Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Mathematics Of or relating to a transformation of coordinates that is equivalent to a linear transformation followed by a translation.
 adj. Mathematics Of or relating to the geometry of affine transformations.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Assigning finite values to finite quantities.
 adj. Describing a function expressible as (which is not linear, but is similar).
 adj. Of or pertaining to a transformation that maps parallel lines to parallel lines and finite points to finite points.
 adj. Having mutual affinity, of two materials.
 n. A relative by marriage.
 v. To refine.
from the GNU version of the Collaborative International Dictionary of English
 transitive v. To refine.
from The Century Dictionary and Cyclopedia
 Related; akin; affined.
 n. A relative by marriage; one akin.
 To refine. Holland.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. (anthropology) kin by marriage
 adj. (anthropology) related by marriage
 adj. (mathematics) of or pertaining to the geometry of affine transformations
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
Examples

We modeled the eating, lying, and standing dynamics of a cow using a piecewise affine dynamical system.

We formulate a mathematical model for daily activities of a cow (eating, lying down, and stand  ing) in terms of a piecewise affine dynamical system.

Thus, for example, the notions of Euclidean geometry are invariant under similarity transformations, those of affine geometry under affine transformations, and those of topology under bicontinuous transformations.

The axioms that Ketonen considers are those of projective and affine geometry, the former taken from Skolem's 1920 paper discussed in the first section above.

Unlike proper length, this generalized affine length depends on some arbitrary choices (roughly speaking, the length will vary depending on the coordinates one chooses).

A maximal spacetime is singular if and only if it contains an inextendible path of finite generalized affine length.

The chief problem facing this definition of singularities is that the physical significance of generalized affine length is opaque, and thus it is unclear what the relevance of singularities, defined in this way, might be.

Thus the question of whether a path has a finite or infinite generalized affine length is a perfectly welldefined question, and that is all we'll need.

To say that a spacetime is singular then is to say that there is at least one maximally extended path that has a bounded (generalized affine) length.

This may involve 11dimensions for the space directions for gauge potentials we measure as affine connections on the base spacetime.
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