双椭圆外切内接N边形问题

数学星空

在http://bbs.emath.ac.cn/thread-5476-1-1.html 中，
mathe 利用二次曲线理论中的射影变换给出了精彩的分析及相关公式，我们现在进一步来讨论并解决对于
双椭圆的外切内接N边形的计算公式。

数学星空

为了便于讨论双椭圆的相关公式，我们先将N边形内接圆并外切于椭圆的相关讨论结果转载如下：

mathe 给出了：

$J=[(1,0,0),(0,1,0),(0,0,-r^2)]$

$K=[(1/{a^2},0,-{x_0}/{a^2}),(0,1/{b^2},-{y_0}/{b^2}),(-{x_0}/{a^2},-{y_0}/{b^2},{x_0^2}/{a^2}+{y_0^2}/{b^2}-1)]$

我们可以求出矩阵$M=J^-1K$有三个特征方程：（三个根分别记为\(r_1,r_2,r_3\)）

$b^2a^2r^2 x^3 +(−a^2b^2 +a^2y_0^2 −a^2r^2+b^2x_0^2 −b^2r^2 )x^2 +(a^2 −x_0^2+b^2 −y_0^2+r^2)x−1$

于是\(\frac{1}{r_1},\frac{1}{r_2},\frac{1}{r_3}\)满足方程

$x^3-(a^2 −x_0^2+b^2 −y_0^2+r^2)x^2+(a^2b^2 -a^2y_0^2 +a^2r^2 -b^2x_0^2 +b^2r^2)x-a^2b^2r^2=0$

其中$a=\sqrt({r_3}/{r_1}),b=\sqrt({r_3}/{r_2})$

在47#mathe 进一步给出结论：

其中将外曲线固定为\(xy=1\),那么如果内曲线方程变化为$(x-a)(x-b)=(1-a)(1-b)$，

那么经计算以后，对应的特征方程为$(x-1)(x-1+a)(x-1+b)=0$

也就是说，那里可以得出，如果固定外曲线，里面有三条内曲线，对应特征值分别为$(1,1-a,1-b),(1,1-at,1-bt),(1,1-as,1-bs)$

如果s,t满足条件

$[s^2,s,1][(a^2b^2,-2ab,1),(-2ab,-2ab+4a+4b-2,-2),(1,-2,1)][(t^2),(t),(1)]=0$

那么它们之间有如链接中复合变换关系，利用这个关系是应该可以推算出k边形的情况

并在49#指出：

设$t_1=1,t_2={4(a-1)(b-1)}/{(ab-1)^2}$而且$t_{n+1}={(t_n-1)^2}/{(abt_n-1)^2t_{n-1}}$然后利用$t_n=0$就可以得出n变形情况特征值的约束方程

数学星空

根据楼上相关的结论：

我们很容易算出如下结果：

\(n=2\)时  

\(4(a-1)(b-1)=0\)  


\(n=3\)时
\((a^2b^2-6ab+4a+4b-3)^2=0\)

\(n=4\)时  
\(16(b-1)(a-1)(ab-2a+1)^2(ab-2b+1)^2(ab-1)^2＝0\)

\(n=5\)时
\((a^6b^6-50a^5b^5+140a^5b^4+140a^4b^5-160a^5b^3-445a^4b^4-160a^3b^5+64a^5b^2+560a^4b^3+560a^3b^4+64a^2b^5-240a^4b^2-780a^3b^3-240a^2b^4+360a^3b^2+360a^2b^3-105a^2b^2-80a^2b-80ab^2+16a^2+94ab+16b^2-20a-20b+5)^2=0\)

\(n=6\)时   
\(4(b-1)(a-1)(a^2b^2-6ab+4a+4b-3)^2(3a^2b^2-4a^2b-4ab^2+6ab-1)^2(a^2b^2-4ab^2+6ab-4a+1)^2(a^2b^2-4a^2b+6ab-4b+1)^2=0\)

\(n=7\)时
\((a^{12}b^{12}-196a^{11}b^{11}+1176a^{11}b^{10}+1176a^{10}b^{11}-3360a^{11}b^9-8022a^{10}b^{10}-3360a^9b^{11}+4928a^{11}b^8+25200a^{10}b^9+25200a^9b^{10}+4928a^8b^{11}-3584a^{11}b^7-39536a^{10}b^8-86660a^9b^9-39536a^8b^{10}-3584a^7b^{11}+1024a^{11}b^6+30208a^{10}b^7+145240a^9b^8+145240a^8b^9+30208a^7b^{10}+1024a^6b^{11}-8960a^{10}b^6-116480a^9b^7-230713a^8b^8-116480a^7b^9-8960a^6b^{10}+35840a^9b^6+108864a^8b^7+108864a^7b^8+35840a^6b^9+111552a^8b^6+265272a^7b^7+111552a^6b^8-164864a^8b^5-739536a^7b^6-739536a^6b^7-164864a^5b^8+89600a^8b^4+765632a^7b^5+1463980a^6b^6+765632a^5b^7+89600a^4b^8-28672a^8b^3-434560a^7b^4-1490272a^6b^5-1490272a^5b^6-434560a^4b^7-28672a^3b^8+4096a^8b^2+144896a^7b^3+890720a^6b^4+1582104a^5b^5+890720a^4b^6+144896a^3b^7+4096a^2b^8-21504a^7b^2-311808a^6b^3-1005200a^5b^4-1005200a^4b^5-311808a^3b^6-21504a^2b^7+48384a^6b^2+372736a^5b^3+683935a^4b^4+372736a^3b^5+48384a^2b^6-60928a^5b^2-271040a^4b^3-271040a^3b^4-60928a^2b^5+47040a^4b^2+113932a^3b^3+47040a^2b^4-19656a^3b^2-19656a^2b^3-672a^3b+1610a^2b^2-672ab^3+64a^3+1136a^2b+1136ab^2+64b^3-112a^2-532ab-112b^2+56a+56b-7)^2=0\)
  
\(n=8\)时
\(64(a-1)(b-1)(ab-2b+1)^2(ab-1)^2(ab-2a+1)^2(a^4b^4-8a^4b^3+8a^4b^2+20a^3b^3-24a^3b^2-32a^2b^3+54a^2b^2+16ab^3-24a^2b-32ab^2+8a^2+20ab-8a+1)^2(a^4b^4-20a^3b^3+32a^3b^2+32a^2b^3-16a^3b-58a^2b^2-16ab^3+32a^2b+32ab^2-20ab+1)^2(a^4b^4-8a^3b^4+20a^3b^3+8a^2b^4-32a^3b^2-24a^2b^3+16a^3b+54a^2b^2-32a^2b-24ab^2+20ab+8b^2-8b+1)^2=0\)

\(n=9\) 时

1. (a^2b^2-6ab+4a+4b-3)^2(a^{18}b^{18}-534a^{17}b^{17}+5540a^{17}b^{16}+5540a^{16}b^{17}-28512a^{17}b^{15}-67947a^{16}b^{16}-28512a^{15}b^{17}+82368a^{17}b^{14}+401280a^{16}b^{15}+401280a^{15}b^{16}+82368a^{14}b^{17}-139776a^{17}b^{13}-1323296a^{16}b^{14}-2709904a^{15}b^{15}-1323296a^{14}b^{16}-139776a^{13}b^{17}+138240a^{17}b^{12}+2621952a^{16}b^{13}+10236000a^{15}b^{14}+10236000a^{14}b^{15}+2621952a^{13}b^{16}+138240a^{12}b^{17}-73728a^{17}b^{11}-3234816a^{16}b^{12}-23825376a^{15}b^{13}-43633260a^{14}b^{14}-23825376a^{13}b^{15}-3234816a^{12}b^{16}-73728a^{11}b^{17}+16384a^{17}b^{10}+2523136a^{16}b^{11}+36575168a^{15}b^{12}+114679040a^{14}b^{13}+114679040a^{13}b^{14}+36575168a^{12}b^{15}+2523136a^{11}b^{16}+16384a^{10}b^{17}-1253376a^{16}b^{10}-39191040a^{15}b^{11}-201315744a^{14}b^{12}-334576872a^{13}b^{13}-201315744a^{12}b^{14}-39191040a^{11}b^{15}-1253376a^{10}b^{16}+393216a^{16}b^9+30723072a^{15}b^{10}+250295808a^{14}b^{11}+649247088a^{13}b^{12}+649247088a^{12}b^{13}+250295808a^{11}b^{14}+30723072a^{10}b^{15}+393216a^9b^{16}-65536a^{16}b^8-17899520a^{15}b^9-228034560a^{14}b^{10}-890780128a^{13}b^{11}-1374888172a^{12}b^{12}-890780128a^{11}b^{13}-228034560a^{10}b^{14}-17899520a^9b^{15}-65536a^8b^{16}+7421952a^{15}b^8+151633920a^{14}b^9+889296576a^{13}b^{10}+2045645952a^{12}b^{11}+2045645952a^{11}b^{12}+889296576a^{10}b^{13}+151633920a^9b^{14}+7421952a^8b^{15}-1966080a^{15}b^7-69918720a^{14}b^8-639452160a^{13}b^9-2198557344a^{12}b^{10}-3276437232a^{11}b^{11}-2198557344a^{10}b^{12}-639452160a^9b^{13}-69918720a^8b^{14}-1966080a^7b^{15}+262144a^{15}b^6+20054016a^{14}b^7+314292224a^{13}b^8+1685263360a^{12}b^9+3766379936a^{11}b^{10}+3766379936a^{10}b^{11}+1685263360a^9b^{12}+314292224a^8b^{13}+20054016a^7b^{14}+262144a^6b^{15}-2752512a^{14}b^6-94789632a^{13}b^7-874137600a^{12}b^8-3063034464a^{11}b^9-4601643954a^{10}b^{10}-3063034464a^9b^{11}-874137600a^8b^{12}-94789632a^7b^{13}-2752512a^6b^{14}+13418496a^{13}b^6+275595264a^{12}b^7+1671042240a^{11}b^8+3935370240a^{10}b^9+3935370240a^9b^{10}+1671042240a^8b^{11}+275595264a^7b^{12}+13418496a^6b^{13}-40312832a^{12}b^6-549262336a^{11}b^7-2217363232a^{10}b^8-3461258500a^9b^9-2217363232a^8b^{10}-549262336a^7b^{11}-40312832a^6b^{12}+82704384a^{11}b^6+720792576a^{10}b^7+1898573400a^9b^8+1898573400a^8b^9+720792576a^7b^{10}+82704384a^6b^{11}+172032a^{11}b^5-82747392a^{10}b^6-481365024a^9b^7-816791082a^8b^8-481365024a^7b^9-82747392a^6b^{10}+172032a^5b^{11}-16384a^{11}b^4-16760832a^{10}b^5-76418752a^9b^6-125153152a^8b^7-125153152a^7b^8-76418752a^6b^9-16760832a^5b^{10}-16384a^4b^{11}+4546560a^{10}b^4+89433600a^9b^5+370474656a^8b^6+574721808a^7b^7+370474656a^6b^8+89433600a^5b^9+4546560a^4b^{10}-786432a^{10}b^3-25482240a^9b^4-210040320a^8b^5-546899808a^7b^6-546899808a^6b^7-210040320a^5b^8-25482240a^4b^9-786432a^3b^{10}+65536a^{10}b^2+4587520a^9b^3+62325760a^8b^4+279571040a^7b^5+451992548a^6b^6+279571040a^5b^7+62325760a^4b^8+4587520a^3b^9+65536a^2b^{10}-393216a^9b^2-11649024a^8b^3-86670528a^7b^4-231141120a^6b^5-231141120a^5b^6-86670528a^4b^7-11649024a^3b^8-393216a^2b^9+1032192a^8b^2+16897536a^7b^3+75396384a^6b^4+123071256a^5b^5+75396384a^4b^6+16897536a^3b^7+1032192a^2b^8-1557504a^7b^2-15450624a^6b^3-42642320a^5b^4-42642320a^4b^5-15450624a^3b^6-1557504a^2b^7+1492992a^6b^2+9276768a^5b^3+15840900a^4b^4+9276768a^3b^5+1492992a^2b^6-948672a^5b^2-3699840a^4b^3-3699840a^3b^4-948672a^2b^5+404768a^4b^2+930928a^3b^3+404768a^2b^4-106656a^3b^2-106656a^2b^3-1056a^3b+9273a^2b^2-1056ab^3+64a^3+1536a^2b+1536ab^2+64b^3-96a^2-534ab-96b^2+36a+36b-3)^2=0

复制代码

\(n=10\)时
\(4(a^6b^6-12a^6b^5+16a^6b^4+50a^5b^5-80a^5b^4-140a^4b^5+335a^4b^4+160a^3b^5-264a^4b^3-464a^3b^4-64a^2b^5+208a^4b^2+508a^3b^3+208a^2b^4-64a^4b-464a^3b^2-264a^2b^3+160a^3b+335a^2b^2-140a^2b-80ab^2+50ab+16b^2-12b+1)^2(5a^6b^6-20a^6b^5-20a^5b^6+16a^6b^4+94a^5b^5+16a^4b^6-80a^5b^4-80a^4b^5-105a^4b^4+360a^4b^3+360a^3b^4-240a^4b^2-780a^3b^3-240a^2b^4+64a^4b+560a^3b^2+560a^2b^3+64ab^4-160a^3b-445a^2b^2-160ab^3+140a^2b+140ab^2-50ab+1)^2(a^6b^6-50a^5b^5+140a^5b^4+140a^4b^5-160a^5b^3-445a^4b^4-160a^3b^5+64a^5b^2+560a^4b^3+560a^3b^4+64a^2b^5-240a^4b^2-780a^3b^3-240a^2b^4+360a^3b^2+360a^2b^3-105a^2b^2-80a^2b-80ab^2+16a^2+94ab+16b^2-20a-20b+5)^2(a^6b^6-12a^5b^6+50a^5b^5+16a^4b^6-140a^5b^4-80a^4b^5+160a^5b^3+335a^4b^4-64a^5b^2-464a^4b^3-264a^3b^4+208a^4b^2+508a^3b^3+208a^2b^4-264a^3b^2-464a^2b^3-64ab^4+335a^2b^2+160ab^3-80a^2b-140ab^2+16a^2+50ab-12a+1)^2(a-1)(b-1)=0\)

数学星空

根据楼上结论：只需作代换\(a\to 1-\frac{r^2}{a^2}, b\to 1-\frac{r^2}{b^2}\)

我们很容易得到\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)，与\(x^2+y^2=r^2\),有一个\(n\)边形内接圆外切椭圆的条件：

\(n=3\)时

\(-r+a+b=0\)

\(n=4\)时

\(a^2+b^2-r^2=0\)

\(n=5\)时

\(a^3-a^2b-a^2r-ab^2-2abr-ar^2+b^3-b^2r-br^2+r^3=0\)

\(n=6\)时

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

\(n=7\)时

\(a^6+2a^5b-2a^5r-a^4b^2+2a^4br-a^4r^2-4a^3b^3+4a^3r^3-a^2b^4+2a^2b^2r^2-a^2r^4+2ab^5+2ab^4r-2abr^4-2ar^5+b^6-2b^5r-b^4r^2+4b^3r^3-b^2r^4-2br^5+r^6=0\)

\(n=8\)时

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

\(n=9\)时

\(a^9-3a^8b-3a^8r+8a^6b^3-4a^6b^2r-4a^6br^2+8a^6r^3-6a^5b^4+12a^5b^2r^2-6a^5r^4-6a^4b^5+14a^4b^4r-8a^4b^3r^2-8a^4b^2r^3+14a^4br^4-6a^4r^5+8a^3b^6-8a^3b^4r^2-8a^3b^2r^4+8a^3r^6-4a^2b^6r+12a^2b^5r^2-8a^2b^4r^3-8a^2b^3r^4+12a^2b^2r^5-4a^2br^6-3ab^8-4ab^6r^2+14ab^4r^4-4ab^2r^6-3ar^8+b^9-3b^8r+8b^6r^3-6b^5r^4-6b^4r^5+8b^3r^6-3br^8+r^9=0\)

\(n=10\)时

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

数学星空

对于外切于椭圆\(\frac{x^2}{a^2}+\frac{y^2}{b^2}＝1\)，内接于圆\((x-x_0)^2+(y-y_0)^2=r^2\) 的n（奇数）边形计算公式

\(n=3\)时

\((a^2+b^2+r^2-x_0^2-y_0^2)^2-4a^2b^2-4a^2r^2+4a^2y_0^2-4b^2r^2+4b^2x_0^2＝0\)


\(n=5\)时

\((a^2+b^2+r^2-x_0^2-y_0^2)^6-12(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)-32(a^2+b^2+r^2-x_0^2-y_0^2)^3a^2b^2r^2+48(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2+(128(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^2b^2r^2-64(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3-256a^4b^4r^4＝0\)


\(n=7\)时

\((a^2+b^2+r^2-x_0^2-y_0^2)^{12}-24(a^2+b^2+r^2-x_0^2-y_0^2)^{10}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)-160(a^2+b^2+r^2-x_0^2-y_0^2)^9a^2b^2r^2+240(a^2+b^2+r^2-x_0^2-y_0^2)^8(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2+2048(a^2+b^2+r^2-x_0^2-y_0^2)^7(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^2b^2r^2-1280(a^2+b^2+r^2-x_0^2-y_0^2)^6(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3-3328(a^2+b^2+r^2-x_0^2-y_0^2)^6a^4b^4r^4-9216(a^2+b^2+r^2-x_0^2-y_0^2)^5(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^2b^2r^2+3840(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4+27648(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^4b^4r^4+16384(a^2+b^2+r^2-x_0^2-y_0^2)^3(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^2b^2r^2-6144(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^5-24576(a^2+b^2+r^2-x_0^2-y_0^2)^3a^6b^6r^6-61440(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^4b^4r^4-(8192(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4a^2b^2r^2+4096(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^6+(98304(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^6b^6r^6+16384(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^4b^4r^4-65536a^8b^8r^8＝0\)


\(n=9\)时

1. -50331648a^{12}b^{12}r^{12}-480(a^2+b^2+r^2-x_0^2-y_0^2)^{15}a^2b^2r^2-19200(a^2+b^2+r^2-x_0^2-y_0^2)^{12}a^4b^4r^4-417792(a^2+b^2+r^2-x_0^2-y_0^2)^9a^6b^6r^6-4718592(a^2+b^2+r^2-x_0^2-y_0^2)^6a^8b^8r^8-25165824(a^2+b^2+r^2-x_0^2-y_0^2)^3a^{10}b^{10}r^{10}-3145728(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^6a^4b^4r^4-16777216(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^8b^8r^8+(1572864(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^7a^2b^2r^2+3342336(a^2+b^2+r^2-x_0^2-y_0^2)^6(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^4b^4r^4+1802240(a^2+b^2+r^2-x_0^2-y_0^2)^5(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^5a^2b^2r^2-16515072(a^2+b^2+r^2-x_0^2-y_0^2)^5(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^6b^6r^6-1376256(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4a^4b^4r^4-2228224(a^2+b^2+r^2-x_0^2-y_0^2)^3(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^6a^2b^2r^2+36700160(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^8b^8r^8+12582912(a^2+b^2+r^2-x_0^2-y_0^2)^3(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^6b^6r^6-1572864(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^5a^4b^4r^4-67108864(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^8b^8r^8+9856(a^2+b^2+r^2-x_0^2-y_0^2)^{13}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^2b^2r^2-83456(a^2+b^2+r^2-x_0^2-y_0^2)^{11}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^2b^2r^2+288768(a^2+b^2+r^2-x_0^2-y_0^2)^{10}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^4b^4r^4+378880(a^2+b^2+r^2-x_0^2-y_0^2)^9(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^2b^2r^2-1560576(a^2+b^2+r^2-x_0^2-y_0^2)^8(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^4b^4r^4-1024000(a^2+b^2+r^2-x_0^2-y_0^2)^7(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4a^2b^2r^2+4718592(a^2+b^2+r^2-x_0^2-y_0^2)^7(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^6b^6r^6+(a^2+b^2+r^2-x_0^2-y_0^2)^18-262144(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^9-36(a^2+b^2+r^2-x_0^2-y_0^2)^{16}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)+576(a^2+b^2+r^2-x_0^2-y_0^2)^{14}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2-5376(a^2+b^2+r^2-x_0^2-y_0^2)^{12}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3+32256(a^2+b^2+r^2-x_0^2-y_0^2)^{10}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4-129024(a^2+b^2+r^2-x_0^2-y_0^2)^8(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^5+344064(a^2+b^2+r^2-x_0^2-y_0^2)^6(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^6-589824(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^7+589824(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^8+(100663296(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^{10}b^{10}r^{10}+(18874368(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4a^6b^6r^6＝0

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数学星空

对于外切于椭圆\(\frac{x^2}{a^2}+\frac{y^2}{b^2}＝1\)，内接于圆\((x-x_0)^2+(y-y_0)^2=r^2\) 的n（偶数）边形计算公式

\(n=4\)时

\(-(a^2+b^2+r^2-x_0^2-y_0^2)^3+(4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2))(a^2+b^2+r^2-x_0^2-y_0^2)-8a^2b^2r^2=0\)


\(n=6\)时

\(3(a^2+b^2+r^2-x_0^2-y_0^2)^6-20(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)+96(a^2+b^2+r^2-x_0^2-y_0^2)^3a^2b^2r^2+16(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2(a^2+b^2+r^2-x_0^2-y_0^2)^2-(384(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^2b^2r^2+64(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3+512a^4b^4r^4＝0\)


\(n=8\)时

\(-(a^2+b^2+r^2-x_0^2-y_0^2)^{12}+16(a^2+b^2+r^2-x_0^2-y_0^2)^{10}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)-80(a^2+b^2+r^2-x_0^2-y_0^2)^9a^2b^2r^2-80(a^2+b^2+r^2-x_0^2-y_0^2)^8(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2+1280(a^2+b^2+r^2-x_0^2-y_0^2)^7(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^2b^2r^2-7680(a^2+b^2+r^2-x_0^2-y_0^2)^5(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^2b^2r^2+1280(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4+2048(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^4b^4r^4+20480(a^2+b^2+r^2-x_0^2-y_0^2)^3(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^2b^2r^2-4096(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^5+8192(a^2+b^2+r^2-x_0^2-y_0^2)^3a^6b^6r^6-16384(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^4b^4r^4-(20480(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4a^2b^2r^2+4096(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^6-(32768(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^6b^6r^6+32768(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^4b^4r^4+32768a^8b^8r^8=0\)

\(n=10\)时

1. 100(a^2+b^2+r^2-x_0^2-y_0^2)^{16}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)-576(a^2+b^2+r^2-x_0^2-y_0^2)^{14}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2-1792(a^2+b^2+r^2-x_0^2-y_0^2)^{12}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3+39424(a^2+b^2+r^2-x_0^2-y_0^2)^{10}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4-215040(a^2+b^2+r^2-x_0^2-y_0^2)^8(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^5+573440(a^2+b^2+r^2-x_0^2-y_0^2)^6(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^6-720896(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^7+196608(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^8+24576(a^2+b^2+r^2-x_0^2-y_0^2)^{13}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^2b^2r^2-210944(a^2+b^2+r^2-x_0^2-y_0^2)^{11}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^2b^2r^2+624640(a^2+b^2+r^2-x_0^2-y_0^2)^{10}(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^4b^4r^4+901120(a^2+b^2+r^2-x_0^2-y_0^2)^9(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^2b^2r^2-4280320(a^2+b^2+r^2-x_0^2-y_0^2)^8(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^4b^4r^4-1802240(a^2+b^2+r^2-x_0^2-y_0^2)^7(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4a^2b^2r^2+5898240(a^2+b^2+r^2-x_0^2-y_0^2)^7(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^6b^6r^6+14090240(a^2+b^2+r^2-x_0^2-y_0^2)^6(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^4b^4r^4+524288(a^2+b^2+r^2-x_0^2-y_0^2)^5(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^5a^2b^2r^2-27525120(a^2+b^2+r^2-x_0^2-y_0^2)^5(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^6b^6r^6-21299200(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4a^4b^4r^4+3670016(a^2+b^2+r^2-x_0^2-y_0^2)^3(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^6a^2b^2r^2+24903680(a^2+b^2+r^2-x_0^2-y_0^2)^4(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^8b^8r^8+52428800(a^2+b^2+r^2-x_0^2-y_0^2)^3(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^6b^6r^6+8912896(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^5a^4b^4r^4-57671680(a^2+b^2+r^2-x_0^2-y_0^2)^2(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^2a^8b^8r^8-5(a^2+b^2+r^2-x_0^2-y_0^2)^{18}+262144(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^9-1152(a^2+b^2+r^2-x_0^2-y_0^2)^{15}a^2b^2r^2-35584(a^2+b^2+r^2-x_0^2-y_0^2)^{12}a^4b^4r^4-450560(a^2+b^2+r^2-x_0^2-y_0^2)^9a^6b^6r^6-2949120(a^2+b^2+r^2-x_0^2-y_0^2)^6a^8b^8r^8-10485760(a^2+b^2+r^2-x_0^2-y_0^2)^3a^{10}b^{10}r^{10}+5242880(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^6a^4b^4r^4+20971520(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^3a^8b^8r^8-16777216a^{12}b^{12}r^{12}-(4194304(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^7a^2b^2r^2-(31457280(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)^4a^6b^6r^6+(41943040(a^2+b^2+r^2-x_0^2-y_0^2))(a^2b^2+a^2r^2-a^2y_0^2+b^2r^2-b^2x_0^2)a^{10}b^{10}r^{10}＝0

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zuijianqiugen

数学星空 发表于 2014-4-27 22:39
对于外切于椭圆\(\frac{x^2}{a^2}+\frac{y^2}{b^2}\)，内接于圆\((x-x_0)^2+(y-y_0)^2=r^2\) 的n（偶数）边 ...

数学星空，你太有才了！一下子搞出这么多有实用价值的一个n边形同时存在外接圆和内切椭圆的条件，特表支持。当n较大时，其条件等式太复杂了，也难记住。能否象约束方程那样？研究出一个递进通式，便于记忆。

数学星空

对于外切椭圆\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\),和内接于椭圆\(\frac{(x-x_0)^2}{m^2}+\frac{(y-y_0)^2}{n^2}=1\)的n边形

$J=[(\frac{1}{m^2},0,0),(0,\frac{1}{n^2},0),(0,0,-1)]$

$K=[(1/{a^2},0,-{x_0}/{a^2}),(0,1/{b^2},-{y_0}/{b^2}),(-{x_0}/{a^2},-{y_0}/{b^2},{x_0^2}/{a^2}+{y_0^2}/{b^2}-1)]$

我们可以求出矩阵$M=J^-1K$有三个特征方程：（三个根分别记为\(r_1,r_2,r_3\)）

$x^3a^2b^2+(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)x^2+(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)x-n^2m^2=0$

于是\(\frac{1}{r_1},\frac{1}{r_2},\frac{1}{r_3}\)满足方程

$-n^2m^2x^3+(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)x^2+(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)x+a^2b^2=0$

其中\(\frac{a}{m}=\sqrt{\frac{r_3}{r_1}},\frac{b}{n}=\sqrt{\frac{r_3}{r_2}}\)  ？？？  (需要mathe确定一下此关系式是否正确）

数学星空

对于双圆的结果：外切于圆\(x^2+y^2=r^2\)，内接于圆 \((x-x_0)^2+(y-y_0)^2=R^2\)的n边形

\(n=3\)时

\(R^2-2Rr-x_0^2-y_0^2=0\)

\(n=4\)时

\(R^4-2R^2r^2-2R^2x_0^2-2R^2y_0^2-2r^2x_0^2-2r^2y_0^2+x_0^4+2x_0^2y_0^2+y_0^4=0\)

\(n=5\)时

\(R^6+2R^5r-4R^4r^2-3R^4x_0^2-3R^4y_0^2-4R^3rx_0^2-4R^3ry_0^2+4R^2r^2x_0^2+4R^2r^2y_0^2+3R^2x_0^4+6R^2x_0^2y_0^2+3R^2y_0^4-8Rr^3x_0^2-8Rr^3y_0^2+2Rrx_0^4+4Rrx_0^2y_0^2+2Rry_0^4-x_0^6-3x_0^4y_0^2-3x_0^2y_0^4-y_0^6=0\)

\(n=6\)时

\(3R^8-4R^6r^2-12R^6x_0^2-12R^6y_0^2+4R^4r^2x_0^2+4R^4r^2y_0^2+18R^4x_0^4+36R^4x_0^2y_0^2+18R^4y_0^4-16R^2r^4x_0^2-16R^2r^4y_0^2+4R^2r^2x_0^4+8R^2r^2x_0^2y_0^2+4R^2r^2y_0^4-12R^2x_0^6-36R^2x_0^4y_0^2-36R^2x_0^2y_0^4-12R^2y_0^6-4r^2x_0^6-12r^2x_0^4y_0^2-12r^2x_0^2y_0^4-4r^2y_0^6+3x_0^8+12x_0^6y_0^2+18x_0^4y_0^4+12x_0^2y_0^6+3y_0^8=0\)

\(n=7\)时

\(R^{12}-4R^{11}r-4R^{10}r^2-6R^{10}x_0^2-6R^{10}y_0^2+8R^9r^3+20R^9rx_0^2+20R^9ry_0^2+16R^8r^2x_0^2+16R^8r^2y_0^2+15R^8x_0^4+30R^8x_0^2y_0^2+15R^8y_0^4-40R^7rx_0^4-80R^7rx_0^2y_0^2-40R^7ry_0^4-16R^6r^4x_0^2-16R^6r^4y_0^2-24R^6r^2x_0^4-48R^6r^2x_0^2y_0^2-24R^6r^2y_0^4-20R^6x_0^6-60R^6x_0^4y_0^2-60R^6x_0^2y_0^4-20R^6y_0^6-32R^5r^5x_0^2-32R^5r^5y_0^2-48R^5r^3x_0^4-96R^5r^3x_0^2y_0^2-48R^5r^3y_0^4+40R^5rx_0^6+120R^5rx_0^4y_0^2+120R^5rx_0^2y_0^4+40R^5ry_0^6+64R^4r^6x_0^2+64R^4r^6y_0^2+32R^4r^4x_0^4+64R^4r^4x_0^2y_0^2+32R^4r^4y_0^4+16R^4r^2x_0^6+48R^4r^2x_0^4y_0^2+48R^4r^2x_0^2y_0^4+16R^4r^2y_0^6+15R^4x_0^8+60R^4x_0^6y_0^2+90R^4x_0^4y_0^4+60R^4x_0^2y_0^6+15R^4y_0^8+64R^3r^3x_0^6+192R^3r^3x_0^4y_0^2+192R^3r^3x_0^2y_0^4+64R^3r^3y_0^6-20R^3rx_0^8-80R^3rx_0^6y_0^2-120R^3rx_0^4y_0^4-80R^3rx_0^2y_0^6-20R^3ry_0^8-16R^2r^4x_0^6-48R^2r^4x_0^4y_0^2-48R^2r^4x_0^2y_0^4-16R^2r^4y_0^6-4R^2r^2x_0^8-16R^2r^2x_0^6y_0^2-24R^2r^2x_0^4y_0^4-16R^2r^2x_0^2y_0^6-4R^2r^2y_0^8-6R^2x_0^{10}-30R^2x_0^8y_0^2-60R^2x_0^6y_0^4-60R^2x_0^4y_0^6-30R^2x_0^2y_0^8-6R^2y_0^{10}+32Rr^5x_0^6+96Rr^5x_0^4y_0^2+96Rr^5x_0^2y_0^4+32Rr^5y_0^6-24Rr^3x_0^8-96Rr^3x_0^6y_0^2-144Rr^3x_0^4y_0^4-96Rr^3x_0^2y_0^6-24Rr^3y_0^8+4Rrx_0^{10}+20Rrx_0^8y_0^2+40Rrx_0^6y_0^4+40Rrx_0^4y_0^6+20Rrx_0^2y_0^8+4Rry_0^{10}+x_0^{12}+6x_0^{10}y_0^2+15x_0^8y_0^4+20x_0^6y_0^6+15x_0^4y_0^8+6x_0^2y_0^{10}+y_0^{12}=0\)


\(n=8\)时

\(R^{16}-8R^{14}r^2-8R^{14}x_0^2-8R^{14}y_0^2+8R^{12}r^4+40R^{12}r^2x_0^2+40R^{12}r^2y_0^2+28R^{12}x_0^4+56R^{12}x_0^2y_0^2+28R^{12}y_0^4+48R^{10}r^4x_0^2+48R^{10}r^4y_0^2-72R^{10}r^2x_0^4-144R^{10}r^2x_0^2y_0^2-72R^{10}r^2y_0^4-56R^{10}x_0^6-168R^{10}x_0^4y_0^2-168R^{10}x_0^2y_0^4-56R^{10}y_0^6-128R^8r^6x_0^2-128R^8r^6y_0^2-264R^8r^4x_0^4-528R^8r^4x_0^2y_0^2-264R^8r^4y_0^4+40R^8r^2x_0^6+120R^8r^2x_0^4y_0^2+120R^8r^2x_0^2y_0^4+40R^8r^2y_0^6+70R^8x_0^8+280R^8x_0^6y_0^2+420R^8x_0^4y_0^4+280R^8x_0^2y_0^6+70R^8y_0^8+128R^6r^8x_0^2+128R^6r^8y_0^2+128R^6r^6x_0^4+256R^6r^6x_0^2y_0^2+128R^6r^6y_0^4+416R^6r^4x_0^6+1248R^6r^4x_0^4y_0^2+1248R^6r^4x_0^2y_0^4+416R^6r^4y_0^6+40R^6r^2x_0^8+160R^6r^2x_0^6y_0^2+240R^6r^2x_0^4y_0^4+160R^6r^2x_0^2y_0^6+40R^6r^2y_0^8-56R^6x_0^{10}-280R^6x_0^8y_0^2-560R^6x_0^6y_0^4-560R^6x_0^4y_0^6-280R^6x_0^2y_0^8-56R^6y_0^{10}+128R^4r^6x_0^6+384R^4r^6x_0^4y_0^2+384R^4r^6x_0^2y_0^4+128R^4r^6y_0^6-264R^4r^4x_0^8-1056R^4r^4x_0^6y_0^2-1584R^4r^4x_0^4y_0^4-1056R^4r^4x_0^2y_0^6-264R^4r^4y_0^8-72R^4r^2x_0^{10}-360R^4r^2x_0^8y_0^2-720R^4r^2x_0^6y_0^4-720R^4r^2x_0^4y_0^6-360R^4r^2x_0^2y_0^8-72R^4r^2y_0^{10}+28R^4x_0^{12}+168R^4x_0^{10}y_0^2+420R^4x_0^8y_0^4+560R^4x_0^6y_0^6+420R^4x_0^4y_0^8+168R^4x_0^2y_0^{10}+28R^4y_0^{12}+128R^2r^8x_0^6+384R^2r^8x_0^4y_0^2+384R^2r^8x_0^2y_0^4+128R^2r^8y_0^6-128R^2r^6x_0^8-512R^2r^6x_0^6y_0^2-768R^2r^6x_0^4y_0^4-512R^2r^6x_0^2y_0^6-128R^2r^6y_0^8+48R^2r^4x_0^{10}+240R^2r^4x_0^8y_0^2+480R^2r^4x_0^6y_0^4+480R^2r^4x_0^4y_0^6+240R^2r^4x_0^2y_0^8+48R^2r^4y_0^{10}+40R^2r^2x_0^{12}+240R^2r^2x_0^{10}y_0^2+600R^2r^2x_0^8y_0^4+800R^2r^2x_0^6y_0^6+600R^2r^2x_0^4y_0^8+240R^2r^2x_0^2y_0^{10}+40R^2r^2y_0^{12}-8R^2x_0^{14}-56R^2x_0^{12}y_0^2-168R^2x_0^{10}y_0^4-280R^2x_0^8y_0^6-280R^2x_0^6y_0^8-168R^2x_0^4y_0^{10}-56R^2x_0^2y_0^{12}-8R^2y_0^{14}+8r^4x_0^{12}+48r^4x_0^{10}y_0^2+120r^4x_0^8y_0^4+160r^4x_0^6y_0^6+120r^4x_0^4y_0^8+48r^4x_0^2y_0^{10}+8r^4y_0^{12}-8r^2x_0^{14}-56r^2x_0^{12}y_0^2-168r^2x_0^{10}y_0^4-280r^2x_0^8y_0^6-280r^2x_0^6y_0^8-168r^2x_0^4y_0^{10}-56r^2x_0^2y_0^{12}-8r^2y_0^{14}+x_0^{16}+8x_0^{14}y_0^2+28x_0^{12}y_0^4+56x_0^{10}y_0^6+70x_0^8y_0^8+56x_0^6y_0^{10}+28x_0^4y_0^{12}+8x_0^2y_0^{14}+y_0^{16}=0\)


\(n=9\)时

1. R^{18}+6R^{17}r-9R^{16}x_0^2-9R^{16}y_0^2-8R^{15}r^3-48R^{15}rx_0^2-48R^{15}ry_0^2+36R^{14}x_0^4+72R^{14}x_0^2y_0^2+36R^{14}y_0^4-8R^{13}r^3x_0^2-8R^{13}r^3y_0^2+168R^{13}rx_0^4+336R^{13}rx_0^2y_0^2+168R^{13}ry_0^4-96R^{12}r^4x_0^2-96R^{12}r^4y_0^2-84R^{12}x_0^6-252R^{12}x_0^4y_0^2-252R^{12}x_0^2y_0^4-84R^{12}y_0^6+32R^{11}r^5x_0^2+32R^{11}r^5y_0^2+216R^{11}r^3x_0^4+432R^{11}r^3x_0^2y_0^2+216R^{11}r^3y_0^4-336R^{11}rx_0^6-1008R^{11}rx_0^4y_0^2-1008R^{11}rx_0^2y_0^4-336R^{11}ry_0^6+256R^{10}r^6x_0^2+256R^{10}r^6y_0^2+480R^{10}r^4x_0^4+960R^{10}r^4x_0^2y_0^2+480R^{10}r^4y_0^4+126R^{10}x_0^8+504R^{10}x_0^6y_0^2+756R^{10}x_0^4y_0^4+504R^{10}x_0^2y_0^6+126R^{10}y_0^8+32R^9r^5x_0^4+64R^9r^5x_0^2y_0^2+32R^9r^5y_0^4-680R^9r^3x_0^6-2040R^9r^3x_0^4y_0^2-2040R^9r^3x_0^2y_0^4-680R^9r^3y_0^6+420R^9rx_0^8+1680R^9rx_0^6y_0^2+2520R^9rx_0^4y_0^4+1680R^9rx_0^2y_0^6+420R^9ry_0^8-256R^8r^8x_0^2-256R^8r^8y_0^2-512R^8r^6x_0^4-1024R^8r^6x_0^2y_0^2-512R^8r^6y_0^4-960R^8r^4x_0^6-2880R^8r^4x_0^4y_0^2-2880R^8r^4x_0^2y_0^4-960R^8r^4y_0^6-126R^8x_0^{10}-630R^8x_0^8y_0^2-1260R^8x_0^6y_0^4-1260R^8x_0^4y_0^6-630R^8x_0^2y_0^8-126R^8y_0^{10}+128R^7r^7x_0^4+256R^7r^7x_0^2y_0^2+128R^7r^7y_0^4-448R^7r^5x_0^6-1344R^7r^5x_0^4y_0^2-1344R^7r^5x_0^2y_0^4-448R^7r^5y_0^6+1000R^7r^3x_0^8+4000R^7r^3x_0^6y_0^2+6000R^7r^3x_0^4y_0^4+4000R^7r^3x_0^2y_0^6+1000R^7r^3y_0^8-336R^7rx_0^{10}-1680R^7rx_0^8y_0^2-3360R^7rx_0^6y_0^4-3360R^7rx_0^4y_0^6-1680R^7rx_0^2y_0^8-336R^7ry_0^{10}+960R^6r^4x_0^8+3840R^6r^4x_0^6y_0^2+5760R^6r^4x_0^4y_0^4+3840R^6r^4x_0^2y_0^6+960R^6r^4y_0^8+84R^6x_0^{12}+504R^6x_0^{10}y_0^2+1260R^6x_0^8y_0^4+1680R^6x_0^6y_0^6+1260R^6x_0^4y_0^8+504R^6x_0^2y_0^{10}+84R^6y_0^{12}-384R^5r^7x_0^6-1152R^5r^7x_0^4y_0^2-1152R^5r^7x_0^2y_0^4-384R^5r^7y_0^6+832R^5r^5x_0^8+3328R^5r^5x_0^6y_0^2+4992R^5r^5x_0^4y_0^4+3328R^5r^5x_0^2y_0^6+832R^5r^5y_0^8-792R^5r^3x_0^{10}-3960R^5r^3x_0^8y_0^2-7920R^5r^3x_0^6y_0^4-7920R^5r^3x_0^4y_0^6-3960R^5r^3x_0^2y_0^8-792R^5r^3y_0^{10}+168R^5rx_0^{12}+1008R^5rx_0^{10}y_0^2+2520R^5rx_0^8y_0^4+3360R^5rx_0^6y_0^6+2520R^5rx_0^4y_0^8+1008R^5rx_0^2y_0^{10}+168R^5ry_0^{12}+512R^4r^6x_0^8+2048R^4r^6x_0^6y_0^2+3072R^4r^6x_0^4y_0^4+2048R^4r^6x_0^2y_0^6+512R^4r^6y_0^8-480R^4r^4x_0^{10}-2400R^4r^4x_0^8y_0^2-4800R^4r^4x_0^6y_0^4-4800R^4r^4x_0^4y_0^6-2400R^4r^4x_0^2y_0^8-480R^4r^4y_0^{10}-36R^4x_0^{14}-252R^4x_0^{12}y_0^2-756R^4x_0^{10}y_0^4-1260R^4x_0^8y_0^6-1260R^4x_0^6y_0^8-756R^4x_0^4y_0^{10}-252R^4x_0^2y_0^{12}-36R^4y_0^{14}-512R^3r^9x_0^6-1536R^3r^9x_0^4y_0^2-1536R^3r^9x_0^2y_0^4-512R^3r^9y_0^6+384R^3r^7x_0^8+1536R^3r^7x_0^6y_0^2+2304R^3r^7x_0^4y_0^4+1536R^3r^7x_0^2y_0^6+384R^3r^7y_0^8-608R^3r^5x_0^{10}-3040R^3r^5x_0^8y_0^2-6080R^3r^5x_0^6y_0^4-6080R^3r^5x_0^4y_0^6-3040R^3r^5x_0^2y_0^8-608R^3r^5y_0^{10}+328R^3r^3x_0^{12}+1968R^3r^3x_0^{10}y_0^2+4920R^3r^3x_0^8y_0^4+6560R^3r^3x_0^6y_0^6+4920R^3r^3x_0^4y_0^8+1968R^3r^3x_0^2y_0^{10}+328R^3r^3y_0^{12}-48R^3rx_0^{14}-336R^3rx_0^{12}y_0^2-1008R^3rx_0^{10}y_0^4-1680R^3rx_0^8y_0^6-1680R^3rx_0^6y_0^8-1008R^3rx_0^4y_0^{10}-336R^3rx_0^2y_0^{12}-48R^3ry_0^{14}+256R^2r^8x_0^8+1024R^2r^8x_0^6y_0^2+1536R^2r^8x_0^4y_0^4+1024R^2r^8x_0^2y_0^6+256R^2r^8y_0^8-256R^2r^6x_0^{10}-1280R^2r^6x_0^8y_0^2-2560R^2r^6x_0^6y_0^4-2560R^2r^6x_0^4y_0^6-1280R^2r^6x_0^2y_0^8-256R^2r^6y_0^{10}+96R^2r^4x_0^{12}+576R^2r^4x_0^{10}y_0^2+1440R^2r^4x_0^8y_0^4+1920R^2r^4x_0^6y_0^6+1440R^2r^4x_0^4y_0^8+576R^2r^4x_0^2y_0^{10}+96R^2r^4y_0^{12}+9R^2x_0^{16}+72R^2x_0^{14}y_0^2+252R^2x_0^{12}y_0^4+504R^2x_0^{10}y_0^6+630R^2x_0^8y_0^8+504R^2x_0^6y_0^{10}+252R^2x_0^4y_0^{12}+72R^2x_0^2y_0^{14}+9R^2y_0^{16}-128Rr^7x_0^{10}-640Rr^7x_0^8y_0^2-1280Rr^7x_0^6y_0^4-1280Rr^7x_0^4y_0^6-640Rr^7x_0^2y_0^8-128Rr^7y_0^{10}+160Rr^5x_0^{12}+960Rr^5x_0^{10}y_0^2+2400Rr^5x_0^8y_0^4+3200Rr^5x_0^6y_0^6+2400Rr^5x_0^4y_0^8+960Rr^5x_0^2y_0^{10}+160Rr^5y_0^{12}-56Rr^3x_0^{14}-392Rr^3x_0^{12}y_0^2-1176Rr^3x_0^{10}y_0^4-1960Rr^3x_0^8y_0^6-1960Rr^3x_0^6y_0^8-1176Rr^3x_0^4y_0^{10}-392Rr^3x_0^2y_0^{12}-56Rr^3y_0^{14}+6Rrx_0^{16}+48Rrx_0^{14}y_0^2+168Rrx_0^{12}y_0^4+336Rrx_0^{10}y_0^6+420Rrx_0^8y_0^8+336Rrx_0^6y_0^{10}+168Rrx_0^4y_0^{12}+48Rrx_0^2y_0^{14}+6Rry_0^{16}-x_0^{18}-9x_0^{16}y_0^2-36x_0^{14}y_0^4-84x_0^{12}y_0^6-126x_0^{10}y_0^8-126x_0^8y_0^{10}-84x_0^6y_0^{12}-36x_0^4y_0^{14}-9x_0^2y_0^{16}-y_0^{18}=0

复制代码

\(n=10\)时

1. 5R^24-20R^22r^2-60R^22x_0^2-60R^22y_0^2+16R^{20}r^4+180R^{20}r^2x_0^2+180R^{20}r^2y_0^2+330R^{20}x_0^4+660R^{20}x_0^2y_0^2+330R^{20}y_0^4-304R^{18}r^4x_0^2-304R^{18}r^4y_0^2-700R^{18}r^2x_0^4-1400R^{18}r^2x_0^2y_0^2-700R^{18}r^2y_0^4-1100R^{18}x_0^6-3300R^{18}x_0^4y_0^2-3300R^{18}x_0^2y_0^4-1100R^{18}y_0^6+1152R^{16}r^6x_0^2+1152R^{16}r^6y_0^2+1872R^{16}r^4x_0^4+3744R^{16}r^4x_0^2y_0^2+1872R^{16}r^4y_0^4+1500R^{16}r^2x_0^6+4500R^{16}r^2x_0^4y_0^2+4500R^{16}r^2x_0^2y_0^4+1500R^{16}r^2y_0^6+2475R^{16}x_0^8+9900R^{16}x_0^6y_0^2+14850R^{16}x_0^4y_0^4+9900R^{16}x_0^2y_0^6+2475R^{16}y_0^8-1792R^{14}r^8x_0^2-1792R^{14}r^8y_0^2-5760R^{14}r^6x_0^4-11520R^{14}r^6x_0^2y_0^2-5760R^{14}r^6y_0^4-5952R^{14}r^4x_0^6-17856R^{14}r^4x_0^4y_0^2-17856R^{14}r^4x_0^2y_0^4-5952R^{14}r^4y_0^6-1800R^{14}r^2x_0^8-7200R^{14}r^2x_0^6y_0^2-10800R^{14}r^2x_0^4y_0^4-7200R^{14}r^2x_0^2y_0^6-1800R^{14}r^2y_0^8-3960R^{14}x_0^{10}-19800R^{14}x_0^8y_0^2-39600R^{14}x_0^6y_0^4-39600R^{14}x_0^4y_0^6-19800R^{14}x_0^2y_0^8-3960R^{14}y_0^{10}+1024R^{12}r^{10}x_0^2+1024R^{12}r^{10}y_0^2+3328R^{12}r^8x_0^4+6656R^{12}r^8x_0^2y_0^2+3328R^{12}r^8y_0^4+10368R^{12}r^6x_0^6+31104R^{12}r^6x_0^4y_0^2+31104R^{12}r^6x_0^2y_0^4+10368R^{12}r^6y_0^6+11424R^{12}r^4x_0^8+45696R^{12}r^4x_0^6y_0^2+68544R^{12}r^4x_0^4y_0^4+45696R^{12}r^4x_0^2y_0^6+11424R^{12}r^4y_0^8+840R^{12}r^2x_0^{10}+4200R^{12}r^2x_0^8y_0^2+8400R^{12}r^2x_0^6y_0^4+8400R^{12}r^2x_0^4y_0^6+4200R^{12}r^2x_0^2y_0^8+840R^{12}r^2y_0^{10}+4620R^{12}x_0^{12}+27720R^{12}x_0^{10}y_0^2+69300R^{12}x_0^8y_0^4+92400R^{12}x_0^6y_0^6+69300R^{12}x_0^4y_0^8+27720R^{12}x_0^2y_0^{10}+4620R^{12}y_0^{12}+2816R^{10}r^8x_0^6+8448R^{10}r^8x_0^4y_0^2+8448R^{10}r^8x_0^2y_0^4+2816R^{10}r^8y_0^6-5760R^{10}r^6x_0^8-23040R^{10}r^6x_0^6y_0^2-34560R^{10}r^6x_0^4y_0^4-23040R^{10}r^6x_0^2y_0^6-5760R^{10}r^6y_0^8-14112R^{10}r^4x_0^{10}-70560R^{10}r^4x_0^8y_0^2-141120R^{10}r^4x_0^6y_0^4-141120R^{10}r^4x_0^4y_0^6-70560R^{10}r^4x_0^2y_0^8-14112R^{10}r^4y_0^{10}+840R^{10}r^2x_0^{12}+5040R^{10}r^2x_0^{10}y_0^2+12600R^{10}r^2x_0^8y_0^4+16800R^{10}r^2x_0^6y_0^6+12600R^{10}r^2x_0^4y_0^8+5040R^{10}r^2x_0^2y_0^{10}+840R^{10}r^2y_0^{12}-3960R^{10}x_0^{14}-27720R^{10}x_0^{12}y_0^2-83160R^{10}x_0^{10}y_0^4-138600R^{10}x_0^8y_0^6-138600R^{10}x_0^6y_0^8-83160R^{10}x_0^4y_0^{10}-27720R^{10}x_0^2y_0^{12}-3960R^{10}y_0^{14}-1024R^8r^{10}x_0^6-3072R^8r^{10}x_0^4y_0^2-3072R^8r^{10}x_0^2y_0^4-1024R^8r^{10}y_0^6-8704R^8r^8x_0^8-34816R^8r^8x_0^6y_0^2-52224R^8r^8x_0^4y_0^4-34816R^8r^8x_0^2y_0^6-8704R^8r^8y_0^8-5760R^8r^6x_0^{10}-28800R^8r^6x_0^8y_0^2-57600R^8r^6x_0^6y_0^4-57600R^8r^6x_0^4y_0^6-28800R^8r^6x_0^2y_0^8-5760R^8r^6y_0^{10}+11424R^8r^4x_0^{12}+68544R^8r^4x_0^{10}y_0^2+171360R^8r^4x_0^8y_0^4+228480R^8r^4x_0^6y_0^6+171360R^8r^4x_0^4y_0^8+68544R^8r^4x_0^2y_0^{10}+11424R^8r^4y_0^{12}-1800R^8r^2x_0^{14}-12600R^8r^2x_0^{12}y_0^2-37800R^8r^2x_0^{10}y_0^4-63000R^8r^2x_0^8y_0^6-63000R^8r^2x_0^6y_0^8-37800R^8r^2x_0^4y_0^{10}-12600R^8r^2x_0^2y_0^{12}-1800R^8r^2y_0^{14}+2475R^8x_0^{16}+19800R^8x_0^{14}y_0^2+69300R^8x_0^{12}y_0^4+138600R^8x_0^{10}y_0^6+173250R^8x_0^8y_0^8+138600R^8x_0^6y_0^{10}+69300R^8x_0^4y_0^{12}+19800R^8x_0^2y_0^{14}+2475R^8y_0^{16}+4096R^6r^{12}x_0^6+12288R^6r^{12}x_0^4y_0^2+12288R^6r^{12}x_0^2y_0^4+4096R^6r^{12}y_0^6-1024R^6r^{10}x_0^8-4096R^6r^{10}x_0^6y_0^2-6144R^6r^{10}x_0^4y_0^4-4096R^6r^{10}x_0^2y_0^6-1024R^6r^{10}y_0^8+2816R^6r^8x_0^{10}+14080R^6r^8x_0^8y_0^2+28160R^6r^8x_0^6y_0^4+28160R^6r^8x_0^4y_0^6+14080R^6r^8x_0^2y_0^8+2816R^6r^8y_0^{10}+10368R^6r^6x_0^{12}+62208R^6r^6x_0^{10}y_0^2+155520R^6r^6x_0^8y_0^4+207360R^6r^6x_0^6y_0^6+155520R^6r^6x_0^4y_0^8+62208R^6r^6x_0^2y_0^{10}+10368R^6r^6y_0^{12}-5952R^6r^4x_0^{14}-41664R^6r^4x_0^{12}y_0^2-124992R^6r^4x_0^{10}y_0^4-208320R^6r^4x_0^8y_0^6-208320R^6r^4x_0^6y_0^8-124992R^6r^4x_0^4y_0^{10}-41664R^6r^4x_0^2y_0^{12}-5952R^6r^4y_0^{14}+1500R^6r^2x_0^{16}+12000R^6r^2x_0^{14}y_0^2+42000R^6r^2x_0^{12}y_0^4+84000R^6r^2x_0^{10}y_0^6+105000R^6r^2x_0^8y_0^8+84000R^6r^2x_0^6y_0^{10}+42000R^6r^2x_0^4y_0^{12}+12000R^6r^2x_0^2y_0^{14}+1500R^6r^2y_0^{16}-1100R^6x_0^{18}-9900R^6x_0^{16}y_0^2-39600R^6x_0^{14}y_0^4-92400R^6x_0^{12}y_0^6-138600R^6x_0^{10}y_0^8-138600R^6x_0^8y_0^{10}-92400R^6x_0^6y_0^{12}-39600R^6x_0^4y_0^{14}-9900R^6x_0^2y_0^{16}-1100R^6y_0^{18}+3328R^4r^8x_0^{12}+19968R^4r^8x_0^{10}y_0^2+49920R^4r^8x_0^8y_0^4+66560R^4r^8x_0^6y_0^6+49920R^4r^8x_0^4y_0^8+19968R^4r^8x_0^2y_0^{10}+3328R^4r^8y_0^{12}-5760R^4r^6x_0^{14}-40320R^4r^6x_0^{12}y_0^2-120960R^4r^6x_0^{10}y_0^4-201600R^4r^6x_0^8y_0^6-201600R^4r^6x_0^6y_0^8-120960R^4r^6x_0^4y_0^{10}-40320R^4r^6x_0^2y_0^{12}-5760R^4r^6y_0^{14}+1872R^4r^4x_0^{16}+14976R^4r^4x_0^{14}y_0^2+52416R^4r^4x_0^{12}y_0^4+104832R^4r^4x_0^{10}y_0^6+131040R^4r^4x_0^8y_0^8+104832R^4r^4x_0^6y_0^{10}+52416R^4r^4x_0^4y_0^{12}+14976R^4r^4x_0^2y_0^{14}+1872R^4r^4y_0^{16}-700R^4r^2x_0^{18}-6300R^4r^2x_0^{16}y_0^2-25200R^4r^2x_0^{14}y_0^4-58800R^4r^2x_0^{12}y_0^6-88200R^4r^2x_0^{10}y_0^8-88200R^4r^2x_0^8y_0^{10}-58800R^4r^2x_0^6y_0^{12}-25200R^4r^2x_0^4y_0^{14}-6300R^4r^2x_0^2y_0^{16}-700R^4r^2y_0^{18}+330R^4x_0^{20}+3300R^4x_0^{18}y_0^2+14850R^4x_0^{16}y_0^4+39600R^4x_0^{14}y_0^6+69300R^4x_0^{12}y_0^8+83160R^4x_0^{10}y_0^{10}+69300R^4x_0^8y_0^{12}+39600R^4x_0^6y_0^{14}+14850R^4x_0^4y_0^{16}+3300R^4x_0^2y_0^{18}+330R^4y_0^{20}+1024R^2r^{10}x_0^{12}+6144R^2r^{10}x_0^{10}y_0^2+15360R^2r^{10}x_0^8y_0^4+20480R^2r^{10}x_0^6y_0^6+15360R^2r^{10}x_0^4y_0^8+6144R^2r^{10}x_0^2y_0^{10}+1024R^2r^{10}y_0^{12}-1792R^2r^8x_0^{14}-12544R^2r^8x_0^{12}y_0^2-37632R^2r^8x_0^{10}y_0^4-62720R^2r^8x_0^8y_0^6-62720R^2r^8x_0^6y_0^8-37632R^2r^8x_0^4y_0^{10}-12544R^2r^8x_0^2y_0^{12}-1792R^2r^8y_0^{14}+1152R^2r^6x_0^{16}+9216R^2r^6x_0^{14}y_0^2+32256R^2r^6x_0^{12}y_0^4+64512R^2r^6x_0^{10}y_0^6+80640R^2r^6x_0^8y_0^8+64512R^2r^6x_0^6y_0^{10}+32256R^2r^6x_0^4y_0^{12}+9216R^2r^6x_0^2y_0^{14}+1152R^2r^6y_0^{16}-304R^2r^4x_0^{18}-2736R^2r^4x_0^{16}y_0^2-10944R^2r^4x_0^{14}y_0^4-25536R^2r^4x_0^{12}y_0^6-38304R^2r^4x_0^{10}y_0^8-38304R^2r^4x_0^8y_0^{10}-25536R^2r^4x_0^6y_0^{12}-10944R^2r^4x_0^4y_0^{14}-2736R^2r^4x_0^2y_0^{16}-304R^2r^4y_0^{18}+180R^2r^2x_0^{20}+1800R^2r^2x_0^{18}y_0^2+8100R^2r^2x_0^{16}y_0^4+21600R^2r^2x_0^{14}y_0^6+37800R^2r^2x_0^{12}y_0^8+45360R^2r^2x_0^{10}y_0^{10}+37800R^2r^2x_0^8y_0^{12}+21600R^2r^2x_0^6y_0^{14}+8100R^2r^2x_0^4y_0^{16}+1800R^2r^2x_0^2y_0^{18}+180R^2r^2y_0^{20}-60R^2x_0^22-660R^2x_0^{20}y_0^2-3300R^2x_0^{18}y_0^4-9900R^2x_0^{16}y_0^6-19800R^2x_0^{14}y_0^8-27720R^2x_0^{12}y_0^{10}-27720R^2x_0^{10}y_0^{12}-19800R^2x_0^8y_0^{14}-9900R^2x_0^6y_0^{16}-3300R^2x_0^4y_0^{18}-660R^2x_0^2y_0^{20}-60R^2y_0^22+16r^4x_0^{20}+160r^4x_0^{18}y_0^2+720r^4x_0^{16}y_0^4+1920r^4x_0^{14}y_0^6+3360r^4x_0^{12}y_0^8+4032r^4x_0^{10}y_0^{10}+3360r^4x_0^8y_0^{12}+1920r^4x_0^6y_0^{14}+720r^4x_0^4y_0^{16}+160r^4x_0^2y_0^{18}+16r^4y_0^{20}-20r^2x_0^22-220r^2x_0^{20}y_0^2-1100r^2x_0^{18}y_0^4-3300r^2x_0^{16}y_0^6-6600r^2x_0^{14}y_0^8-9240r^2x_0^{12}y_0^{10}-9240r^2x_0^{10}y_0^{12}-6600r^2x_0^8y_0^{14}-3300r^2x_0^6y_0^{16}-1100r^2x_0^4y_0^{18}-220r^2x_0^2y_0^{20}-20r^2y_0^22+5x_0^24+60x_0^22y_0^2+330x_0^{20}y_0^4+1100x_0^{18}y_0^6+2475x_0^{16}y_0^8+3960x_0^{14}y_0^{10}+4620x_0^{12}y_0^{12}+3960x_0^{10}y_0^{14}+2475x_0^8y_0^{16}+1100x_0^6y_0^{18}+330x_0^4y_0^{20}+60x_0^2y_0^22+5y_0^24=0

复制代码

若进一步要求\( x_0=0, y_0=0\),结果进一步简化为

\(n=3\)时

\(R-2r=0\)


\(n=4\)时

\(R^2-2r^2=0\)


\(n=5\)时

\(R^2+2Rr-4r^2=0\)


\(n=6\)时

\(3R^2-4r^2=0\)


\(n=7\)时

\(R^3-4R^2r-4Rr^2+8r^3=0\)


\(n=8\)时

\(R^4-8R^2r^2+8r^4=0\)


\(n=9\)时

\(R^3+6R^2r-8r^3=0\)


\(n=10\)时

\(5R^4-20R^2r^2+16r^4=0\)

mathe

可以直接用两曲线的特征方程刻画比较方便。我查看了一本关于椭圆曲线的书，里面就有n重点满足方程的递推式，只是里面已经将二次系数变换成0了，不然我们可以直接使用那里的递推式