By I. Madsen, B. Oliver
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Additional resources for Algebraic Topology Aarhus 1982. Proc. conf. Aarhus, 1982
Ra) = R-a0Rt = ^- ^n Example 3, Ra and Rp are not globally the ruling involutions, but they are so in each component, so that the example can be constructed using them. Now, suppose that Qa and Qb have double baseball intersection. Then, each of the ruling involutions keeps the components of C fixed (as opposed to Example 3 where they transpose them) because now they are inessential. Therefore, they act as (topological) reflections on C° and C1. Consider one of the components, C° say. The composition of two ruling involutions (corresponding to the two hyperboloids, say R^ o 72+) is then an orientation preserving homeomorphism in CQ = S1, with a rotation angle a.
They are all intervals or points. So that the case follows from the classical Hadwiger's Theorem (with "generosity 0"). Indeed, for each three of the intersections, /j n P, adding /i,/2,/3 to them we get a compatibly ordered transversal on the plane. Observe that the argument extends to the case p ^ /i fl /2, because then /i(/i, I^) is contained in the lines of the plane P and we can apply Hadwiger (with "generosity 1"). Case 3. We can assume that p — I\ Pi /2, and the case extends to the existence of 73 such that 1% n P = 0.
13 (1978), 453-461.  G. Fejes-Toth, New results in the theory of packing and covering, Convexity and its Applications(Ed. by P. M. Gruber and J. M. Wills), 318-359, Birkhauser, Basel Boston Stuttgart (1983).  L. Fejes-Toth, On the permeability of a circle layer, Jour. Studia Sci. Math. Hung. 1 (1966), 5-10. M. Mitchell, Geometric shortest paths and network optimization, Handbook of Comp. Geometry (ED. -R. Sack and J. Urrutia), pp. 633-701. V. North-Holland, Amsterdam, 2000.  C. H. Papadimitriou, and M.
Algebraic Topology Aarhus 1982. Proc. conf. Aarhus, 1982 by I. Madsen, B. Oliver