A New Proof of Pappus's Theorem
msra(2007)
摘要
Any stretching of Ringel's non-Pappus pseudoline arrangement when projected
into the Euclidean plane, implicitly contains a particular arrangement of nine
triangles. This arrangement has a complex constraint involving the sines of its
angles. These constraints cannot be satisfied by any projection of the initial
arrangement. This is sufficient to prove Pappus's theorem. The derivation of
the constraint is via systems of inequalities arising from the polar
coordinates of the lines. These systems are linear in r for any given theta,
and their solubility can be analysed in terms of the signs of determinants. The
evaluation of the determinants is via a normal form for sums of products of
sines, giving a powerful system of trigonometric identities. The particular
result is generalized to arrangements derived from three edge connected totally
cyclic directed graphs, conjectured to be sufficient for a complete analysis of
angle constraining arrangements of lines, and thus a full response to Ringel's
slope conjecture. These methods are generally applicable to the realizability
problem for rank 3 oriented matroids.
更多查看译文
关键词
sine,. pseudoline stretching,pappus,oriented matroid realizability,multiset.,polar coordinates,power system,polar coordinate,satisfiability,normal form,sum of products,oriented matroid,directed graph
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要