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Affine


John Daggett asks:

  I have seen Affine scaling, affine logic, affine transformations as
  well as affine lines, affine spaces and affine planes.

  But, can you tell me the mathematical significance of the word:
  affine.

The word is the adjective form of "affinity". Its mathematical use
began in geometry: figures can be congruent, they can be similar, they
can be affine. A congruence is a distance-preserving map. A similarity
is a map that preserves lines and angles. An affinity is a map that
preserves lines. (Any two triangles are affine, and any two
ellipses. But the only quadrilaterals affine to a square are the
parallelograms.)

A function between vector spaces is affine if it preserves affine
combinations, that is, linear combinations in which the coefficients
add up to 1. For coordianted Euclidean spaces of dimension greater
than one the invertable continuous transformtions that preserve lines
are precisely those transformations that preserve affine combinations.

The reason that we usually don't learn the phrase "affine
transformation" as undergraduates is that any affine transformation
is a linear transformation followed by a translation (i.e. \x.x+v).

Note that we are stuck with the wrong words here. By rights, a linear
transformation should be one that preserves lines. As it is, it is a
transformation that preserves lines and the origin.

The notion of affine space has been described as what's left when one
removes the origin from a linear space. (Given a field, K, one can rig
up an equational algebraic theory for its affine spaces: for each
element k of K there's a binary operation \xy.kx + (1-k)y. The
equations are a mess.)

If one has a measurement scale for which it makes sense to say that
one measurement is twice another, then one can build a linear scale
(e.g. Kelvin). If one can't do that but can make sense out of saying
that one difference of two measurements is twice another difference of
two measurements, then one can build an affine scale (e.g. Celsius).
Linear scales are common. The best known dimension that demands an
affine scale is time.
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