超越笛卡尔圆定理
Beyond the Descartes circle theorem
摘要：
笛卡尔圆定理指出，如果四个圆以不相交的交点相切，则它们的曲率（或“弯曲”）b_j = 1/r_j满足关系式（b_1 + b_2 + b_3 + b_4）^2 = 2（b_1^2 + b_2^2 + b_3^2 + b_4^2）。我们发现，类似的关系举行涉及在这样的配置，协调为复数，产生一个复杂的笛卡尔定理的圆心。这些关系有矩阵推广到n维的情况下，在每一个欧几里德，球面和双曲几何，他们包括一个笛卡尔圆定理的球面和双曲空间。
英文摘要：
The Descartes circle theorem states that if four circles are mutually tangent with disjoint intersion, then their curvatures (or "bends) b_j = 1/r_j satisfy the relation (b_1 + b_2 + b_3 + b_4)^2 = 2(b_1^2 + b_2^2 + b_3^2 + b_4^2). We show that similar relations hold involving the centers of the circles in such a configuration, coordinatized as complex numbers, yielding a complex Descartes theorem. These relations have matrix generalizations to the n-dimensional case, in each of Euclidean, spherical and hyperbolic geometries, and they include a Descartes circle theorem for spherical and hyperbolic space.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
