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

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

These can be found by applying the extended Euclidean algorithm.

These coordinates are known as local coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean.

These restrictions correspond roughly to a particular mathematical model which differs from Euclidean space in its manifest symmetry.

These states are necessarily perpendicular to each other using the Euclidean notion of perpendicularity which comes from sums-of-squares length, except that they also must not be i multiples of each other.

These formulas contain the " slope " of the line connecting P and Q, hence involve divisions between residue classes modulo n, which can be performed using the extended Euclidean algorithm.

These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general.

These are finite groups generated by reflections which act on a finite dimensional Euclidean space.

These inhabitants would in fact determine that their universe is not ruled by Euclidean geometry, but instead by hyperbolic geometry.

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.

These axioms axiomatize Euclidean solid geometry.

These are combinations of one step of the simple Euclidean algorithm, which uses subtraction at each step, and an application of step 3 above.

These functions are called the Schwinger functions, named after Julian Schwinger, and they are analytic, symmetric under the permutation of arguments ( antisymmetric for fermionic fields ), Euclidean covariant and satisfy a property known as reflection positivity.

These can be carried out in Euclidean space, particularly in dimensions 2 and 3.

These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane.

These axioms are a more elegant version of a set Tarski devised in the 1920s as part of his investigation of the metamathematical properties of Euclidean plane geometry.

These theories are related by transformations that are called dualities.

These components can be modified and manipulated by two-dimensional geometric transformations such as translation, rotation, scaling.

These transformations may also be written as biquaternions ( quaternions with complex elements ), where the elements are related to the Jones matrix in the same way that the Stokes parameters are related to the coherency matrix.

These transformations are determined by solutions to the Schrödinger equation.

These transformations displace the triangle in the plane without changing the angle at each vertex or the distances between vertices.

These homogeneous transforms perform rigid transformations on the points in the plane z = 1, that is on points with coordinates p =( x, y, 1 ).

These transformations are guaranteed to only linearly increase the size of the formula, but introduce new variables.

These measures were publicly billed as " reforms " rather than socioeconomic transformations.

These elicited experiences may include perceptions of transformations of the patient's arms and legs ( complex somatosensory responses ) and whole-body displacements ( vestibular responses ).

These topographical transformations are often as rapid as they are drastic.

These transformations are capable of locally warping the target image to align with the reference image.

These transformations in the fado production would necessarily drift it apart from improvise, losing some of its original performing contexts diversity and imposing the specialization of interpreters, authors and musicians.

These transformations leave invariant a Hermitian norm of signature ( 2, 2 ).

These trees are then transformed by a sequence of tree rewriting operations (" transformations ") into surface structures.

These transformations may also be local.

These transformations are threefold:

These transformations are examples of affine involutions.

These transformation laws are automorphisms of the state space, that is bijective transformations which preserve some mathematical property.

These relativistic transformations are applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

These transformations are threefold:

These transformations preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line ( loosely speaking, a circle with infinite radius ).

These transformations generate a subgroup, of the planar affine group, called the Lorentz group of the plane.

These transformations have their own special features and allow the player to access areas that were previously inaccessible ; for instance, the walrus can resist the effects of icy-cold water.

These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.