Möbius plane

A Möbius plane or inversive plane is a particular kind of plane geometry, built upon some affine planes by adding one point, called the ideal point or point at infinity. In a Möbius plane straight lines are a special case of circles; they are the circles that pass through the ideal point. Möbius planes have a simple axiomatization by incidence properties. Thus a Möbius plane is analogous to a projective plane (where an entire line at infinity is added to an affine plane) but the purpose is different. The Möbius plane provides a natural setting for inversive geometry of two dimensions and for properties shared by lines and circles in an affine plane such as the real or complex affine plane.

Abstract definition

A Möbius plane M is a pair of sets, $(\mathcal P, \mathcal C)$, and an incidence relation between them, satisfying the following four axioms. The elements of $\mathcal P$ are called the points of M and the elements of $\mathcal C$ are the circles of M. If a point P and a circle C are incident we say "P is on C ", or "C is on P ". The intersection of two circles is the set of points on both. The axioms are:

(M1) There exist four points not incident with any one circle.

(M2) Any three points are incident with exactly one circle.

(M3) Every circle is incident with at least three points.

(M4) For any circle C, point P on C, and point Q not on C, there is a unique circle on both P and Q and having exactly one point of intersection with C.

Connection with affine planes

In an abstract Möbius plane M choose an arbitrary point I; call it the "ideal point". Let $A := (\mathcal P \backslash I, \mathcal C_I)$, where $\mathcal C_I$ is the set of all circles on I. Then A is an affine plane in which a line is the set of nonideal points on a circle that contains I.

Geometric constructions

In the real or complex affine plane, A = A(R) or A(C), a circle is the set of solutions (x,y) of a quadratic equation (x − a)² + (y − b)² = r² that has at least one solution. A characteristic of circles is that it takes three points to determine a circle, in contrast to the two points that determine a straight line. If we add one point I to A, which is on every straight line, and we call the extended lines "circles" along with the affine circles, then in the enlarged geometry it takes three points to determine any "circle", since two points of a line l lie on many circles as well as on l, so to single out l one must specify either a third affine point or the ideal point. Thus, the extended geometry is a Möbius plane, called the real or complex Möbius plane, M(R) or M(C).

Conversely, given M(R) or M(C), by deleting any point P in it one gets the real or complex affine plane with its circle structure. Its lines are the circles of M that are on P (with P deleted) and its circles are the remaining circles of M. All such planes are isomorphic as incidence structures.

The real Möbius plane is one way of looking at the Riemann sphere. Forgetting the complex structure of the Riemann sphere, rest the sphere on R² so they are tangent at the south pole of the sphere and the origin of the plane. Under stereographic projection, the circles through the north pole correspond to straight lines and the other circles to plane circles. If we extend R² by an ideal point to form a Möbius plane, the north pole projects to the ideal point, making the sphere a perfect model for M(R).

Not every Möbius plane is real or complex. Circles can be defined in the affine plane over any field and a Möbius plane constructed in an analogous manner.

Inversion

Inversion in a circle of a Möbius plane M is the generalization of reflection in a straight line in an affine plane. This explains the name "inversive plane".

External links

• Michiel Hazewinkel, editor, Encyclopaedia of Mathematics, article "Möbius plane". Springer-Verlag, Berlin–Heidelberg–New York. ISBN 1-4020-0609-8