By Leuschner A. O.
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Additional info for Comparison of Theory with Observation for the Minor Planets 10 Hygiea and 175 Andromache with Respect to Perturbations by Jupiter
Proof. Let ÁQ be a maximal continuous curve in H obtained by joining successive lifts of Á. There is some T 2 that preserves ÁQ and corresponds to moving through one period of Á. This T must be hyperbolic, or else Á would be cuspidal. T/, in the free homotopy class of Á. T/ is the unique geodesic in H fixed by T, there is no other geodesic in the free homotopy class of Á. 7) that is the shortest curve in this class. 24 reveals an association between closed geodesics and axes of hyperbolic elements of which will turn out to be of great importance to us.
First consider a triangle with at least one point on @H. We can apply a Möbius transformation to map this point to 1 while sending the opposite side to an arc of the unit circle. Let ˛, ˇ be the interior angles at the other two vertices. 11 we can see that the two sides meeting at infinity are the vertical lines Re z D cos ˛ and Re z D cos ˇ. T/ D Z D cos ˇ cos ˛ cos ˇ cos ˛ D ˛ Z 1 p 1 x2 dy dx y2 dx p 1 x2 ˇ: α β α β Fig. 11 Triangle with vertex at 1. 6 Hyperbolic Trigonometry A 35 α β B θ γ C D Fig.
This is easy if 0 is a hyperbolic fixed point. In this case contains a dilation T W z 7! e z for some > 0. Since . / contains at least three points, there must exist some q 2 . /, not equal to 0 or 1, Then T k q ! 0, showing that 0 is a limit point. Suppose that 0 2 . / is not a hyperbolic fixed point. We know that contains some hyperbolic element T with fixed points p1 ; p2 . T/. 12, since 0 2 . / we can assume that Rj w ! 0 for some sequence fRj g . Given " > 0, we can insist that jRj wj < " for all j, by passing to a subsequence if needed.
Comparison of Theory with Observation for the Minor Planets 10 Hygiea and 175 Andromache with Respect to Perturbations by Jupiter by Leuschner A. O.