Bundle theorem

In geometry, the bundle theorem is in the simplest case a statement on six circles and eight points in the real Euclidean plane. In general it is a property of a Möbius plane that is fulfilled by ovoidal Möbius planes only.

The bundle theorem should not be confused with Miquel's theorem.

An ovoidal Möbius plane in real Euclidean space may be considered as the geometry of the plane sections of an egglike surface, like a sphere or an ellipsoid or a half of a sphere glued to a suitable half of an ellipsoid or the surface with equation ${\displaystyle x^{4}+y^{4}+z^{4}=1}$, .... If the egglike surface is just a sphere one gets the space model of the classical real Möbius plane, the circle geometry on the sphere.

The essential property of an ovoidal Möbius plane is the existence of a space model via an ovoid. An ovoid in a 3-dimensional projective space is a set of points, which a) is intersected by lines in 0, 1, or 2 points and b) its tangents at an arbitrary point covers a plane (tangent plane). The geometry of an ovoid in projective 3-space is a Möbius plane, called ovoidal Möbius plane. The point set of the geometry consists of the points of the ovoid and the curves (cycles) are the plane sections of the ovoid. A suitable stereographical projection shows: For any ovoidal Möbius plane there exists a plane model. In the classical case the plane model is the geometry of the circles and lines (any line is completed by a point ${\displaystyle \infty }$). The bundle theorem has a planar and a spatial interpretation. In the planar model there may be lines involved. The proof of the bundle theorem is performed within the spatial model.

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