

The Euclidean distance formula is simply the Pythagorean theorem.

## Use the Euclidean distance formula to find the similarity between the input vector and the map's node's weight vector

which is the analog of the formula in Euclidean geometry expressing the sine of an angle as the opposite side divided by the hypotenuse.

Given a vector a in Euclidean space R < sup > n </ sup >, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by

The Poisson summation formula similarly provides a connection between Fourier analysis on Euclidean spaces and on the tori of the corresponding dimensions.

The Poisson summation formula may be used to derive Landau's asymptotic formula for the number of lattice points in a large Euclidean sphere.

The Poisson summation formula holds in Euclidean space of arbitrary dimension.

The formula for the absolute value of z is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space:

In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:

Euclidean division can also be extended to negative integers using the same formula ; for example − 9

The formula for distance between two points is a fundamental property of a Euclidean space, it is called the Euclidean metric tensor ( or simply the Euclidean metric ).

In general, for any c > 0, where d is the number of dimensions in the Euclidean space, there is a polynomial-time algorithm that finds a tour of length at most ( 1 + 1 / c ) times the optimal for geometric instances of TSP in time ; this is called a polynomial-time approximation scheme ( PTAS ).

Besides finding the greatest common divisor of integers a and b, as the Euclidean algorithm does, it also finds integers x and y ( one of which is typically negative ) that satisfy Bézout's identity

Specifically, for Euclidean R < sup > 3 </ sup >, one easily finds that

; Non-metric multidimensional scaling: In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space.

However, when a prominent surveyor finds a Triangle with more than 180 degrees, he is fired from his job and generally considered a crackpot, since such a construction is not possible in Euclidean geometry.