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.

Thus Euclidean geometry corresponds to the choice of the group E ( 3 ) of distance-preserving transformations of the Euclidean space R < sup > 3 </ sup >, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.

* Euclidean geometry, under Peano's axiom system the primitive notions are point, segment and motion.

This means that surface area is invariant under the group of Euclidean motions.

* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication

For example, the real line is Tychonoff under the standard Euclidean topology.

An alternative characterization of Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under a coordinate transformation.

Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space.

Perhaps best known is high school Euclidean geometry where planar triangles are studied under congruent transformations, also called isometries or rigid transformations.

In mechanics and geometry, the 3D rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R < sup > 3 </ sup > under the operation of composition.

For example, lengths, angles and areas are preserved with respect to the Euclidean group of symmetries, while only the incidence structure and the cross-ratio are preserved under the most general projective transformations.

It is not possible to talk about angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing.

Projective geometry, like affine and Euclidean geometry, can also be developed from the Erlangen program of Felix Klein ; projective geometry is characterized by invariants under transformations of the projective group.

* The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group.

It is characterized by Euclidean distance or angle, which are invariant under the subgroup of rotations.

In the simple case of a nonrelativistic particle moving in Euclidean space under a force field with coordinates and momenta, Liouville's theorem can be written

In geometry the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R < sup > 3 </ sup > under the operation of composition.

The affine subspaces are model surfaces — they are the simplest surfaces in R < sup > 3 </ sup >, and are homogeneous under the Euclidean group of the plane, hence they are Klein geometries in the sense of Felix Klein's Erlangen programme.

For example, in a 2-dimensional space, the distance between the point ( 1, 0 ) and the origin ( 0, 0 ) is always 1 according to the usual norms, but the distance between the point ( 1, 1 ) and the origin ( 0, 0 ) can be 2, or 1 under Manhattan distance, Euclidean distance or maximum distance respectively.

He completed his PhD at Caltech in mathematics and physics in 1925 under Harry Bateman, with the dissertation, “ On Dynamical Space-Times Which Contain a Conformal Euclidean 3-Space ”.< ref name = cit >

This is easily confirmed under modern expansions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts of the time.

In the point of view of Cartan connections, however, the affine subspaces of Euclidean space are model surfaces — they are the simplest surfaces in Euclidean 3-space, and are homogeneous under the affine group of the plane — and every smooth surface has a unique model surface tangent to it at each point.