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 Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object.

Some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles.

Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids.

The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in ( d + 1 )- dimensional space, by giving each point p an extra coordinate equal to | p |< sup > 2 </ sup >, taking the bottom side of the convex hull, and mapping back to d-dimensional space by deleting the last coordinate.

In Euclidean space R < sup > n </ sup >, or any convex subset of R < sup > n </ sup >, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element.

In Euclidean geometry, a Platonic solid is a regular, convex polyhedron.

In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or Euclidean space is the smallest convex set that contains X.

With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces ; they may also be generalized further, to oriented matroids.

The algorithmic problem of finding the convex hull of a finite set of points in the plane or in low-dimensional Euclidean spaces is one of the fundamental problems of computational geometry.

Suppose that Γ is a lattice in n-dimensional Euclidean space R < sup > n </ sup > and K is a convex centrally symmetric body.

* Convex geometry, the branch of geometry studying convex sets, mainly in Euclidean space

A linear program may be specified by a system of real variables ( the coordinates for a point in Euclidean space R < sup > n </ sup >), a system of linear constraints ( specifying that the point lie in a halfspace ; the intersection of these halfspaces is a convex polytope, the feasible region of the program ), and a linear function ( what to optimize ).

Hadwiger's theorem in integral geometry classifies the possible isotropic measures on compact convex sets in d-dimensional Euclidean space.

The k-Helly property is the property of being a Helly family of order k. These concepts are named after Eduard Helly ( 1884-1943 ); Helly's theorem on convex sets, which gave rise to this notion, states that convex sets in Euclidean space of dimension n are a Helly family of order n + 1.

* Carathéodory's theorem ( convex hull ), about the convex hulls of sets in Euclidean space

The fourth angle of a Lambert quadrilateral is acute if the geometry is hyperbolic, a right angle if the geometry is Euclidean or obtuse if the geometry is elliptic.

The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry is Euclidean and obtuse angles if the geometry is elliptic.

In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

In Euclidean geometry, a parallelogram is a simple ( non self-intersecting ) quadrilateral with two pairs of parallel sides.

In Euclidean geometry, a rhombus (◊), plural rhombi or rhombuses, is a simple ( non self-intersecting ) quadrilateral whose four sides all have the same length.

In Euclidean geometry, Brahmagupta's formula finds the area of any cyclic quadrilateral ( one that can be inscribed in a circle ) given the lengths of the sides.

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle.

The close axiomatic study of Euclidean geometry led to the construction of the Lambert quadrilateral and the Saccheri quadrilateral.

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral ( a quadrilateral whose vertices lie on a common circle ).