双圆锥曲线内接外切N边形问题

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

数学星空

对于12#结果，若令\(r=0,R=0\)即\(x_0=0,y_0=0\),进一步化简得到如下结果，由于不太理解具体的几何意义和具体的数值解例子，因此无法筛选出有用的因子，

或许对于\(N\)边形（星形和凹形是下面结果中的两种不同的因子）

当\(N=3\)时

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

当\(N=4\)时

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

当\(N=5\)时

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

当\(N=6\)时

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

当\(N=7\)时

\((a^{12}n^{12}+4a^{11}mn^{12}+6a^{10}b^2m^2n^{10}+2a^{10}m^2n^{12}-4a^9b^2m^3n^{10}-12a^9m^3n^{12}+15a^8b^4m^4n^8-2a^8b^2m^4n^{10}-17a^8m^4n^{12}-24a^7b^4m^5n^8-16a^7b^2m^5n^{10}+8a^7m^5n^{12}+20a^6b^6m^6n^6-12a^6b^4m^6n^8-4a^6b^2m^6n^{10}+28a^6m^6n^{12}-8a^5b^6m^7n^6+24a^5b^4m^7n^8+40a^5b^2m^7n^{10}+8a^5m^7n^{12}+15a^4b^8m^8n^4-4a^4b^6m^8n^6-6a^4b^4m^8n^8-4a^4b^2m^8n^{10}-17a^4m^8n^{12}+20a^3b^8m^9n^4+48a^3b^6m^9n^6+24a^3b^4m^9n^8-16a^3b^2m^9n^{10}-12a^3m^9n^{12}+6a^2b^{10}m^{10}n^2+10a^2b^8m^{10}n^4-4a^2b^6m^{10}n^6-12a^2b^4m^{10}n^8-2a^2b^2m^{10}n^{10}+2a^2m^{10}n^{12}+12ab^{10}m^{11}n^2+20ab^8m^{11}n^4-8ab^6m^{11}n^6-24ab^4m^{11}n^8-4ab^2m^{11}n^{10}+4am^{11}n^{12}+b^{12}m^{12}+6b^{10}m^{12}n^2+15b^8m^{12}n^4+20b^6m^{12}n^6+15b^4m^{12}n^8+6b^2m^{12}n^{10}+m^{12}n^{12})(a^{12}n^{12}-4a^{11}mn^{12}+6a^{10}b^2m^2n^{10}+2a^{10}m^2n^{12}+4a^9b^2m^3n^{10}+12a^9m^3n^{12}+15a^8b^4m^4n^8-2a^8b^2m^4n^{10}-17a^8m^4n^{12}+24a^7b^4m^5n^8+16a^7b^2m^5n^{10}-8a^7m^5n^{12}+20a^6b^6m^6n^6-12a^6b^4m^6n^8-4a^6b^2m^6n^{10}+28a^6m^6n^{12}+8a^5b^6m^7n^6-24a^5b^4m^7n^8-40a^5b^2m^7n^{10}-8a^5m^7n^{12}+15a^4b^8m^8n^4-4a^4b^6m^8n^6-6a^4b^4m^8n^8-4a^4b^2m^8n^{10}-17a^4m^8n^{12}-20a^3b^8m^9n^4-48a^3b^6m^9n^6-24a^3b^4m^9n^8+16a^3b^2m^9n^{10}+12a^3m^9n^{12}+6a^2b^{10}m^{10}n^2+10a^2b^8m^{10}n^4-4a^2b^6m^{10}n^6-12a^2b^4m^{10}n^8-2a^2b^2m^{10}n^{10}+2a^2m^{10}n^{12}-12ab^{10}m^{11}n^2-20ab^8m^{11}n^4+8ab^6m^{11}n^6+24ab^4m^{11}n^8+4ab^2m^{11}n^{10}-4am^{11}n^{12}+b^{12}m^{12}+6b^{10}m^{12}n^2+15b^8m^{12}n^4+20b^6m^{12}n^6+15b^4m^{12}n^8+6b^2m^{12}n^{10}+m^{12}n^{12})=0\)

当\(N=8\)时

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

当\(N=9\)时

1. (a^{18}n^{18}-6a^{17}mn^{18}+9a^{16}b^2m^2n^{16}+9a^{16}m^2n^{18}+8a^{15}b^2m^3n^{16}+16a^{15}m^3n^{18}+36a^{14}b^4m^4n^{14}-24a^{14}b^2m^4n^{16}-60a^{14}m^4n^{18}+64a^{13}b^4m^5n^{14}+88a^{13}b^2m^5n^{16}+24a^{13}m^5n^{18}+84a^{12}b^6m^6n^{12}-68a^{12}b^4m^6n^{14}-36a^{12}b^2m^6n^{16}+116a^{12}m^6n^{18}+8a^{11}b^6m^7n^{12}-160a^{11}b^4m^7n^{14}-312a^{11}b^2m^7n^{16}-144a^{11}m^7n^{18}+126a^{10}b^8m^8n^{10}+88a^{10}b^6m^8n^{12}+244a^{10}b^4m^8n^{14}+216a^{10}b^2m^8n^{16}-66a^{10}m^8n^{18}-220a^9b^8m^9n^{10}-664a^9b^6m^9n^{12}-448a^9b^4m^9n^{14}+216a^9b^2m^9n^{16}+220a^9m^9n^{18}+126a^8b^{10}m^{10}n^8+246a^8b^8m^{10}n^{10}+172a^8b^6m^{10}n^{12}-212a^8b^4m^{10}n^{14}-330a^8b^2m^{10}n^{16}-66a^8m^{10}n^{18}-296a^7b^{10}m^{11}n^8-368a^7b^8m^{11}n^{10}+656a^7b^6m^{11}n^{12}+1088a^7b^4m^{11}n^{14}+216a^7b^2m^{11}n^{16}-144a^7m^{11}n^{18}+84a^6b^{12}m^{12}n^6+88a^6b^{10}m^{12}n^8-372a^6b^8m^{12}n^{10}-688a^6b^6m^{12}n^{12}-212a^6b^4m^{12}n^{14}+216a^6b^2m^{12}n^{16}+116a^6m^{12}n^{18}-112a^5b^{12}m^{13}n^6+296a^5b^{10}m^{13}n^8+1176a^5b^8m^{13}n^{10}+656a^5b^6m^{13}n^{12}-448a^5b^4m^{13}n^{14}-312a^5b^2m^{13}n^{16}+24a^5m^{13}n^{18}+36a^4b^{14}m^{14}n^4-68a^4b^{12}m^{14}n^6-428a^4b^{10}m^{14}n^8-372a^4b^8m^{14}n^{10}+172a^4b^6m^{14}n^{12}+244a^4b^4m^{14}n^{14}-36a^4b^2m^{14}n^{16}-60a^4m^{14}n^{18}+24a^3b^{14}m^{15}n^4+256a^3b^{12}m^{15}n^6+296a^3b^{10}m^{15}n^8-368a^3b^8m^{15}n^{10}-664a^3b^6m^{15}n^{12}-160a^3b^4m^{15}n^{14}+88a^3b^2m^{15}n^{16}+16a^3m^{15}n^{18}+9a^2b^{16}m^{16}n^2-24a^2b^{14}m^{16}n^4-68a^2b^{12}m^{16}n^6+88a^2b^{10}m^{16}n^8+246a^2b^8m^{16}n^{10}+88a^2b^6m^{16}n^{12}-68a^2b^4m^{16}n^{14}-24a^2b^2m^{16}n^{16}+9a^2m^{16}n^{18}+18ab^{16}m^{17}n^2+24ab^{14}m^{17}n^4-112ab^{12}m^{17}n^6-296ab^{10}m^{17}n^8-220ab^8m^{17}n^{10}+8ab^6m^{17}n^{12}+64ab^4m^{17}n^{14}+8ab^2m^{17}n^{16}-6am^{17}n^{18}+b^{18}m^{18}+9b^{16}m^{18}n^2+36b^{14}m^{18}n^4+84b^{12}m^{18}n^6+126b^{10}m^{18}n^8+126b^8m^{18}n^{10}+84b^6m^{18}n^{12}+36b^4m^{18}n^{14}+9b^2m^{18}n^{16}+m^{18}n^{18})(a^{18}n^{18}+6a^{17}mn^{18}+9a^{16}b^2m^2n^{16}+9a^{16}m^2n^{18}-8a^{15}b^2m^3n^{16}-16a^{15}m^3n^{18}+36a^{14}b^4m^4n^{14}-24a^{14}b^2m^4n^{16}-60a^{14}m^4n^{18}-64a^{13}b^4m^5n^{14}-88a^{13}b^2m^5n^{16}-24a^{13}m^5n^{18}+84a^{12}b^6m^6n^{12}-68a^{12}b^4m^6n^{14}-36a^{12}b^2m^6n^{16}+116a^{12}m^6n^{18}-8a^{11}b^6m^7n^{12}+160a^{11}b^4m^7n^{14}+312a^{11}b^2m^7n^{16}+144a^{11}m^7n^{18}+126a^{10}b^8m^8n^{10}+88a^{10}b^6m^8n^{12}+244a^{10}b^4m^8n^{14}+216a^{10}b^2m^8n^{16}-66a^{10}m^8n^{18}+220a^9b^8m^9n^{10}+664a^9b^6m^9n^{12}+448a^9b^4m^9n^{14}-216a^9b^2m^9n^{16}-220a^9m^9n^{18}+126a^8b^{10}m^{10}n^8+246a^8b^8m^{10}n^{10}+172a^8b^6m^{10}n^{12}-212a^8b^4m^{10}n^{14}-330a^8b^2m^{10}n^{16}-66a^8m^{10}n^{18}+296a^7b^{10}m^{11}n^8+368a^7b^8m^{11}n^{10}-656a^7b^6m^{11}n^{12}-1088a^7b^4m^{11}n^{14}-216a^7b^2m^{11}n^{16}+144a^7m^{11}n^{18}+84a^6b^{12}m^{12}n^6+88a^6b^{10}m^{12}n^8-372a^6b^8m^{12}n^{10}-688a^6b^6m^{12}n^{12}-212a^6b^4m^{12}n^{14}+216a^6b^2m^{12}n^{16}+116a^6m^{12}n^{18}+112a^5b^{12}m^{13}n^6-296a^5b^{10}m^{13}n^8-1176a^5b^8m^{13}n^{10}-656a^5b^6m^{13}n^{12}+448a^5b^4m^{13}n^{14}+312a^5b^2m^{13}n^{16}-24a^5m^{13}n^{18}+36a^4b^{14}m^{14}n^4-68a^4b^{12}m^{14}n^6-428a^4b^{10}m^{14}n^8-372a^4b^8m^{14}n^{10}+172a^4b^6m^{14}n^{12}+244a^4b^4m^{14}n^{14}-36a^4b^2m^{14}n^{16}-60a^4m^{14}n^{18}-24a^3b^{14}m^{15}n^4-256a^3b^{12}m^{15}n^6-296a^3b^{10}m^{15}n^8+368a^3b^8m^{15}n^{10}+664a^3b^6m^{15}n^{12}+160a^3b^4m^{15}n^{14}-88a^3b^2m^{15}n^{16}-16a^3m^{15}n^{18}+9a^2b^{16}m^{16}n^2-24a^2b^{14}m^{16}n^4-68a^2b^{12}m^{16}n^6+88a^2b^{10}m^{16}n^8+246a^2b^8m^{16}n^{10}+88a^2b^6m^{16}n^{12}-68a^2b^4m^{16}n^{14}-24a^2b^2m^{16}n^{16}+9a^2m^{16}n^{18}-18ab^{16}m^{17}n^2-24ab^{14}m^{17}n^4+112ab^{12}m^{17}n^6+296ab^{10}m^{17}n^8+220ab^8m^{17}n^{10}-8ab^6m^{17}n^{12}-64ab^4m^{17}n^{14}-8ab^2m^{17}n^{16}+6am^{17}n^{18}+b^{18}m^{18}+9b^{16}m^{18}n^2+36b^{14}m^{18}n^4+84b^{12}m^{18}n^6+126b^{10}m^{18}n^8+126b^8m^{18}n^{10}+84b^6m^{18}n^{12}+36b^4m^{18}n^{14}+9b^2m^{18}n^{16}+m^{18}n^{18})=0

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数学星空

现将圆锥曲线所有可能的组合结果公布如下：


1.  外切于圆\(x^2+y^2=r^2\),内接于圆\((x-x_0)^2+(y-y_0)^2=R^2\)

    特征根对称式为:\(u_0 = -r^2/R^2, u_1 = (R^2-d^2+2r^2)/R^2, u_2 = (-2R^2+d^2-r^2)/R^2\), 其中： \(x_0^2+y_0^2 = d^2\)
   

2.  外切于抛物线 \(y^2=2px\) ,内接于圆\((x-x_0)^2+(y-y_0)^2=R^2\)

    特征对称式为:\(u_0 = -p^2/R^2, u_1 = (p^2+2py_0)/R^2, u_2 = (-R^2-2py_0+x_0^2)/R^2\)


3.  外切于椭圆\(x^2/a^2+y^2/b^2=1\),内接于圆\((x-x_0)^2+(y-y_0)^2=R^2\)

    特征对称式为：\(u_0 = -1/(a^2b^2R^2), u_1 = (R^2+a^2+b^2-d^2)/(a^2b^2R^2), u_2 = (a^2b^2r^2-R^2a^2-R^2b^2-a^2b^2)/(a^2b^2R^2)\),其中：\(y_0^2/a^2+x_0^2/b^2 = r^2, x_0^2+y_0^2 = d^2\)


4.  外切于双曲线\(x^2/a^2-y^2/b^2=1\),内接于圆\((x-x_0)^2+(y-y_0)^2=R^2\)
   
    特征对称式为：\(u_0 = 1/(a^2b^2R^2), u_1 = (-R^2-a^2+b^2+d^2)/(R^2b^2a^2), u_2 =(a^2b^2r^2+R^2a^2-R^2b^2-a^2b^2)/(R^2b^2a^2)\),其中：\(-y_0^2/b^2+x_0^2/a^2 = r^2, x_0^2+y_0^2 = d^2\)


5.  外切于圆\(x^2+y^2=r^2\),内接于抛物线\((y-y_0)^2=2q(x-x_0)\)

    特征对称式为：\(u_0 = -r^2/q^2, u_1 = (-2qy_0+r^2-x_0^2)/q^2, u_2 = (-q^2+2qy_0)/q^2\)


6.  外切于抛物线 \(y^2=2px\),内接于抛物线\((y-y_0)^2=2q(x-x_0)\)
   
    特征对称式为：\(u_0 = -p^2/q^2, u_1 = (p^2+2pq)/q^2, u_2 = (-2pq-q^2)/q^2\)


7.  外切于椭圆\(x^2/a^2+y^2/b^2=1\),内接于抛物线\((y-y_0)^2=2q(x-x_0)\)

    特征对称式为：\(u_0 = -1/(q^2a^2b^2), u_1 = (a^2-2qy_0-x_0^2)/(q^2a^2b^2), u_2 = (2a^2qy_0-b^2q^2)/(q^2a^2b^2)\)



8.  外切于双曲线\(x^2/a^2-y^2/b^2=1\),内接于抛物线\((y-y_0)^2=2q(x-x_0)\)

    特征对称式为：\(u_0 = 1/(q^2a^2b^2), u_1 = (-a^2+2qy_0+x_0^2)/(q^2a^2b^2), u_2 = (-2a^2qy_0-b^2q^2)/(q^2a^2b^2)\)


9.  外切于圆\(x^2+y^2=r^2\),内接于椭圆\((x-x_0)^2/m^2+(y-y_0)^2/n^2=1\)

    特征对称式为：\(u_0 = -m^2n^2r^2, u_1 =-R^2m^2n^2+m^2n^2+m^2r^2+n^2r^2, u_2 = d^2-m^2-n^2-r^2\), 其中：\(y_0^2/m^2+x_0^2/n^2 = R^2, x_0^2+y_0^2 = d^2\)


10.  外切于抛物线 \(y^2=2px\),内接于椭圆\((x-x_0)^2/m^2+(y-y_0)^2/n^2=1\)

     特征对称式为：\(u_0 = -m^2n^2p^2, u_1 = 2m^2py_0+n^2p^2, u_2 = -m^2-2py_0+x_0^2\)


11.  外切于椭圆\(x^2/a^2+y^2/b^2=1\),内接于椭圆\((x-x_0)^2/m^2+(y-y_0)^2/n^2=1\)
     
     特征对称式为：\(u_0 = -m^2n^2/(a^2b^2), u_1 = (-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)/(a^2b^2), u_2 = (a^2b^2r^2-a^2b^2-a^2n^2-b^2m^2)/(a^2b^2)\)，其中：\(y_0^2/b^2+x_0^2/a^2 = r^2, y_0^2/n^2+x_0^2/m^2 = R^2\)


12.  外切于双曲线\(x^2/a^2-y^2/b^2=1\),内接于椭圆\((x-x_0)^2/m^2+(y-y_0)^2/n^2=1\)

     特征对称式为：\(u_0 = m^2n^2/(a^2b^2), u_1 = (R^2m^2n^2-a^2n^2+b^2m^2-m^2n^2)/(a^2b^2), u_2 = (a^2b^2r^2-a^2b^2+a^2n^2-b^2m^2)/(a^2b^2)\)， 其中：\(-y_0^2/b^2+x_0^2/a^2 = r^2, y_0^2/m^2+x_0^2/n^2 = R^2\)


13.  外切于圆\(x^2+y^2=r^2\),内接于双曲线\((x-x_0)^2/m^2-(y-y_0)^2/n^2=1\)

     特征对称式为：\(u_0 = m^2n^2r^2, u_1 = R^2m^2n^2-m^2n^2+m^2r^2-n^2r^2, u_2 = d^2-m^2+n^2-r^2\)， 其中：\(-y_0^2/n^2+x_0^2/m^2 = R^2, x_0^2+y_0^2 = d^2\)


14.  外切于抛物线\(y^2=2px\),内接于双曲线\((x-x_0)^2/m^2-(y-y_0)^2/n^2=1\)

     特征对称式为：\(u_0 = m^2n^2p^2, u_1 = 2m^2py_0-n^2p^2, u_2 = -m^2-2py_0+x_0^2\)


15.  外切于椭圆\(x^2/a^2+y^2/b^2=1\),内接于双曲线\((x-x_0)^2/m^2-(y-y_0)^2/n^2=1\)

     特征对称式为：\(u_0 = m^2n^2/(a^2b^2), u_1 = (R^2m^2n^2-a^2n^2+b^2m^2-m^2n^2)/(a^2b^2), u_2 = (a^2b^2r^2-a^2b^2+a^2n^2-b^2m^2)/(a^2b^2)\)， 其中：\(y_0^2/b^2+x_0^2/a^2 = r^2, -y_0^2/n^2+x_0^2/m^2 = R^2\)


16.  外切于双曲线\(x^2/a^2-y^2/b^2=1\),内接于双曲线\((x-x_0)^2/m^2-(y-y_0)^2/n^2=1\)

     特征对称式为：\(u_0 = -m^2n^2/(a^2b^2), u_1 = (-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)/(a^2b^2), u_2 = (a^2b^2r^2-a^2b^2-a^2n^2-b^2m^2)/(a^2b^2)\)，其中：\(-y_0^2/b^2+x_0^2/a^2 = r^2, -y_0^2/n^2+x_0^2/m^2 = R^2\)

zuijianqiugen

数学星空 发表于 2014-5-5 22:56
现将圆锥曲线所有可能的组合结果公布如下：

不知“特征对称式”是何意？如何应用？与“弦切n边形”有何关系？