双椭圆外切内接N边形问题

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

zuijianqiugen

数学星空 发表于 2014-4-28 01:59
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
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对于最后一个问题，有一个相当简单的通式：r=Rcos(π/n)

数学星空

若8#所列的关系式正确，我们可以得到双椭圆计算公式

外切于椭圆\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\),内接于椭圆\(\frac{(x-x_0)^2}{m^2}+\frac{(y-y_1)}{n^2}=1\)

\(n=3\)时

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

\(n=4\)时

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


\(n=5\)时

\((a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6+12n^2m^2(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)-32n^4m^4(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3a^2b^2+48n^4m^4(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2-128(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)a^2b^2m^6n^6+64m^6n^6(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3-256a^4b^4n^8m^8=0\)

若\(x_0=0,y_0=0\),则有下面结论：


\(n=3\)时

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

\(n=4\)时

\(-(a^2n^2+b^2m^2+m^2n^2)^3-4n^2m^2(-a^2b^2-a^2n^2-b^2m^2)(a^2n^2+b^2m^2+m^2n^2)-8a^2b^2n^4m^4=0\)

\(n=5\)时

\((a^2n^2+b^2m^2+m^2n^2)^6+12n^2m^2(a^2n^2+b^2m^2+m^2n^2)^4(-a^2b^2-a^2n^2-b^2m^2)-32n^4m^4(a^2n^2+b^2m^2+m^2n^2)^3a^2b^2+48n^4m^4(a^2n^2+b^2m^2+m^2n^2)^2(-a^2b^2-a^2n^2-b^2m^2)^2-128(a^2n^2+b^2m^2+m^2n^2)(-a^2b^2-a^2n^2-b^2m^2)a^2b^2m^6n^6+64m^6n^6(-a^2b^2-a^2n^2-b^2m^2)^3-256a^4b^4n^8m^8=0\)

谁有兴趣验证一下上面的公式是否正确？

数学星空

对于楼上n=3~5的结论我们进一步分解得到（若\(x_0=0,y_0=0\)）：

\(n=3\)时

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

\(n=4\)时

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

\(n=5\)时

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

根据 http://bbs.cnool.net/cthread-65224240.html  黄利兵 教授的精彩分析：







我们可以验证楼上结论的正确性

数学星空

现将双椭圆内接外切的计算结果公布如下：

\(n=3\)时

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


\(n=4\)时

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


\(n=5\)时

\((a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6+12n^2m^2(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)-32n^4m^4(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3a^2b^2+48n^4m^4(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2-(128(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)a^2b^2m^6n^6+64m^6n^6(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3-256a^4b^4n^8m^8=0\)


\(n=6\)时

\(3(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6+20n^2m^2(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)+96n^4m^4(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3a^2b^2+16n^4m^4(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2+(384(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)a^2b^2m^6n^6-64m^6n^6(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3+512a^4b^4n^8m^8=0\)

\(n=7\)时

\((a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{12}+24(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{10}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^2m^2-160(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^9a^2b^2n^4m^4+240(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^8(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^4m^4-2048(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^7(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^6m^6a^2b^2+1280(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^6m^6-3328(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6a^4b^4n^8m^8-9216(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^5(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^8m^8a^2b^2+3840(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^8m^8-27648(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^{10}m^{10}a^4b^4-16384(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^{10}m^{10}a^2b^2+6144(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^5n^{10}m^{10}-24576(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3a^6b^6n^{12}m^{12}-61440(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^{12}m^{12}a^4b^4-(8192(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^{12}m^{12}a^2b^2+4096(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^6n^{12}m^{12}-(98304(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^{14}m^{14}a^6b^6-16384(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^{14}m^{14}a^4b^4-65536a^8b^8n^{16}m^{16}=0\)


\(n=8\)时

1. -(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{12}-16(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{10}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^2m^2-80(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^9a^2b^2n^4m^4-80(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^8(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^4m^4-1280(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^7(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^6m^6a^2b^2-7680(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^5(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^8m^8a^2b^2+1280(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^8m^8-2048(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^{10}m^{10}a^4b^4-20480(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^{10}m^{10}a^2b^2+4096(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^5n^{10}m^{10}+8192(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3a^6b^6n^{12}m^{12}-16384(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^{12}m^{12}a^4b^4-(20480(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^{12}m^{12}a^2b^2+4096(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^6n^{12}m^{12}+(32768(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^{14}m^{14}a^6b^6-32768(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^{14}m^{14}a^4b^4+32768a^8b^8n^{16}m^{16}=0

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\(n=9\)时

1. 262144(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^9n^18m^18-83456(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{11}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^8m^8a^2b^2-1560576(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^8(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^{12}m^{12}a^4b^4-1024000(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^7(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^{12}m^{12}a^2b^2-16515072(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^5(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^{16}m^{16}a^6b^6-1376256(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^{16}m^{16}a^4b^4-2228224(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^6n^{16}m^{16}a^2b^2-67108864(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^20m^20a^8b^8-480(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{15}a^2b^2n^4m^4+576(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{14}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^4m^4-19200(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{12}a^4b^4n^8m^8+32256(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{10}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^8m^8-417792(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^9a^6b^6n^{12}m^{12}+344064(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^6n^{12}m^{12}-4718592(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6a^8b^8n^{16}m^{16}+589824(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^8n^{16}m^{16}-25165824(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3a^{10}b^{10}n^20m^20+16777216(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^22m^22a^8b^8-(1572864(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^7n^18m^18a^2b^2-3145728(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^6n^20m^20a^4b^4+36(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{16}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^2m^2+5376(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{12}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^6m^6+129024(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^8(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^5n^{10}m^{10}+589824(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^7n^{14}m^{14}-50331648a^{12}b^{12}n^24m^24-(100663296(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^22m^22a^{10}b^{10}+(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^18-9856(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{13}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^6m^6a^2b^2-288768(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{10}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^{10}m^{10}a^4b^4-378880(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^9(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^{10}m^{10}a^2b^2-4718592(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^7(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^{14}m^{14}a^6b^6-3342336(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^{14}m^{14}a^4b^4-1802240(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^5(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^5n^{14}m^{14}a^2b^2-36700160(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^18m^18a^8b^8-12582912(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^18m^18a^6b^6+1572864(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^5n^18m^18a^4b^4+(18874368(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^20m^20a^6b^6=0

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\(n=10\)时

1. -262144(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^9n^18m^18-210944(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{11}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^8m^8a^2b^2-4280320(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^8(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^{12}m^{12}a^4b^4-1802240(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^7(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^{12}m^{12}a^2b^2-27525120(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^5(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^{16}m^{16}a^6b^6-21299200(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^{16}m^{16}a^4b^4+3670016(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^6n^{16}m^{16}a^2b^2-57671680(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^20m^20a^8b^8-1152(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{15}a^2b^2n^4m^4-576(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{14}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^2n^4m^4-35584(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{12}a^4b^4n^8m^8+39424(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{10}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^8m^8-450560(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^9a^6b^6n^{12}m^{12}+573440(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^6n^{12}m^{12}-2949120(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6a^8b^8n^{16}m^{16}+196608(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^8n^{16}m^{16}-10485760(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3a^{10}b^{10}n^20m^20-20971520(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^22m^22a^8b^8+(4194304(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^7n^18m^18a^2b^2-(31457280(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^4n^20m^20a^6b^6+5242880(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^6n^20m^20a^4b^4-100(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{16}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^2m^2+1792(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{12}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^6m^6+215040(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^8(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^5n^{10}m^{10}+720896(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^7n^{14}m^{14}-16777216a^{12}b^{12}n^24m^24-5(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^18-24576(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{13}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^6m^6a^2b^2-624640(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^{10}(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^{10}m^{10}a^4b^4-901120(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^9(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^{10}m^{10}a^2b^2-5898240(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^7(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^{14}m^{14}a^6b^6-14090240(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^6(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^{14}m^{14}a^4b^4-524288(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^5(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^5n^{14}m^{14}a^2b^2-24903680(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^4(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^18m^18a^8b^8-52428800(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^3(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^3n^18m^18a^6b^6-8912896(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2)^2(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)^5n^18m^18a^4b^4-(41943040(a^2n^2+b^2m^2+m^2n^2-m^2y_0^2-n^2x_0^2))(-a^2b^2-a^2n^2+a^2y_0^2-b^2m^2+b^2x_0^2)n^22m^22a^{10}b^{10}=0

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数学星空

若\(x_0=0,y_0=0\),我们进一步分解楼上的结果如下：

\(n=3\)时

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


\(n=4\)时

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


\(n=5\)时

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


\(n=6\)时

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


\(n=7\)时

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


\(n=8\)时

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


\(n=9\)时

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

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\(n=10\)时

1. -(5a^{12}n^{12}-10a^{10}b^2m^2n^{10}-10a^{10}m^2n^{12}-9a^8b^4m^4n^8+34a^8b^2m^4n^{10}-9a^8m^4n^{12}+36a^6b^6m^6n^6-36a^6b^4m^6n^8-36a^6b^2m^6n^{10}+36a^6m^6n^{12}-29a^4b^8m^8n^4+4a^4b^6m^8n^6+50a^4b^4m^8n^8+4a^4b^2m^8n^{10}-29a^4m^8n^{12}+6a^2b^{10}m^{10}n^2+14a^2b^8m^{10}n^4-20a^2b^6m^{10}n^6-20a^2b^4m^{10}n^8+14a^2b^2m^{10}n^{10}+6a^2m^{10}n^{12}+b^{12}m^{12}-6b^{10}m^{12}n^2+15b^8m^{12}n^4-20b^6m^{12}n^6+15b^4m^{12}n^8-6b^2m^{12}n^{10}+m^{12}n^{12})(a^{12}n^{12}-6a^{10}b^2m^2n^{10}+6a^{10}m^2n^{12}+15a^8b^4m^4n^8+14a^8b^2m^4n^{10}-29a^8m^4n^{12}-20a^6b^6m^6n^6-20a^6b^4m^6n^8+4a^6b^2m^6n^{10}+36a^6m^6n^{12}+15a^4b^8m^8n^4-20a^4b^6m^8n^6+50a^4b^4m^8n^8-36a^4b^2m^8n^{10}-9a^4m^8n^{12}-6a^2b^{10}m^{10}n^2+14a^2b^8m^{10}n^4+4a^2b^6m^{10}n^6-36a^2b^4m^{10}n^8+34a^2b^2m^{10}n^{10}-10a^2m^{10}n^{12}+b^{12}m^{12}+6b^{10}m^{12}n^2-29b^8m^{12}n^4+36b^6m^{12}n^6-9b^4m^{12}n^8-10b^2m^{12}n^{10}+5m^{12}n^{12})(a^{12}n^{12}+6a^{10}b^2m^2n^{10}-6a^{10}m^2n^{12}-29a^8b^4m^4n^8+14a^8b^2m^4n^{10}+15a^8m^4n^{12}+36a^6b^6m^6n^6+4a^6b^4m^6n^8-20a^6b^2m^6n^{10}-20a^6m^6n^{12}-9a^4b^8m^8n^4-36a^4b^6m^8n^6+50a^4b^4m^8n^8-20a^4b^2m^8n^{10}+15a^4m^8n^{12}-10a^2b^{10}m^{10}n^2+34a^2b^8m^{10}n^4-36a^2b^6m^{10}n^6+4a^2b^4m^{10}n^8+14a^2b^2m^{10}n^{10}-6a^2m^{10}n^{12}+5b^{12}m^{12}-10b^{10}m^{12}n^2-9b^8m^{12}n^4+36b^6m^{12}n^6-29b^4m^{12}n^8+6b^2m^{12}n^{10}+m^{12}n^{12})=0

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数学星空

我们下一步的任务就是筛选楼上有用的因子，那么得到的就是N边形内接外切于双椭圆的条件，也就是黄利兵所述的封闭指数为N的条件。

数学星空

根据 http://bbs.emath.ac.cn/thread-3740-5-1.html  中数值计算的结果可以得到n=3~8,双椭圆内接外切N边形的条件或者说封闭指数为N的条件

\(n=3\)时

\(an+bm-mn=0\)

\(n=4\)时

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

\(n=5\)时

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

\(n=6\)时

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

\(n=7\)时

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

\(n=8\)时

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

mathe

8#红色公式如何得出的，我看不出来

mathe

换一种计算方法，设\(J^{-1}K\)的特征方程为$f(x)=u_0(x^3+u_2x^2+u_1x+u_0)$.

我们通过线性变换将三次方系数变为\(1\)，二次方系数变为\(0\)得到标准方程\(y^2=x^3+Ax+B\),

其中\(A=\frac{3u_1-u_2^2}{3u_0^2},B=\frac{27u_0+2u_2^3-9u_1u_2}{27u_0^3}\).而单位变换对应点\((\frac{u_2}{3u_0},\frac{1}{u_0})\).

于是这个点在这曲线上\(n\)次自相加为无穷远点就是原俩曲线分别有内外\(n\)边形相接相切的充要条件.