Positive infinity is one of the non real number elements in the affinely extended real number system.

In particular, x t is a geodesic if and only if its development is an affinely parametrized straight line in T x 0 M.If M is a surface in R 3, it is easy to see that M has a natural affine connection.…

The IEEE standard has four different rounding modes; the first is the default; the others are called directed roundings.The IEEE standard employs (and extends) the affinely extended real number system, with separate positive and negative infinities.During drafting, there was a proposal for the standard to incorporate the projectively extended real number system, with a single unsigned infinity, by providing programmers with a mode selection option.…

An affine combination is a linear combination in which the sum of the coefficients is 1.Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others.The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace.…

Similarly they are affinely dependent if in addition the sum of coefficients is zero: a condition that ensures that the combination makes sense as a displacement vector.

One application of the torsion of a connection involves the geodesic spray of the connection: roughly the family of all affinely parametrized geodesics.

A manifold having a distinguished affine structure is called an affine manifold and the charts which are affinely related to those of the affine structure are called affine charts.In each affine coordinate domain the coordinate vector fields form a parallelization of that domain, so there is an associated connection on each domain.…

Every possible bit combination is either a NaN or a number with a unique value in the affinely extended real number system with its associated order, except for the two bit combinations negative zero and positive zero, which sometimes require special attention (see below).

A convex polytope can be decomposed into a simplicial complex, or union of simplices, satisfying certain properties.Given a convex r-dimensional polytope P, a subset of its vertices containing (r+1) affinely independent points defines an r-simplex.It is possible to form a collection of subsets such that the union of the corresponding simplices is equal to P, and the intersection of any two simplices is either empty or a lower-dimensional simplex.…

In geometry, an affine-regular polygon or affinely regular polygon is a polygon that is related to a regular polygon by an affine transformation.Affine transformations include translations, uniform and non-uniform scaling, reflections, rotations, shears, and other similarities and some, but not all linear maps.…

In the theory of general relativity, and differential geometry more generally, Schild's ladder is a first-order method for approximating parallel transport of a vector along a curve using only affinely parametrized geodesics.The method is named for Alfred Schild, who introduced the method during lectures at Princeton University.…

In more generality, a set containing k points, for arbitrary k, is in general linear position if and only if no (k-2) -dimensional flat contains all k points.A set of points in general linear position is also said to be affinely independent (this is the affine analog of linear independence of vectors, or more precisely of maximal rank), and d+1 points in general linear position in affine d-space are an affine basis.…

Accordingly, solutions of are called geodesics with affine parameter.An affine connection is determined by its family of affinely parameterized geodesics, up to torsion, The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection.…