## 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,BC, 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 BC (while AC 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/(x2+y2), y/(x2+y2)).
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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