By Jean Meeus

ISBN-10: 0943396352

ISBN-13: 9780943396354

Said in book," almost each prior guide on celestial calcuations was once pressured to depend upon formaulae for sunlight Moon and planets that have been built within the final century or sooner than 1920.

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Since the above also holds if J is a polygonal arc and q is an endpoint of J , R2 \ J has only one connected region in this case. Returning to the case when J is a polygon, it remains to prove that R2 \ J has more than one region. For any p ∈ R2 \ J and any angle α we consider the ray lα starting at p with angle α. J ∩ lα is a set of points or closed intervals. g. 4 we have cr ( p, lα ) = 2). J J p lα J Fig. 4. Note that for any angle α, lim →0, >0 cr ( p, lα+ ) − cr ( p, lα ) + lim →0, <0 cr ( p, lα+ ) − cr ( p, lα ) is twice the number of intervals of J ∩ lα that J enters from the same side as to which it leaves.

X has a neighbour y in C, because otherwise C is a connected component of G − {v, w} (but G is 3-connected). e. there exists a vertex z such that G − {x, y, z} is disconnected. Since {v, w} ∈ E(G), there exists a connected component D of G − {x, y, z} which contains neither v nor w. But D contains a neighbour d of y, since otherwise D is a connected component of G − {x, z} (again contradicting the fact that G is 3-connected). So d ∈ V (D) ∩ V (C), and thus D is a subgraph of C. Since y ∈ V (C) \ V (D), we have a contradiction to the minimality of |V (C)|.

There exists a vertex z such that G − {x, y, z} is disconnected. Since {v, w} ∈ E(G), there exists a connected component D of G − {x, y, z} which contains neither v nor w. But D contains a neighbour d of y, since otherwise D is a connected component of G − {x, z} (again contradicting the fact that G is 3-connected). So d ∈ V (D) ∩ V (C), and thus D is a subgraph of C. Since y ∈ V (C) \ V (D), we have a contradiction to the minimality of |V (C)|. 37. (Kuratowski [1930], Wagner [1937]) A 3-connected graph is planar if and only if it contains neither K 5 nor K 3,3 as a minor.

### Astronomical algorithms by Jean Meeus

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