双圆锥曲线内接外切N边形问题

数学星空

由于楼上的特征方程适用所有的圆锥曲线？？（mathe对此论断是否正确 最有权威！）

是否可以直接将6#的特征根代入7#特征方程化简即可？

另外，抛物线的矩阵方程应如何表示？

mathe

直接将特征方程最高次系数变成1（也就是除最高次系数），余下系数即各u_i.抛物线其实完全相同，只是标准化后的y^2系数为0而已。
比如y=x^2的矩阵是\(\begin{bmatrix}1&0&0\\0&0&-\frac12\\0&-\frac12&0\end{bmatrix}\), (注意y的系数是\(2*(-\frac12)\))

mathe

比如6楼特征方程\(x^3a^2b^2+(-a^2b^2+a^2n^2-a^2y_0^2-b^2m^2+b^2x_0^2)x^2+(-a^2n^2+b^2m^2-m^2n^2+m^2y_0^2+n^2x_0^2)x+m^2n^2=0\)
可以写成
\(x^3+\frac{-a^2b^2+a^2n^2-a^2y_0^2-b^2m^2+b^2x_0^2}{a^2b^2}x^2+\frac{-a^2n^2+b^2m^2-m^2n^2+m^2y_0^2+n^2x_0^2}{a^2b^2}x+\frac{m^2n^2}{a^2b^2}=0\)
于是我们有
\(u_2=\frac{a^2b^2-a^2n^2+a^2y_0^2+b^2m^2-b^2x_0^2}{a^2b^2}, u_1=\frac{-a^2n^2+b^2m^2-m^2n^2+m^2y_0^2+n^2x_0^2}{a^2b^2}, u_0=-\frac{m^2n^2}{a^2b^2}\)

mathe

不过你6#的矩阵给的是
外切于双曲线\(\frac{(x-x_0)^2}{a^2}-\frac{(y-y_0)^2}{b^2}=1\),和内接于椭圆\(\frac{x^2}{m^2}+\frac{y^2}{n^2}=1\)的n边形,当然，两者区别不大

数学星空

对于外切于双曲线\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\),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

我们设

\(\frac{x_0^2}{m^2}+\frac{y_0^2}{n^2}=R^2\)

\(\frac{x_0^2}{a^2}-\frac{y_0^2}{b^2}=r^2\)

当\(N=3\)时

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

当\(N=4\)时

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

当\(N=5\)时

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

当\(N=6\)时

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

当\(N=7\)时

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

当\(N=8\)时

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

当\(N=9\)时

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

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当\(N=10\)时

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

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数学星空

突然想到：N(大于等于4）边形如何外切双曲线？？？

似乎不存在啊？

数学星空

谁能给出n=4.5.6.7边形外切双曲线，内接椭圆的图形及相关各边长，各个角度值？

mathe

从射影几何的角度，这个多边形可以是无界的，也就是其中某些边可以跨越无穷远点。（也就是两条平行的射线构成)

数学星空

TO mathe  11#

对于外切于双曲线\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\),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

A: (我按照 http://bbs.emath.ac.cn/forum.php ... 95&fromuid=1455 类比给出的表达式）

\(\D J=\begin{bmatrix}\frac{1}{m^2}&0&0\\0&\frac{1}{n^2}&0\\0&0&-1\end{bmatrix}\)

\(\D K=\begin{bmatrix}\frac1{a^2}&0&-\frac{x_0}{a^2}\\0&-\frac1{b^2}&\frac{y_0}{b^2}\\-\frac{x_0}{a^2}&\frac{y_0}{b^2}&\frac{x_0^2}{a^2}-\frac{y_0^2}{b^2}-1\end{bmatrix}\)

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

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



B:若按照11# mathe的说法

对于外切于双曲线\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\),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

特征方程应修改为：

\(\D K=\begin{bmatrix}\frac{1}{a^2}&0&0\\0&-\frac{1}{b^2}&0\\0&0&-1\end{bmatrix}\)

\(\D J=\begin{bmatrix}\frac1{m^2}&0&-\frac{x_0}{m^2}\\0&\frac1{n^2}&-\frac{y_0}{n^2}\\-\frac{x_0}{m^2}&-\frac{y_0}{n^2}&\frac{x_0^2}{m^2}+\frac{y_0^2}{n^2}-1\end{bmatrix}\)

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

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

A和 B的结果的确是一样，多谢mathe指点迷津！

我觉得B的特征方程更自然一些！

mathe

后面的矩阵有几个元符号弄错了