椭圆内接n边形周长最大值

zuijianqiugen

zuijianqiugen 发表于 2014-3-22 20:16
用6棱长的表达式太复杂了，肯定的是：外内心距d与外半径R和内半径r不存在表达式。

只能说如何用最简单的式子来表达d，包括基本参数(6个棱长)和复合参数(R、r……等等)

zuijianqiugen

chendu 发表于 2011-12-2 20:15
四面体的Euler线
――欢迎渐行渐近的“四面体几何学”

向量ＯE＝(1/2)*∑向量OA，  向量OT＝(1/3)*∑向量OA， 这是有问题的。
参见《多个定点的位移矢量独立线性相关的充要条件》
http://zuijianqiugen.blog.163.co ... 062201422285927964/

补充内容 (2014-3-31 13:35):
应该说明O是外心。

zuijianqiugen

是不是有问题？

数学星空

根据 http://bbs.emath.ac.cn/forum.php ... 05&fromuid=1455 结论：

我们可以很容易计算\(a,b,m,n\)之间的关系，先将参数统一，即\(a \longleftrightarrow  m  ,b \longleftrightarrow  n\) 得到


当\(k=3\)时

\(-ab+an+bm=0\)


当\(k=4\)时

\(-a^2b^2+a^2n^2+b^2m^2=0\)


当\(k=5\)时

\(-a^3b^3-a^3b^2n+a^3bn^2+a^3n^3-a^2b^3m+2a^2b^2mn-a^2bmn^2+ab^3m^2-ab^2m^2n+b^3m^3=0\)


当\(k=6\)时

\(-3a^4b^4+2a^4b^2n^2+a^4n^4+2a^2b^4m^2-2a^2b^2m^2n^2+b^4m^4=0\)

当\(k=7\)时

\(a^6b^6-2a^6b^5n-a^6b^4n^2+4a^6b^3n^3-a^6b^2n^4-2a^6bn^5+a^6n^6-2a^5b^6m-2a^5b^5mn+2a^5b^2mn^4+2a^5bmn^5-a^4b^6m^2+2a^4b^4m^2n^2-a^4b^2m^2n^4+4a^3b^6m^3-4a^3b^3m^3n^3-a^2b^6m^4+2a^2b^5m^4n-a^2b^4m^4n^2-2ab^6m^5+2ab^5m^5n+b^6m^6=0\)

当\(k=8\)时

\(a^8b^8-4a^8b^6n^2+6a^8b^4n^4-4a^8b^2n^6+a^8n^8-4a^6b^8m^2-4a^6b^6m^2n^2+4a^6b^4m^2n^4+4a^6b^2m^2n^6+6a^4b^8m^4+4a^4b^6m^4n^2-10a^4b^4m^4n^4-4a^2b^8m^6+4a^2b^6m^6n^2+b^8m^8=0\)

当\(k=9\)时

\(-a^9b^9-3a^9b^8n+8a^9b^6n^3+6a^9b^5n^4-6a^9b^4n^5-8a^9b^3n^6+3a^9bn^8+a^9n^9-3a^8b^9m-4a^8b^7mn^2+14a^8b^5mn^4-4a^8b^3mn^6-3a^8bmn^8-4a^7b^8m^2n-12a^7b^7m^2n^2-8a^7b^6m^2n^3+8a^7b^5m^2n^4+12a^7b^4m^2n^5+4a^7b^3m^2n^6+8a^6b^9m^3-8a^6b^7m^3n^2-8a^6b^5m^3n^4+8a^6b^3m^3n^6+6a^5b^9m^4+14a^5b^8m^4n+8a^5b^7m^4n^2-8a^5b^6m^4n^3-14a^5b^5m^4n^4-6a^5b^4m^4n^5-6a^4b^9m^5+12a^4b^7m^5n^2-6a^4b^5m^5n^4-8a^3b^9m^6-4a^3b^8m^6n+4a^3b^7m^6n^2+8a^3b^6m^6n^3+3ab^9m^8-3ab^8m^8n+b^9m^9=0\)

当\(k=10\)时

\(5a^{12}b^{12}-10a^{12}b^{10}n^2-9a^{12}b^8n^4+36a^{12}b^6n^6-29a^{12}b^4n^8+6a^{12}b^2n^{10}+a^{12}n^{12}-10a^{10}b^{12}m^2+34a^{10}b^{10}m^2n^2-36a^{10}b^8m^2n^4+4a^{10}b^6m^2n^6+14a^{10}b^4m^2n^8-6a^{10}b^2m^2n^{10}-9a^8b^{12}m^4-36a^8b^{10}m^4n^2+50a^8b^8m^4n^4-20a^8b^6m^4n^6+15a^8b^4m^4n^8+36a^6b^{12}m^6+4a^6b^{10}m^6n^2-20a^6b^8m^6n^4-20a^6b^6m^6n^6-29a^4b^{12}m^8+14a^4b^{10}m^8n^2+15a^4b^8m^8n^4+6a^2b^{12}m^{10}-6a^2b^{10}m^{10}n^2+b^{12}m^{12}=0\)

当\(k=11\)时

1. -a^{15}b^{15}+3a^{15}b^{14}n+3a^{15}b^{13}n^2-17a^{15}b^{12}n^3+3a^{15}b^{11}n^4+39a^{15}b^{10}n^5-25a^{15}b^9n^6-45a^{15}b^8n^7+45a^{15}b^7n^8+25a^{15}b^6n^9-39a^{15}b^5n^{10}-3a^{15}b^4n^{11}+17a^{15}b^3n^{12}-3a^{15}b^2n^{13}-3a^{15}bn^{14}+a^{15}n^{15}+3a^{14}b^{15}m+6a^{14}b^{14}mn+a^{14}b^{13}mn^2-4a^{14}b^{12}mn^3-21a^{14}b^{11}mn^4-38a^{14}b^{10}mn^5+17a^{14}b^9mn^6+72a^{14}b^8mn^7+17a^{14}b^7mn^8-38a^{14}b^6mn^9-21a^{14}b^5mn^{10}-4a^{14}b^4mn^{11}+a^{14}b^3mn^{12}+6a^{14}b^2mn^{13}+3a^{14}bmn^{14}+3a^{13}b^{15}m^2+a^{13}b^{14}m^2n-14a^{13}b^{13}m^2n^2-2a^{13}b^{12}m^2n^3+25a^{13}b^{11}m^2n^4-5a^{13}b^{10}m^2n^5-20a^{13}b^9m^2n^6+20a^{13}b^8m^2n^7+5a^{13}b^7m^2n^8-25a^{13}b^6m^2n^9+2a^{13}b^5m^2n^{10}+14a^{13}b^4m^2n^{11}-a^{13}b^3m^2n^{12}-3a^{13}b^2m^2n^{13}-17a^{12}b^{15}m^3-4a^{12}b^{14}m^3n-2a^{12}b^{13}m^3n^2+76a^{12}b^{12}m^3n^3+33a^{12}b^{11}m^3n^4-72a^{12}b^{10}m^3n^5-28a^{12}b^9m^3n^6-72a^{12}b^8m^3n^7+33a^{12}b^7m^3n^8+76a^{12}b^6m^3n^9-2a^{12}b^5m^3n^{10}-4a^{12}b^4m^3n^{11}-17a^{12}b^3m^3n^{12}+3a^{11}b^{15}m^4-21a^{11}b^{14}m^4n+25a^{11}b^{13}m^4n^2+33a^{11}b^{12}m^4n^3-50a^{11}b^{11}m^4n^4-34a^{11}b^{10}m^4n^5+34a^{11}b^9m^4n^6+50a^{11}b^8m^4n^7-33a^{11}b^7m^4n^8-25a^{11}b^6m^4n^9+21a^{11}b^5m^4n^{10}-3a^{11}b^4m^4n^{11}+39a^{10}b^{15}m^5-38a^{10}b^{14}m^5n-5a^{10}b^{13}m^5n^2-72a^{10}b^{12}m^5n^3-34a^{10}b^{11}m^5n^4+220a^{10}b^{10}m^5n^5-34a^{10}b^9m^5n^6-72a^{10}b^8m^5n^7-5a^{10}b^7m^5n^8-38a^{10}b^6m^5n^9+39a^{10}b^5m^5n^{10}-25a^9b^{15}m^6+17a^9b^{14}m^6n-20a^9b^{13}m^6n^2-28a^9b^{12}m^6n^3+34a^9b^{11}m^6n^4-34a^9b^{10}m^6n^5+28a^9b^9m^6n^6+20a^9b^8m^6n^7-17a^9b^7m^6n^8+25a^9b^6m^6n^9-45a^8b^{15}m^7+72a^8b^{14}m^7n+20a^8b^{13}m^7n^2-72a^8b^{12}m^7n^3+50a^8b^{11}m^7n^4-72a^8b^{10}m^7n^5+20a^8b^9m^7n^6+72a^8b^8m^7n^7-45a^8b^7m^7n^8+45a^7b^{15}m^8+17a^7b^{14}m^8n+5a^7b^{13}m^8n^2+33a^7b^{12}m^8n^3-33a^7b^{11}m^8n^4-5a^7b^{10}m^8n^5-17a^7b^9m^8n^6-45a^7b^8m^8n^7+25a^6b^{15}m^9-38a^6b^{14}m^9n-25a^6b^{13}m^9n^2+76a^6b^{12}m^9n^3-25a^6b^{11}m^9n^4-38a^6b^{10}m^9n^5+25a^6b^9m^9n^6-39a^5b^{15}m^{10}-21a^5b^{14}m^{10}n+2a^5b^{13}m^{10}n^2-2a^5b^{12}m^{10}n^3+21a^5b^{11}m^{10}n^4+39a^5b^{10}m^{10}n^5-3a^4b^{15}m^{11}-4a^4b^{14}m^{11}n+14a^4b^{13}m^{11}n^2-4a^4b^{12}m^{11}n^3-3a^4b^{11}m^{11}n^4+17a^3b^{15}m^{12}+a^3b^{14}m^{12}n-a^3b^{13}m^{12}n^2-17a^3b^{12}m^{12}n^3-3a^2b^{15}m^{13}+6a^2b^{14}m^{13}n-3a^2b^{13}m^{13}n^2-3ab^{15}m^{14}+3ab^{14}m^{14}n+b^{15}m^{15}=0

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当\(k=12\)时

1. a^{16}b^{16}-8a^{16}b^{14}n^2+28a^{16}b^{12}n^4-56a^{16}b^{10}n^6+70a^{16}b^8n^8-56a^{16}b^6n^{10}+28a^{16}b^4n^{12}-8a^{16}b^2n^{14}+a^{16}n^{16}-8a^{14}b^{16}m^2-40a^{14}b^{14}m^2n^2+56a^{14}b^{12}m^2n^4+88a^{14}b^{10}m^2n^6-88a^{14}b^8m^2n^8-56a^{14}b^6m^2n^{10}+40a^{14}b^4m^2n^{12}+8a^{14}b^2m^2n^{14}+28a^{12}b^{16}m^4+56a^{12}b^{14}m^4n^2-316a^{12}b^{12}m^4n^4+144a^{12}b^{10}m^4n^6+228a^{12}b^8m^4n^8-72a^{12}b^6m^4n^{10}-68a^{12}b^4m^4n^{12}-56a^{10}b^{16}m^6+88a^{10}b^{14}m^6n^2+144a^{10}b^{12}m^6n^4-400a^{10}b^{10}m^6n^6+40a^{10}b^8m^6n^8+184a^{10}b^6m^6n^{10}+70a^8b^{16}m^8-88a^8b^{14}m^8n^2+228a^8b^{12}m^8n^4+40a^8b^{10}m^8n^6-250a^8b^8m^8n^8-56a^6b^{16}m^{10}-56a^6b^{14}m^{10}n^2-72a^6b^{12}m^{10}n^4+184a^6b^{10}m^{10}n^6+28a^4b^{16}m^{12}+40a^4b^{14}m^{12}n^2-68a^4b^{12}m^{12}n^4-8a^2b^{16}m^{14}+8a^2b^{14}m^{14}n^2+b^{16}m^{16}=0

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当\(k=13\)时

1. -a^{21}b^{21}-3a^{21}b^{20}n+6a^{21}b^{19}n^2+26a^{21}b^{18}n^3-9a^{21}b^{17}n^4-99a^{21}b^{16}n^5-24a^{21}b^{15}n^6+216a^{21}b^{14}n^7+126a^{21}b^{13}n^8-294a^{21}b^{12}n^9-252a^{21}b^{11}n^{10}+252a^{21}b^{10}n^{11}+294a^{21}b^9n^{12}-126a^{21}b^8n^{13}-216a^{21}b^7n^{14}+24a^{21}b^6n^{15}+99a^{21}b^5n^{16}+9a^{21}b^4n^{17}-26a^{21}b^3n^{18}-6a^{21}b^2n^{19}+3a^{21}bn^{20}+a^{21}n^{21}-3a^{20}b^{21}m+12a^{20}b^{20}mn-10a^{20}b^{19}mn^2-20a^{20}b^{18}mn^3+57a^{20}b^{17}mn^4-80a^{20}b^{16}mn^5+8a^{20}b^{15}mn^6+240a^{20}b^{14}mn^7-310a^{20}b^{13}mn^8-152a^{20}b^{12}mn^9+516a^{20}b^{11}mn^{10}-152a^{20}b^{10}mn^{11}-310a^{20}b^9mn^{12}+240a^{20}b^8mn^{13}+8a^{20}b^7mn^{14}-80a^{20}b^6mn^{15}+57a^{20}b^5mn^{16}-20a^{20}b^4mn^{17}-10a^{20}b^3mn^{18}+12a^{20}b^2mn^{19}-3a^{20}bmn^{20}+6a^{19}b^{21}m^2-10a^{19}b^{20}m^2n-22a^{19}b^{19}m^2n^2+26a^{19}b^{18}m^2n^3+40a^{19}b^{17}m^2n^4+40a^{19}b^{16}m^2n^5-104a^{19}b^{15}m^2n^6-232a^{19}b^{14}m^2n^7+260a^{19}b^{13}m^2n^8+356a^{19}b^{12}m^2n^9-356a^{19}b^{11}m^2n^{10}-260a^{19}b^{10}m^2n^{11}+232a^{19}b^9m^2n^{12}+104a^{19}b^8m^2n^{13}-40a^{19}b^7m^2n^{14}-40a^{19}b^6m^2n^{15}-26a^{19}b^5m^2n^{16}+22a^{19}b^4m^2n^{17}+10a^{19}b^3m^2n^{18}-6a^{19}b^2m^2n^{19}+26a^{18}b^{21}m^3-20a^{18}b^{20}m^3n+26a^{18}b^{19}m^3n^2+192a^{18}b^{18}m^3n^3-264a^{18}b^{17}m^3n^4-240a^{18}b^{16}m^3n^5+376a^{18}b^{15}m^3n^6-448a^{18}b^{14}m^3n^7-164a^{18}b^{13}m^3n^8+1032a^{18}b^{12}m^3n^9-164a^{18}b^{11}m^3n^{10}-448a^{18}b^{10}m^3n^{11}+376a^{18}b^9m^3n^{12}-240a^{18}b^8m^3n^{13}-264a^{18}b^7m^3n^{14}+192a^{18}b^6m^3n^{15}+26a^{18}b^5m^3n^{16}-20a^{18}b^4m^3n^{17}+26a^{18}b^3m^3n^{18}-9a^{17}b^{21}m^4+57a^{17}b^{20}m^4n+40a^{17}b^{19}m^4n^2-264a^{17}b^{18}m^4n^3-44a^{17}b^{17}m^4n^4+460a^{17}b^{16}m^4n^5-8a^{17}b^{15}m^4n^6-344a^{17}b^{14}m^4n^7-70a^{17}b^{13}m^4n^8+70a^{17}b^{12}m^4n^9+344a^{17}b^{11}m^4n^{10}+8a^{17}b^{10}m^4n^{11}-460a^{17}b^9m^4n^{12}+44a^{17}b^8m^4n^{13}+264a^{17}b^7m^4n^{14}-40a^{17}b^6m^4n^{15}-57a^{17}b^5m^4n^{16}+9a^{17}b^4m^4n^{17}-99a^{16}b^{21}m^5-80a^{16}b^{20}m^5n+40a^{16}b^{19}m^5n^2-240a^{16}b^{18}m^5n^3+460a^{16}b^{17}m^5n^4+1200a^{16}b^{16}m^5n^5-616a^{16}b^{15}m^5n^6-880a^{16}b^{14}m^5n^7+430a^{16}b^{13}m^5n^8-880a^{16}b^{12}m^5n^9-616a^{16}b^{11}m^5n^{10}+1200a^{16}b^{10}m^5n^{11}+460a^{16}b^9m^5n^{12}-240a^{16}b^8m^5n^{13}+40a^{16}b^7m^5n^{14}-80a^{16}b^6m^5n^{15}-99a^{16}b^5m^5n^{16}-24a^{15}b^{21}m^6+8a^{15}b^{20}m^6n-104a^{15}b^{19}m^6n^2+376a^{15}b^{18}m^6n^3-8a^{15}b^{17}m^6n^4-616a^{15}b^{16}m^6n^5+264a^{15}b^{15}m^6n^6+360a^{15}b^{14}m^6n^7-360a^{15}b^{13}m^6n^8-264a^{15}b^{12}m^6n^9+616a^{15}b^{11}m^6n^{10}+8a^{15}b^{10}m^6n^{11}-376a^{15}b^9m^6n^{12}+104a^{15}b^8m^6n^{13}-8a^{15}b^7m^6n^{14}+24a^{15}b^6m^6n^{15}+216a^{14}b^{21}m^7+240a^{14}b^{20}m^7n-232a^{14}b^{19}m^7n^2-448a^{14}b^{18}m^7n^3-344a^{14}b^{17}m^7n^4-880a^{14}b^{16}m^7n^5+360a^{14}b^{15}m^7n^6+2176a^{14}b^{14}m^7n^7+360a^{14}b^{13}m^7n^8-880a^{14}b^{12}m^7n^9-344a^{14}b^{11}m^7n^{10}-448a^{14}b^{10}m^7n^{11}-232a^{14}b^9m^7n^{12}+240a^{14}b^8m^7n^{13}+216a^{14}b^7m^7n^{14}+126a^{13}b^{21}m^8-310a^{13}b^{20}m^8n+260a^{13}b^{19}m^8n^2-164a^{13}b^{18}m^8n^3-70a^{13}b^{17}m^8n^4+430a^{13}b^{16}m^8n^5-360a^{13}b^{15}m^8n^6+360a^{13}b^{14}m^8n^7-430a^{13}b^{13}m^8n^8+70a^{13}b^{12}m^8n^9+164a^{13}b^{11}m^8n^{10}-260a^{13}b^{10}m^8n^{11}+310a^{13}b^9m^8n^{12}-126a^{13}b^8m^8n^{13}-294a^{12}b^{21}m^9-152a^{12}b^{20}m^9n+356a^{12}b^{19}m^9n^2+1032a^{12}b^{18}m^9n^3+70a^{12}b^{17}m^9n^4-880a^{12}b^{16}m^9n^5-264a^{12}b^{15}m^9n^6-880a^{12}b^{14}m^9n^7+70a^{12}b^{13}m^9n^8+1032a^{12}b^{12}m^9n^9+356a^{12}b^{11}m^9n^{10}-152a^{12}b^{10}m^9n^{11}-294a^{12}b^9m^9n^{12}-252a^{11}b^{21}m^{10}+516a^{11}b^{20}m^{10}n-356a^{11}b^{19}m^{10}n^2-164a^{11}b^{18}m^{10}n^3+344a^{11}b^{17}m^{10}n^4-616a^{11}b^{16}m^{10}n^5+616a^{11}b^{15}m^{10}n^6-344a^{11}b^{14}m^{10}n^7+164a^{11}b^{13}m^{10}n^8+356a^{11}b^{12}m^{10}n^9-516a^{11}b^{11}m^{10}n^{10}+252a^{11}b^{10}m^{10}n^{11}+252a^{10}b^{21}m^{11}-152a^{10}b^{20}m^{11}n-260a^{10}b^{19}m^{11}n^2-448a^{10}b^{18}m^{11}n^3+8a^{10}b^{17}m^{11}n^4+1200a^{10}b^{16}m^{11}n^5+8a^{10}b^{15}m^{11}n^6-448a^{10}b^{14}m^{11}n^7-260a^{10}b^{13}m^{11}n^8-152a^{10}b^{12}m^{11}n^9+252a^{10}b^{11}m^{11}n^{10}+294a^9b^{21}m^{12}-310a^9b^{20}m^{12}n+232a^9b^{19}m^{12}n^2+376a^9b^{18}m^{12}n^3-460a^9b^{17}m^{12}n^4+460a^9b^{16}m^{12}n^5-376a^9b^{15}m^{12}n^6-232a^9b^{14}m^{12}n^7+310a^9b^{13}m^{12}n^8-294a^9b^{12}m^{12}n^9-126a^8b^{21}m^{13}+240a^8b^{20}m^{13}n+104a^8b^{19}m^{13}n^2-240a^8b^{18}m^{13}n^3+44a^8b^{17}m^{13}n^4-240a^8b^{16}m^{13}n^5+104a^8b^{15}m^{13}n^6+240a^8b^{14}m^{13}n^7-126a^8b^{13}m^{13}n^8-216a^7b^{21}m^{14}+8a^7b^{20}m^{14}n-40a^7b^{19}m^{14}n^2-264a^7b^{18}m^{14}n^3+264a^7b^{17}m^{14}n^4+40a^7b^{16}m^{14}n^5-8a^7b^{15}m^{14}n^6+216a^7b^{14}m^{14}n^7+24a^6b^{21}m^{15}-80a^6b^{20}m^{15}n-40a^6b^{19}m^{15}n^2+192a^6b^{18}m^{15}n^3-40a^6b^{17}m^{15}n^4-80a^6b^{16}m^{15}n^5+24a^6b^{15}m^{15}n^6+99a^5b^{21}m^{16}+57a^5b^{20}m^{16}n-26a^5b^{19}m^{16}n^2+26a^5b^{18}m^{16}n^3-57a^5b^{17}m^{16}n^4-99a^5b^{16}m^{16}n^5+9a^4b^{21}m^{17}-20a^4b^{20}m^{17}n+22a^4b^{19}m^{17}n^2-20a^4b^{18}m^{17}n^3+9a^4b^{17}m^{17}n^4-26a^3b^{21}m^{18}-10a^3b^{20}m^{18}n+10a^3b^{19}m^{18}n^2+26a^3b^{18}m^{18}n^3-6a^2b^{21}m^{19}+12a^2b^{20}m^{19}n-6a^2b^{19}m^{19}n^2+3ab^{21}m^{20}-3ab^{20}m^{20}n+b^{21}m^{21}=0

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数学星空

下一步：由于共焦点，因此有关系式\(t=a^2-m^2=b^2-n^2\),将楼上的结果与此式消元\(m,n\)即可以得到下面结论：

对于\(k\)边形：

当\(k=3\)时

\(t=\frac{(-a^2-b^2+2\sqrt{a^4-a^2b^2+b^4})b^2a^2}{a^4-2a^2b^2+b^4}\)

当\(k=4\)时

\(t=\frac{a^2b^2}{a^2+b^2}\)

当\(k=5\)时

\(5a^{12}b^{12}-10a^{10}b^{10}(a^2+b^2)t-a^8b^8(3a^2-4ab-3b^2)(3a^2+4ab-3b^2)t^2+36a^6b^6(a^2+b^2)(a-b)^2(a+b)^2t^3-a^4b^4(29a^4+54a^2b^2+29b^4)(a-b)^2(a+b)^2t^4+2a^2b^2(3a^2+b^2)(a^2+3b^2)(a^2+b^2)(a-b)^2(a+b)^2t^5+(a-b)^6(a+b)^6t^6=0\)

当\(k=6\)时

\(t=\frac{a^2b^2}{a^2+2ab+b^2}\)

当\(k=7\)时

\(-7a^{24}b^{24}+28a^{22}b^{22}(a^2+b^2)t+14a^{20}b^{20}(3a^4-14a^2b^2+3b^4)t^2-4a^{18}b^{18}(a^2+b^2)(121a^4-250a^2b^2+121b^4)t^3+3a^{16}b^{16}(437a^4+726a^2b^2+437b^4)(a-b)^2(a+b)^2t^4-24a^{14}b^{14}(a^2+b^2)(75a^4+106a^2b^2+75b^4)(a-b)^2(a+b)^2t^5+12a^{12}b^{12}(105a^8+420a^6b^2+422a^4b^4+420a^2b^6+105b^8)(a-b)^2(a+b)^2t^6-8a^{10}b^{10}(a^2+b^2)(21a^8+420a^6b^2-50a^4b^4+420a^2b^6+21b^8)(a-b)^2(a+b)^2t^7-a^8b^8(7a^4-30a^2b^2+7b^4)(63a^8+84a^6b^2-38a^4b^4+84a^2b^6+63b^8)(a-b)^2(a+b)^2t^8+28a^6b^6(a^2+b^2)(13a^4+38a^2b^2+13b^4)(a-b)^6(a+b)^6t^9-2a^4b^4(59a^8+332a^6b^2+626a^4b^4+332a^2b^6+59b^8)(a-b)^6(a+b)^6t^{10}+4a^2b^2(3a^2+b^2)(a^2+3b^2)(a^2+b^2)(a^4+6a^2b^2+b^4)(a-b)^6(a+b)^6t^{11}+(a-b)^{12}(a+b)^{12}t^{12}=0\)

当\(k=8\)时

\(a^8b^8-4a^6b^6(a^2+b^2)t+2a^4b^4(3a^4-2a^2b^2+3b^4)t^2-4a^2b^2(a^2+b^2)(a-b)^2(a+b)^2t^3+(a^4+6a^2b^2+b^4)(a-b)^2(a+b)^2t^4=0\)

当\(k=9\)时

\(-3a^36b^36+18a^34b^34(a^2+b^2)t+3a^32b^32(19a^4-70a^2b^2+19b^4)t^2-16a^{30}b^{30}(a^2+b^2)(61a^4-124a^2b^2+61b^4)t^3+60a^{28}b^{28}(79a^4+122a^2b^2+79b^4)(a-b)^2(a+b)^2t^4-168a^{26}b^{26}(a^2+b^2)(77a^4+82a^2b^2+77b^4)(a-b)^2(a+b)^2t^5+28a^{24}b^{24}(767a^8+2236a^6b^2+2186a^4b^4+2236a^2b^6+767b^8)(a-b)^2(a+b)^2t^6-16a^{22}b^{22}(a^2+b^2)(1131a^8+5226a^6b^2+118a^4b^4+5226a^2b^6+1131b^8)(a-b)^2(a+b)^2t^7-2a^{20}b^{20}(3861a^{12}-57486a^{10}b^2-65413a^8b^4+16892a^6b^6-65413a^4b^8-57486a^2b^{10}+3861b^{12})(a-b)^2(a+b)^2t^8+4a^{18}b^{18}(a^2+b^2)(12155a^{12}-32890a^{10}b^2-25579a^8b^4+59860a^6b^6-25579a^4b^8-32890a^2b^{10}+12155b^{12})(a-b)^2(a+b)^2t^9-2a^{16}b^{16}(39897a^{16}-10296a^{14}b^2-148388a^{12}b^4-29960a^{10}b^6+264726a^8b^8-29960a^6b^{10}-148388a^4b^{12}-10296a^2b^{14}+39897b^{16})(a-b)^2(a+b)^2t^{10}+16a^{14}b^{14}(a^2+b^2)(5109a^{12}+7176a^{10}b^2-2045a^8b^4-20992a^6b^6-2045a^4b^8+7176a^2b^{10}+5109b^{12})(a-b)^4(a+b)^4t^{11}-28a^{12}b^{12}(2093a^{12}+10218a^{10}b^2+23747a^8b^4+32332a^6b^6+23747a^4b^8+10218a^2b^{10}+2093b^{12})(a-b)^6(a+b)^6t^{12}+56a^{10}b^{10}(a^2+b^2)(537a^{12}+2334a^{10}b^2+4391a^8b^4+6980a^6b^6+4391a^4b^8+2334a^2b^{10}+537b^{12})(a-b)^6(a+b)^6t^{13}-4a^8b^8(2715a^{16}+16080a^{14}b^2+32516a^{12}b^4+56560a^{10}b^6+79170a^8b^8+56560a^6b^{10}+32516a^4b^{12}+16080a^2b^{14}+2715b^{16})(a-b)^6(a+b)^6t^{14}+16a^6b^6(a^2+b^2)(163a^{16}+950a^{14}b^2+864a^{12}b^4+234a^{10}b^6+3770a^8b^8+234a^6b^{10}+864a^4b^{12}+950a^2b^{14}+163b^{16})(a-b)^6(a+b)^6t^{15}-a^4b^4(363a^{16}+3912a^{14}b^2+13300a^{12}b^4+12280a^{10}b^6+5826a^8b^8+12280a^6b^{10}+13300a^4b^{12}+3912a^2b^{14}+363b^{16})(a-b)^8(a+b)^8t^{16}+2a^2b^2(a^2+b^2)(a^4+14a^2b^2+b^4)(3a^2+b^2)^2(a^2+3b^2)^2(a-b)^{10}(a+b)^{10}t^{17}+(a-b)^{18}(a+b)^{18}t^{18}=0\)

当\(k=10\)时

\(a^{12}b^{12}-6a^{10}b^{10}(a^2+b^2)t+a^8b^8(15a^4-14a^2b^2+15b^4)t^2-20a^6b^6(a^2+b^2)(a-b)^2(a+b)^2t^3+5a^4b^4(3a^2+b^2)(a^2+3b^2)(a-b)^2(a+b)^2t^4-2a^2b^2(3a^2+b^2)(a^2+3b^2)(a^2+b^2)(a-b)^2(a+b)^2t^5+(a-b)^6(a+b)^6t^6=0\)

当\(k=11\)时

1. 2a^42b^42(a^2+b^2)(5281375a^{12}-31890650a^{10}b^2-31760623a^8b^4+21065428a^6b^6-31760623a^4b^8-31890650a^2b^{10}+5281375b^{12})(a-b)^2(a+b)^2t^9+110a^58b^58(a^2+b^2)t+11a^56b^56(33a^4-178a^2b^2+33b^4)t^2+11a^52b^52(9995a^8-4060a^6b^2-12126a^4b^4-4060a^2b^6+9995b^8)t^4+176a^{30}b^{30}(a^2+b^2)(1300075a^{16}+10400600a^{14}b^2+29060500a^{12}b^4+56198376a^{10}b^6+76154754a^8b^8+56198376a^6b^{10}+29060500a^4b^{12}+10400600a^2b^{14}+1300075b^{16})(a-b)^6(a+b)^6t^{15}-8a^46b^46(a^2+b^2)(429505a^8+1501100a^6b^2+734758a^4b^4+1501100a^2b^6+429505b^8)(a-b)^2(a+b)^2t^7+2a^34b^34(a^2+b^2)(145531925a^{20}+174830590a^{18}b^2-380207311a^{16}b^4-687408920a^{14}b^6-324641350a^{12}b^8+2117575732a^{10}b^{10}-324641350a^8b^{12}-687408920a^6b^{14}-380207311a^4b^{16}+174830590a^2b^{18}+145531925b^{20})(a-b)^2(a+b)^2t^{13}+4a^38b^38(a^2+b^2)(30720525a^{16}-14218600a^{14}b^2-106020244a^{12}b^4-42256856a^{10}b^6+203453838a^8b^8-42256856a^6b^{10}-106020244a^4b^{12}-14218600a^2b^{14}+30720525b^{16})(a-b)^2(a+b)^2t^{11}-8a^{14}b^{14}(a^2+b^2)(350625a^{20}+5125670a^{18}b^2+28761293a^{16}b^4+98023112a^{14}b^6+201932434a^{12}b^8+249117732a^{10}b^{10}+201932434a^8b^{12}+98023112a^6b^{14}+28761293a^4b^{16}+5125670a^2b^{18}+350625b^{20})(a-b)^{12}(a+b)^{12}t^{23}+a^44b^44(2437425a^{12}+34639610a^{10}b^2+48707679a^8b^4+27782892a^6b^6+48707679a^4b^8+34639610a^2b^{10}+2437425b^{12})(a-b)^2(a+b)^2t^8+a^48b^48(1716575a^8+5006500a^6b^2+5616378a^4b^4+5006500a^2b^6+1716575b^8)(a-b)^2(a+b)^2t^6+4a^6b^6(a^2+b^2)(1603a^{12}+21934a^{10}b^2+95213a^8b^4+155716a^6b^6+95213a^4b^8+21934a^2b^{10}+1603b^{12})(a-b)^{20}(a+b)^{20}t^{27}-11a^60b^60+2a^{10}b^{10}(a^2+b^2)(9053a^{24}-637692a^{22}b^2-4984518a^{20}b^4-16945676a^{18}b^6-55339053a^{16}b^8-107480824a^{14}b^{10}-115781844a^{12}b^{12}-107480824a^{10}b^{14}-55339053a^8b^{16}-16945676a^6b^{18}-4984518a^4b^{20}-637692a^2b^{22}+9053b^{24})(a-b)^{12}(a+b)^{12}t^{25}-2a^{18}b^{18}(a^2+b^2)(13997225a^{28}+58961650a^{26}b^2-19226413a^{24}b^4-449053164a^{22}b^6-161755503a^{20}b^8+1522357038a^{18}b^{10}-296694797a^{16}b^{12}-1605607528a^{14}b^{14}-296694797a^{12}b^{16}+1522357038a^{10}b^{18}-161755503a^8b^{20}-449053164a^6b^{22}-19226413a^4b^{24}+58961650a^2b^{26}+13997225b^{28})(a-b)^6(a+b)^6t^{21}-4a^{22}b^{22}(a^2+b^2)(19134275a^{24}+15973500a^{22}b^2-250142458a^{20}b^4-421411444a^{18}b^6+282365837a^{16}b^8-598765064a^{14}b^{10}-1835628460a^{12}b^{12}-598765064a^{10}b^{14}+282365837a^8b^{16}-421411444a^6b^{18}-250142458a^4b^{20}+15973500a^2b^{22}+19134275b^{24})(a-b)^6(a+b)^6t^{19}-2a^{26}b^{26}(a^2+b^2)(2576115a^{20}-386688190a^{18}b^2-1538799433a^{16}b^4-2389229800a^{14}b^6-4728583338a^{12}b^8-7206009524a^{10}b^{10}-4728583338a^8b^{12}-2389229800a^6b^{14}-1538799433a^4b^{16}-386688190a^2b^{18}+2576115b^{20})(a-b)^6(a+b)^6t^{17}+2a^2b^2(3a^2+b^2)(a^2+3b^2)(a^2+b^2)(a^4+14a^2b^2+b^4)(5a^4+10a^2b^2+b^4)(a^4+10a^2b^2+5b^4)(a-b)^{20}(a+b)^{20}t^{29}-a^40b^40(49609505a^{16}-37919640a^{14}b^2-286113284a^{12}b^4-112685480a^{10}b^6+257073222a^8b^8-112685480a^6b^{10}-286113284a^4b^{12}-37919640a^2b^{14}+49609505b^{16})(a-b)^2(a+b)^2t^{10}-a^8b^8(26235a^{28}+19030a^{26}b^2-1310727a^{24}b^4-3567300a^{22}b^6-8961997a^{20}b^8-35638262a^{18}b^{10}-58013351a^{16}b^{12}-53542712a^{14}b^{14}-58013351a^{12}b^{16}-35638262a^{10}b^{18}-8961997a^8b^{20}-3567300a^6b^{22}-1310727a^4b^{24}+19030a^2b^{26}+26235b^{28})(a-b)^{12}(a+b)^{12}t^{26}-a^4b^4(745a^{16}+14088a^{14}b^2+99228a^{12}b^4+295864a^{10}b^6+425334a^8b^8+295864a^6b^{10}+99228a^4b^{12}+14088a^2b^{14}+745b^{16})(a-b)^{20}(a+b)^{20}t^{28}+a^{12}b^{12}(416075a^{24}+9817500a^{22}b^2+67130646a^{20}b^4+274745900a^{18}b^6+775935237a^{16}b^8+1400052792a^{14}b^{10}+1671467316a^{12}b^{12}+1400052792a^{10}b^{14}+775935237a^8b^{16}+274745900a^6b^{18}+67130646a^4b^{20}+9817500a^2b^{22}+416075b^{24})(a-b)^{12}(a+b)^{12}t^{24}+a^{16}b^{16}(10601965a^{20}+136576990a^{18}b^2+764787881a^{16}b^4+2479730280a^{14}b^6+4884025962a^{12}b^8+6070532468a^{10}b^{10}+4884025962a^8b^{12}+2479730280a^6b^{14}+764787881a^4b^{16}+136576990a^2b^{18}+10601965b^{20})(a-b)^{12}(a+b)^{12}t^{22}-2a^50b^50(a^2+b^2)(268923a^8-226588a^6b^2-84926a^4b^4-226588a^2b^6+268923b^8)t^5-44a^54b^54(a^2+b^2)(295a^4-622a^2b^2+295b^4)t^3+a^{20}b^{20}(54378555a^{28}+203439830a^{26}b^2-212728519a^{24}b^4-1803503044a^{22}b^6-409192973a^{20}b^8+3400677258a^{18}b^{10}-1722780519a^{16}b^{12}-7342080312a^{14}b^{14}-1722780519a^{12}b^{16}+3400677258a^{10}b^{18}-409192973a^8b^{20}-1803503044a^6b^{22}-212728519a^4b^{24}+203439830a^2b^{26}+54378555b^{28})(a-b)^6(a+b)^6t^{20}+a^{24}b^{24}(68051825a^{24}-170665980a^{22}b^2-2159513598a^{20}b^4-4331465228a^{18}b^6-5101010433a^{16}b^8-12157968888a^{14}b^{10}-18581635812a^{12}b^{12}-12157968888a^{10}b^{14}-5101010433a^8b^{16}-4331465228a^6b^{18}-2159513598a^4b^{20}-170665980a^2b^{22}+68051825b^{24})(a-b)^6(a+b)^6t^{18}-7a^{28}b^{28}(15415175a^{20}+214326650a^{18}b^2+815037115a^{16}b^4+1779333560a^{14}b^6+3197535422a^{12}b^8+4067025820a^{10}b^{10}+3197535422a^8b^{12}+1779333560a^6b^{14}+815037115a^4b^{16}+214326650a^2b^{18}+15415175b^{20})(a-b)^6(a+b)^6t^{16}-a^32b^32(300317325a^{24}+972456100a^{22}b^2-150791718a^{20}b^4-1862255340a^{18}b^6-4156413789a^{16}b^8+629657160a^{14}b^{10}+8517283308a^{12}b^{12}+629657160a^{10}b^{14}-4156413789a^8b^{16}-1862255340a^6b^{18}-150791718a^4b^{20}+972456100a^2b^{22}+300317325b^{24})(a-b)^2(a+b)^2t^{14}-a^36b^36(216723595a^{20}+300485570a^{18}b^2-556866321a^{16}b^4-1495241704a^{14}b^6+144050566a^{12}b^8+2489143884a^{10}b^{10}+144050566a^8b^{12}-1495241704a^6b^{14}-556866321a^4b^{16}+300485570a^2b^{18}+216723595b^{20})(a-b)^2(a+b)^2t^{12}+(a-b)^{30}(a+b)^{30}t^{30}=0

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数学星空

下面论文中提出了计算椭圆内接N边形的公式：（不知我理解的是否正确，请mathe帮忙看一下？）

内接椭圆公式.pdf (148.04 KB, 下载次数: 25)

2014-5-11 11:56 上传

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对于共心共焦点的双椭圆


\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)


\(\frac{x^2}{a^2-\mu^2}+\frac{y^2}{b^2-\mu^2}=1\)


对于N边形外切内接上面双椭圆，则有计算公式：


\(k^2=\frac{a^2-b^2}{a^2-\mu^2}\)


\(\frac{\delta}{2}=\int_0^{\arcsin(\frac{\mu}{b})} (1-k^2\sin(\phi )^2)^{-\frac{1}{2}}  d{\phi}\)


\(t_j=\int_0^{\mu_j} (1-k^2\sin(\phi )^2)^{-\frac{1}{2}} d{\phi}\)


\(sn(t_j)=\sin(\mu_j),  cn(t_j)=\cos(\mu_j)\)


\(q_j=(a\cos(\mu_j),b\sin(\mu_j))=(a\*cn(t_j),b\*sn(t_j))\)


\(t_{j+1}=t_j+\delta\)


==================================＝＝＝＝＝＝＝＝

经计算并画图，以上计算公式并不适用于共焦点双椭圆的N边形边长计算

数学星空

对于\(N=3\),我们设双椭圆分别为：


\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)


\(\frac{x^2}{a^2-t}+\frac{y^2}{b^2-t}=1\)


设\(\triangle ABC\)内接外切于双椭圆，第一个点为\(A[a\cos(\alpha_1),b\sin(\alpha_1)]\)，且\(s_1=\tan(\frac{\alpha_1}{2}), i=1,2,3   AB=L_1,BC=L_2,CA=L_3\)

则有\(s_i,L_i （i=1,2,3)\) 均满足下列方程：

\(9(b^2s_i^4+4a^2s_i^2-2b^2s_i^2+b^2)^8-12(s_i^2+1)^2(-2a^2b^4s_i^{12}+b^6s_i^{12}+32a^4b^2s_i^{10}+10b^6s_i^{10}+64a^6s_i^8-176a^4b^2s_i^8+146a^2b^4s_i^8-49b^6s_i^8-64a^6s_i^6+160a^4b^2s_i^6-288a^2b^4s_i^6+76b^6s_i^6+64a^6s_i^4-176a^4b^2s_i^4+146a^2b^4s_i^4-49b^6s_i^4+32a^4b^2s_i^2+10b^6s_i^2-2a^2b^4+b^6)(b^2s_i^4+4a^2s_i^2-2b^2s_i^2+b^2)^4L_i^2+2(s_i^2+1)^4(8a^4b^8s_i^{24}-8a^2b^{10}s_i^{24}-b^{12}s_i^{24}+128a^6b^6s_i^{22}-304a^4b^8s_i^{22}+328a^2b^{10}s_i^{22}+28b^{12}s_i^{22}+1664a^6b^6s_i^{20}-64a^4b^8s_i^{20}-800a^2b^{10}s_i^{20}+286b^{12}s_i^{20}+14336a^{10}b^2s_i^{18}-32768a^8b^4s_i^{18}+36224a^6b^6s_i^{18}-22512a^4b^8s_i^{18}+10152a^2b^{10}s_i^{18}-3156b^{12}s_i^{18}+8192a^{12}s_i^{16}-53248a^{10}b^2s_i^{16}+153856a^8b^4s_i^{16}-209920a^6b^6s_i^{16}+150712a^4b^8s_i^{16}-60472a^2b^{10}s_i^{16}+11921b^{12}s_i^{16}-28672a^{12}s_i^{14}+143360a^{10}b^2s_i^{14}-373760a^8b^4s_i^{14}+516608a^6b^6s_i^{14}-391904a^4b^8s_i^{14}+155408a^2b^{10}s_i^{14}-24520b^{12}s_i^{14}+36864a^{12}s_i^{12}-241664a^{10}b^2s_i^{12}+538112a^8b^4s_i^{12}-689408a^6b^6s_i^{12}+528128a^4b^8s_i^{12}-209216a^2b^{10}s_i^{12}+30884b^{12}s_i^{12}-28672a^{12}s_i^{10}+143360a^{10}b^2s_i^{10}-373760a^8b^4s_i^{10}+516608a^6b^6s_i^{10}-391904a^4b^8s_i^{10}+155408a^2b^{10}s_i^{10}-24520b^{12}s_i^{10}+8192a^{12}s_i^8-53248a^{10}b^2s_i^8+153856a^8b^4s_i^8-209920a^6b^6s_i^8+150712a^4b^8s_i^8-60472a^2b^{10}s_i^8+11921b^{12}s_i^8+14336a^{10}b^2s_i^6-32768a^8b^4s_i^6+36224a^6b^6s_i^6-22512a^4b^8s_i^6+10152a^2b^{10}s_i^6-3156b^{12}s_i^6+1664a^6b^6s_i^4-64a^4b^8s_i^4-800a^2b^{10}s_i^4+286b^{12}s_i^4+128a^6b^6s_i^2-304a^4b^8s_i^2+328a^2b^{10}s_i^2+28b^{12}s_i^2+8a^4b^8-8a^2b^{10}-b^{12})L_i^4-4(s_i^2+1)^6(2a^2b^8s_i^{20}-b^{10}s_i^{20}+32a^6b^4s_i^{18}-32a^4b^6s_i^{18}-16a^2b^8s_i^{18}+42b^{10}s_i^{18}+256a^{10}s_i^{16}-768a^8b^2s_i^{16}+928a^6b^4s_i^{16}-256a^4b^6s_i^{16}-6a^2b^8s_i^{16}-45b^{10}s_i^{16}-512a^{10}s_i^{14}+4352a^8b^2s_i^{14}-8256a^6b^4s_i^{14}+6912a^4b^6s_i^{14}-2816a^2b^8s_i^{14}+504b^{10}s_i^{14}+1792a^{10}s_i^{12}-10240a^8b^2s_i^{12}+22112a^6b^4s_i^{12}-22784a^4b^6s_i^{12}+11524a^2b^8s_i^{12}-2258b^{10}s_i^{12}-4096a^{10}s_i^{10}+15360a^8b^2s_i^{10}-29632a^6b^4s_i^{10}+32320a^4b^6s_i^{10}-17376a^2b^8s_i^{10}+3516b^{10}s_i^{10}+1792a^{10}s_i^8-10240a^8b^2s_i^8+22112a^6b^4s_i^8-22784a^4b^6s_i^8+11524a^2b^8s_i^8-2258b^{10}s_i^8-512a^{10}s_i^6+4352a^8b^2s_i^6-8256a^6b^4s_i^6+6912a^4b^6s_i^6-2816a^2b^8s_i^6+504b^{10}s_i^6+256a^{10}s_i^4-768a^8b^2s_i^4+928a^6b^4s_i^4-256a^4b^6s_i^4-6a^2b^8s_i^4-45b^{10}s_i^4+32a^6b^4s_i^2-32a^4b^6s_i^2-16a^2b^8s_i^2+42b^{10}s_i^2+2a^2b^8-b^{10})L_i^6+(s_i^2+1)^8(b^4s_i^8+16a^4s_i^6-24a^2b^2s_i^6+12b^4s_i^6-16a^4s_i^4+48a^2b^2s_i^4-26b^4s_i^4+16a^4s_i^2-24a^2b^2s_i^2+12b^4s_i^2+b^4)^2L_i^8=0\)

且\(s_2,s_3\)为下列关于\(y\)的方程的正实根，且\(s_2<s_3\)

\(b^2(-b^2s_1^4+4a^2s_1^2-2b^2s_1^2+3b^2)+8s_1(2a^4s_1^2-2a^2b^2s_1^2+b^4s_1^2+2a^2b^2-b^4)y+(2(2a^2-b^2))(b^2s_1^4+4a^2s_1^2-2b^2s_1^2+b^2)y^2+8s_1(2a^2b^2s_1^2-b^4s_1^2+2a^4-2a^2b^2+b^4)y^3+b^2(3b^2s_1^4+4a^2s_1^2-2b^2s_1^2-b^2)y^4=0\)

数学星空

对于\(N=4\),我们设双椭圆分别为：


\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)


\(\frac{x^2}{a^2-t}+\frac{y^2}{b^2-t}=1\)


设凸四边形\(ABCD\)内接外切于双椭圆，第一个点为\(A[a\cos(\alpha_1),b\sin(\alpha_1)]\)，且\(s_i=\tan(\frac{\alpha_i}{2}), i=1,2,3,4\)   \(AB=L_1,BC=L_2,CD=L_3,DA=L_4\),

则有\(s_i,L_i （i=1,2,3,4)\) 均满足下列方程：

\((a^2+b^2)^2(b^2s_i^4+4a^2s_i^2-2b^2s_i^2+b^2)^4-2(s_i^2+1)^2(a^2+b^2)(b^4s_i^8+8a^4s_i^6-8a^2b^2s_i^6+4b^4s_i^6+16a^2b^2s_i^4-10b^4s_i^4+8a^4s_i^2-8a^2b^2s_i^2+4b^4s_i^2+b^4)(b^4s_i^4+4a^4s_i^2-2b^4s_i^2+b^4)L_i^2+(s_i^2+1)^4(b^4s_i^4+4a^4s_i^2-2b^4s_i^2+b^4)^2L_i^4=0\)

且\(s_1,s_2,s_3,s_4\)的关系式如下：

\(s_2 = -\frac{2a^2s_1+\sqrt{b^4s_1^4+4a^4s_1^2-2b^4s_1^2+b^4}}{b^2(s_1^2-1)}\)

\( s_3 = -\frac{1}{s_1}\)

\( s_4 =\frac{-2a^2s_1+\sqrt{b^4s_1^4+4a^4s_1^2-2b^4s_1^2+b^4}}{b^2(s_1^2-1)}=-\frac{1}{s_2}\)

数学星空

对于\(N=6\),我们设双椭圆分别为：


\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)


\(\frac{x^2}{a^2-t}+\frac{y^2}{b^2-t}=1\)


设凸六边形\(ABCDEF\)内接外切于双椭圆，第一个点为\(A[a\cos(\alpha_1),b\sin(\alpha_1)]\)，且\(s_i=\tan(\frac{\alpha_i}{2}), i=1,2,3,4,5,6\)   \(AB=L_1,BC=L_2,CD=L_3,DE=L_4,EF=L_5,FA=L_6\),

则有\(s_i,L_i （i=1,2,3,4,5,6)\) 均满足下列方程：

\((b^2s_i^4+4a^2s_i^2-2b^2s_i^2+b^2)^2-2(s_i^2+1)^2(b^3s_i^4+4a^3s_i^2-2b^3s_i^2+b^3)L_i+(s_i^2+1)^2(b^2s_i^4+4a^2s_i^2-2b^2s_i^2+b^2)L_i^2=0\)

且\(s_1,s_2,s_3,s_4,s_5,s_6\)的关系式如下：

\(s_2 = \frac{2a^2s_1+2abs_1+\sqrt{2ab^3s_1^4+b^4s_1^4+4a^4s_1^2+8a^3bs_1^2-4ab^3s_1^2-2b^4s_1^2+2ab^3+b^4}}{b(-bs_1^2+2a+b)}\)

\(s_3 = -\frac{2a^2s_1+2abs_1+\sqrt{2ab^3s_1^4+b^4s_1^4+4a^4s_1^2+8a^3bs_1^2-4ab^3s_1^2-2b^4s_1^2+2ab^3+b^4}}{b(2as_1^2+bs_1^2-b)}\)

\(s_4 = -\frac{1}{s_1}\)

\(s_5 = -\frac{b(-bs_1^2+2a+b)}{2a^2s_1+2abs_1+\sqrt{2ab^3s_1^4+b^4s_1^4+4a^4s_1^2+8a^3bs_1^2-4ab^3s_1^2-2b^4s_1^2+2ab^3+b^4}}=-\frac{1}{s_2}\)

\(s_6 = \frac{b(2as_1^2+bs_1^2-b)}{2a^2s_1+2abs_1+\sqrt{2ab^3s_1^4+b^4s_1^4+4a^4s_1^2+8a^3bs_1^2-4ab^3s_1^2-2b^4s_1^2+2ab^3+b^4}}=-\frac{1}{s_3}\)

数学星空

对于\(N=k\),我们设双椭圆分别为：


\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)


\(\frac{x^2}{a^2-t}+\frac{y^2}{b^2-t}=1\)


设凸\(k\)边形\(A_1 A_2 A_3... A_K\)内接外切于双椭圆，第一个点为\(A_1[a\cos(\alpha_1),b\sin(\alpha_1)]\)，且\(s_i=\tan(\frac{\alpha_i}{2}), i=1,2,3...,k\)   \(A_1 A_2=L_1,A_2 A_3=L_2,A_3 A_4=L_3...,A_k A_1=L_k\),

则有\(s_i,L_i （i=1,2,3...,k)\) 均满足下列方程：

\(4(s_i-s_{i+1})^2(b^2s_i^2s_{i+1}^2+a^2s_i^2+2a^2s_is_{i+1}+a^2s_{i+1}^2-2b^2s_is_{i+1}+b^2)-(s_{i+1}^2+1)^2(s_i^2+1)^2L_i^2=0\)

即

\(L_i=\frac{2(\sqrt{b^2s_i^2s_{i+1}^2+a^2s_i^2+2a^2s_is_{i+1}+a^2s_{i+1}^2-2b^2s_is_{i+1}+b^2})(s_{i+1}-s_i)}{(s_{i+1}^2+1)(s_i^2+1)}\)


且\(s_1,s_2,s_3...,s_k\)的关系式如下：


\(s_{i+1}=\frac{(a^2b^2s_i+a^2ts_i-b^2ts_i+ab\sqrt{t(b^2s_i^4-s_i^4t+4a^2s_i^2-2b^2s_i^2-2s_i^2t+b^2-t)}}{-b^2s_i^2t+a^2b^2-a^2t}\)


且\(i \in[1,k-1]\)


\(s_2=\frac{a^2b^2s_1+a^2ts_1-b^2ts_1+ab\sqrt{t(b^2s_1^4-s_1^4t+4a^2s_1^2-2b^2s_1^2-2s_1^2t+b^2-t)}}{-b^2s_1^2t+a^2b^2-a^2t}\)


\(s_k=-\frac{-(a^2b^2s_1+a^2ts_1-b^2ts_1)+ab\sqrt{t(b^2s_1^4-s_1^4t+4a^2s_1^2-2b^2s_1^2-2s_1^2t+b^2-t)}}{-b^2s_1^2t+a^2b^2-a^2t}\)


特别地，当\(k=2p\)为偶数时，有\(s_i=-\frac{1}{s_{p+i}},L_i=L_{p+i},i \in[1,p]\)

注：\(t \)满足55#关系式