# affine

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I have seen Affine scaling, affine logic, affine transformations as wellas affine lines, affine spaces and affice planes.
But, can you tell me the mathematical significance of the word: affine.

It comes from affine geometry.  Geometries can be divided up according
to the axioms they satisfy (the Euclidean axiomatic approach) and/or
the transformations which their theorems are immune to or *invariant
under* (the invariant approach, Felix Klein's 1872 Erlangen program).

Euclidean geometry regards the relative notions of both angle and
linearity as significant, but not the absolute notions of position,
orientation, or scale, i.e. it is invariant under translation,
rotation, and dilatation but not under e.g. shearing (which can change
angles though not linearity) or conformal mapping (which can change
linearity though not angles).  Regular polygons (equilateral triangles,
squares, cubes, etc.), regular polytopes (e.g. the five Platonic
solids), circles, ellipses, parabolas, and hyperbolas are among the
multitude of interesting figures that exist in Euclidean geometry, in
the sense of being meaningful or *invariant* concepts.

Affine geometry is a generalization of Euclidean geometry that is also
invariant under scaling in only one direction (distances orthogonal to
that direction are unchanged, and the possibility of negative scale
permits reflection), and shearing.  Transformations obtainable as
compositions of these and the Euclidean transformations are called
affinities.  Affinities preserve parallel lines, whence parallelism
(and hence in a suitable sense zero angle) remains meaningful for
affine geometry.  Neither directional scaling nor shearing preserves
nonzero angle however, which is therefore a meaningless notion in
affine geometry.  Parallelograms, ellipses, parabolas, and hyperbolas
exist in affine geometry, being invariant under affinities, but not
regular polygons, circles, isosceles triangles, or the Platonic solids,
which cannot be defined.

Among the theorems of affine geometry are that the midpoints of the
sides of a quadrilateral form a parallelogram, and a hexagon whose
three diameters are concurrent at their midpoints has parallel opposite
sides.  However it is not a theorem of affine geometry that the base
angles of an isosceles triangle are equal, not because it is false but
because it is meaningless, involving noninvariant concepts.

Like Euclidean geometry, affine geometry is *ordered* in the sense that
the notion of "between" for collinear points remains defined.

Projective geometry is a further weakening of affine geometry that is
invariant under projection ("projectivities"), which makes parallelism
meaningless.  Conic sections, triangles, quadrilaterals, and hexagons
continue to exist in projective geometry but not parallelograms or
hexagons with parallel opposite sides.  Furthermore the distinction
between ellipses (bounded connected), parabolas (unbounded connected),
and hyperbolas (unbounded disconnected) disappears, there is just the
class of conic sections.  And "between" is no longer meaningful.

(Here's a bit more about why the ellipse-hyperbola distinction
disappears.  Projectively, an ellipse drawn on the ground looks like an
ellipse when you look down on it, but becomes a hyperbola when the
plane of your retina intersects the ellipse.  The half of the ellipse
cut off by that plane has to projected by your imagination backwards
onto your retina, where it then appears as the other branch of a
hyperbola.

Conversely if you draw a hyperbola on the ground and then look at the
horizon where one branch of the hyperbola vanishes, it vanishes at two
points on the horizon.  On your retina the hyperbola shows up as an
ellipse truncated by the horizon.  If the other branch of the
hyperbola, behind you, is projected backwards through your retina, it
completes that ellipse above the horizon, connecting to the first
branch at the two points on the horizon.

A parabola vanishes at a single point on the horizon, like railroad
tracks except that instead of forming two lines converging at the
horizon on your retina it forms an ellipse tangent to the horizon.
Ellipses on the ground don't reach the horizon, while hyperbolas on the
ground look like ellipses that have overshot the horizon.)

Affine transformations, those mappings of space under which affine
geometry is invariant, are closely related to linear transformations.
The main difference is that there is no origin, whence affine
transformations need not have a fixpoint, permitting translations in

Just as every linear transformation is uniquely determined by its
effect on an arbitrarily chosen basis, so is every affine
transformation uniquely determined by its effect on an arbitrarily
chosen nondegenerate (nonzero volume) simplex (triangle in the plane,
tetrahedron in 3D space, etc.).  (In d dimensions a basis has d points
while a simplex has d+1; the extra point can explained by saying that
when specifying a "basis" in affine geometry one must explicitly name
an "origin", though which of those points one chooses to view as *the*
origin for that "basis" is up to the viewer, normally one does not
single one out for that distinction.)  And just as a basis linearly
coordinatizes a vector space, uniquely assigning d scalars to every
point, so does a simplex linearly coordinatize an affine space,
uniquely assigning d+1 scalars (which sum to 1) to every point, with the
vertices of the simplex being coordinatized as (1,0,0,...,0),
(0,1,0,...,0), ..., (0,0,...,0,1).

Vaughan Pratt

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