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.

Bézout's lemma is a consequence of the Euclidean division defining property, namely that the division by a nonzero integer b has a remainder strictly less than | b |.

For example a Euclidean straight line has no width, but any real drawn line will.

The Hutchinsonian niche is defined more technically as a " Euclidean hyperspace whose dimensions are defined as environmental variables and whose size is a function of the number of values that the environmental values may assume for which an organism has positive fitness.

This approach brings the tools of algebra and calculus to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept to arbitrary dimensions.

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID.

The Euclidean algorithm has been generalized further to other mathematical structures, such as knots and multivariate polynomials.

A straight line, for instance, is self-similar but not fractal because it lacks detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and is fully defined without a need for recursion.

Space, in this construction, still has the ordinary Euclidean geometry.

In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements ( so an analogue of the fundamental theorem of arithmetic holds ); any two elements of a PID have a greatest common divisor ( although it may not be possible to find it using the Euclidean algorithm ).

A ( topological ) surface is a nonempty second countable Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E < sup > 2 </ sup >.

* The space is a cube with Euclidean metric ; the figures include cubes of the same size as the space, with colors or patterns on the faces ; the automorphisms of the space are the 48 isometries ; the figure is a cube of which one face has a different color ; the figure has a symmetry group of 8 isometries, there are 6 equivalence classes of 8 isometries, for 6 isomorphic versions of the figure.

From a Euclidean space perspective, the universe has three dimensions of space and one of time.

This metric has only two undetermined parameters: an overall length scale R that can vary with time, and a curvature index k that can be only 0, 1 or − 1, corresponding to flat Euclidean geometry, or spaces of positive or negative curvature.

In mathematics, physics, and engineering, a Euclidean vector ( sometimes called a geometric or spatial vector, or — as here — simply a vector ) is a geometric object that has a magnitude ( or length ) and direction and can be added to other vectors according to vector algebra.

Every rational number / has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to.

Moreover, it has the same number of points as a square of area 1, as a cube of volume 1, and even as an unbounded n-dimensional Euclidean space ( see Space filling curve ).

It has a standard inner product, making it a Euclidean space.

In the Euclidean case, equality occurs only if the triangle has a 180 ° angle and two 0 ° angles, making the three vertices collinear, as shown in the bottom example.