Introduction to Inversive geometry
The notion (second-order structure) of circle or sphere can also
be equivalently expressed as the 4-ary relation of circularity,
(the relation between 4 points saying they belong to the same
circle or straight line) suffices to define angles of
intersection, for the following intuitive reason:
In n-dimensional geometry, let us start from the concept
of (n-1)-dimensional sphere, which are circles if n=2. A
space with such a concept is smooth (as small parts of non-small
spheres or circles are approximately straight) and thus can
compare nearby small volumes (areas). Then, to measure the angle
of intersection between 2 spheres (circles) S and S' at a given
point, consider a small ball (disk) around this intersection,
divided by S and S' into 4 pieces, such that each has the same
volume (area) as the opposite piece. Then the ratio of the volume
of one piece to that of the whole ball, multiplied by 2π,
approaches the angle in radians of that piece, when the volume of
the whole ball approaches zero.
In an affine plane, measures of angles suffice to define the
concept of circle centered on a point O, as a curve C
equivalently satisfying:
∀A,B∈C, the angle between the tangent
lines to C at A and B, equals (with the
same sign) the angle between straight lines (OA) and (OB).
For a variable B∈C (while A∈C may
vary or not), the angle between straight lines (OA) and (OB)
is proportional to the area of the region limited by [OA],
[OB] and the part of C between A and B.
Now you may ask, how can we have a concept of circles but no affine
structure ? The alignment relation may fail to be defined by "to not
be contained in a same circle" if straight lines are confused among
circles : straight lines are the limit cases of circles that "open"
up to infinity. For any two circles (or one straight line and one
circle) that intersect at one point, either it is a point
of tangency (the angle of intersection is zero), or they
intersect at one other point with the same angle. Thus the confusion
of straight lines among circles requires that any two intersecting
straight lines also intersect at another point, that is located "at
infinity" (think of a fixed line intersecting a moving circle at a
fixed point with a fixed angle, and let that circle open up to
become a straight line, and again a circle on the other side : the
other intersection goes through infinity).
A typical automorphism of this geometry, is the inversion,
defined by
(x,y)↦ (x/(x^{2}+y^{2}),
y/(x^{2}+y^{2})).
It is called this way because the image of every point is a point in
the same direction from the chosen center, and with a distance from
the center given by the inverse of the original distance. (The
inversion reverses the orientation; we can get an automorphism
preserving the orientation, by composing it with a reflection). It
can be similarly defined in Euclidean spaces of any dimension, with
the same properties.
Thus, n-dimensional inversive geometry is the
geometry where:
- The set of points can be seen as those of an n-dimensional
Euclidean space, with one more point "at infinity",
- The main structures are the notion of circle (where a straight
line is seen as a circle among others) or sphere of any
dimension (same remark)
- Automorphisms can be described (from the viewpoint of
Euclidean geometry) as composed of Euclidean moves, inversions
and dilations. (Dilations can be obtained by composing enough
Euclidean moves and inversions). Precisely, they are either
Euclidean moves, or composites of a translation and the
inversion with fixed center (or equivalently, an inversion with
variable center; the goal is to choose which point goes to
infinity), a dilation and an Euclidean move (i.e. an affine
transformation preserving circularity, to preserve the chosen
infinity point).
The notion of straight line from the initial Euclidean space is
expressible there with one parameter : the choice of the infinity
point. Namely, they are the circles that contain this point.
Structures of alignment and angles, together suffice to restore
affine structures (thus also circularity). For example, 2 lines are
parallel if they have the same angle of intersection with a third
one.
Thus, inversive geometry together with a symbol of constant point,
is equivalent to Euclidean geometry without unit of distance.
Aside the difference of structures, the construction of the
inversive space from the Euclidean space already differs from that
of the projective space, as follows:
- The projective plane has a whole line of infinity points (the
horizon) ; two straight lines only intersect at one of them if
they are parallel.
- But now we have only one point at infinity, seen as belonging
to all straight lines: any two intersecting lines also intersect
there, but parallel lines are tangent at this infinity
point.
Affine geometry can be restored from either projective geometry
(alignment) or inversive geometry (circularity) together with the
operation of ratio of volumes.
Proof of equivalence of expressibility :
From the concepts of circle and measure of area, we can define
d(A,B) by the fact that the minimum area of
a circle going through A and B is πd(A,B)^{2}/4.
From the concept of distance, we can fill a given surface by a
crystal of a convenient kind so that its density is determined by
the requirement for each atom to have a specific distance to its
closest neighbors. Then the area of the surface is approximately
proportional to the number of atoms of this crystal included
there.
Another way to see it:
A point is at infinity if the measure of any volume
around it (a neighborhood) is infinite. And we saw that the
distinction of points at infinity suffices. (Thus, volumes cannot
be compared in either inversive or projective geometry alone).
In the absence of the alignment relation, the question of
definability of circles from angles, is more difficult to resolve.
In dimension 2, functions that preserve angles with orientation
(signs of angles) can be understood as "functions of one complex
variable", given (or approached) by ordinary formulas. (Then,
orientation can be reversed by composing them with complex
conjugation.)
Among them, those preserving circularity are Möbius
transformations.
But it turns out that the only case where angles can be preserved
without circularity, is the local views of 2-dimensional spaces.
Complex functions other than Möbius transformations, cannot remain
injective when extended to the whole set of complex numbers.
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