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
(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
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：\(-y_0^2/b^2+x_0^2/a^2 = r^2, -y_0^2/n^2+x_0^2/m^2 = R^2\)
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？
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？