双圆锥曲线内接外切N边形问题

数学星空 · 发表于 2014-5-3 12:23:47

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我们在 http://bbs.emath.ac.cn/forum.php ... 21&fromuid=1455
中根据mathe提供的椭圆曲线理论解决了双椭圆内接外切N边形问题的条件解。
现在我们进一步讨论抛物线，双曲线，椭圆，圆之间内接外切N边形问题的条件解？

数学星空 · 发表于 2014-5-3 12:40:33

为了便于讨论和交流，先将已得到结果转载过来：

根据 http://bbs.emath.ac.cn/forum.php ... 56&fromuid=1455

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

若记\(x_0^2+y_0^2=d^2\),即可以得到双圆内接外切N边形的圆心心距公式：

当\(n=3\)时

\(R^2-2Rr-d^2＝0\)

当\(n=4\)时

\(d^4+(-2R^2-2r^2)d^2+R^4-2r^2R^2＝0\)

当\(n=5\)时

\(-d^6+(3R^2+2Rr)d^4+(-3R^4-4R^3r+4R^2r^2-8Rr^3)d^2+R^6+2R^5r-4R^4r^2＝0\)

当\(n=6\)时

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

当\(n=7\)时

\(d^{12}+(-6R^2+4Rr)d^{10}+(15R^4-20R^3r-4R^2r^2-24Rr^3)d^8+(-20R^6+40R^5r+16R^4r^2+64R^3r^3-16R^2r^4+32Rr^5)d^6+(15R^8-40R^7r-24R^6r^2-48R^5r^3+32R^4r^4)d^4+(-6R^{10}+20R^9r+16R^8r^2-16R^6r^4-32R^5r^5+64R^4r^6)d^2+R^{12}-4R^{11}r-4R^{10}r^2+8R^9r^3＝0\)

当\(n=8\)时

\(d^{16}+(-8R^2-8r^2)d^{14}+(28R^4+40R^2r^2+8r^4)d^{12}+(-56R^6-72R^4r^2+48R^2r^4)d^{10}+(70R^8+40R^6r^2-264R^4r^4-128R^2r^6)d^8+(-56R^{10}+40R^8r^2+416R^6r^4+128R^4r^6+128R^2r^8)d^6+(28R^{12}-72R^{10}r^2-264R^8r^4+128R^6r^6)d^4+(-8R^{14}+40R^{12}r^2+48R^{10}r^4-128R^8r^6+128R^6r^8)d^2+R^{16}-8R^{14}r^2+8R^{12}r^4＝0\)

当\(n=9\)时

\(-d^{18}+(9R^2+6Rr)d^{16}+(-36R^4-48R^3r-56Rr^3)d^{14}+(84R^6+168R^5r+328R^3r^3+96R^2r^4+160Rr^5)d^{12}+(-126R^8-336R^7r-792R^5r^3-480R^4r^4-608R^3r^5-256R^2r^6-128Rr^7)d^{10}+(126R^{10}+420R^9r+1000R^7r^3+960R^6r^4+832R^5r^5+512R^4r^6+384R^3r^7+256R^2r^8)d^8+(-84R^{12}-336R^{11}r-680R^9r^3-960R^8r^4-448R^7r^5-384R^5r^7-512R^3r^9)d^6+(36R^{14}+168R^{13}r+216R^{11}r^3+480R^{10}r^4+32R^9r^5-512R^8r^6+128R^7r^7)d^4+(-9R^{16}-48R^{16}r-8R^{13}r^3-96R^{12}r^4+32R^{11}r^5+256R^{10}r^6-256R^8r^8)d^2+R^{18}+6R^{17}r-8R^{15}r^3＝0\)

当\(n=10\)时

\(5d^{24}+(-60R^2-20r^2)d^{22}+(330R^4+180R^2r^2+16r^4)d^{20}+(-1100R^6-700R^4r^2-304R^2r^4)d^{18}+(2475R^8+1500R^6r^2+1872R^4r^4+1152R^2r^6)d^{16}+(-3960R^{10}-1800R^8r^2-5952R^6r^4-5760R^4r^6-1792R^2r^8)d^{14}+(4620R^{12}+840R^{10}r^2+11424R^8r^4+10368R^6r^6+3328R^4r^8+1024R^2r^{10})d^{12}+(-3960R^{14}+840R^{12}r^2-14112R^{10}r^4-5760R^8r^6+2816R^6r^8)d^{10}+(2475R^{16}-1800R^{14}r^2+11424R^{12}r^4-5760R^{10}r^6-8704R^8r^8-1024R^6r^{10})d^8+(-1100R^{18}+1500R^{16}r^2-5952R^{14}r^4+10368R^{12}r^6+2816R^{10}r^8-1024R^8r^{10}+4096R^6r^{12})d^6+(330R^{20}-700R^{18}r^2+1872R^{16}r^4-5760R^{14}r^6+3328R^{12}r^8)d^4+(-60R^{22}+180R^{20}r^2-304R^{18}r^4+1152R^{16}r^6-1792R^{14}r^8+1024R^{12}r^{10})d^2+5R^{24}-20R^{22}r^2+16R^{20}r^4＝0\)

数学星空 · 发表于 2014-5-3 12:42:25

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\(n\)边形计算公式

\(n=3\)时

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

\(n=4\)时

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

\(n=5\)时

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

\(n=6\)时

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

\(n=7\)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

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\(n=8\)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

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\(n=9\)时

-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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\(n=10\)时

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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数学星空 · 发表于 2014-5-3 12:43:43

http://bbs.emath.ac.cn/forum.php ... 47&fromuid=1455

按照mathe的方案，我们直接利用递推式确定\(\psi[n]\)即可

\(A=\frac{(-u_2^2+3u_1)}{3u_0^2}, B=\frac{(2u_2^3-9u_1u_2+27u_0)}{27u_0^3}, x=\frac{u_2}{3u_0}, y=\frac{1}{u_0}\)

\(\psi[0] =0\)

\(\psi[1] =1\)

\(\psi[2] =\frac{2}{u_0}\)

\(\psi[3] =\frac{4u_0u_2-u_1^2}{u_0^4}\)

\(\psi[4] =-\frac{4(-4u_0u_1u_2+u_1^3+8u_0^2)}{u_0^7}\)

\(\psi[2k] = \frac{u_0\psi[k](-\psi[k-2]\psi[k+1]^2+\psi[k-1]^2\psi[k+2])}{2}\)

\(\psi[2k+1] = \psi[k]^3\psi[k+2]-\psi[k-1]\psi[k+1]^3\)

数学星空 · 发表于 2014-5-3 12:44:54

http://bbs.emath.ac.cn/forum.php ... 48&fromuid=1455

通过计算，我们可以得到双椭圆心距公式( \(N\)边形）：

外椭圆 \(\frac{(x-x_0)^2}{m^2}+\frac{(y-y_0)^2}{n^2}=1\),内椭圆： \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

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

\(-32m^4n^4a^2b^2-16m^2n^2(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)(a^2b^2r^2-a^2b^2-a^2n^2-b^2m^2)-4(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^3=0\)

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

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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

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当\(N=9\)时

4718592m^{16}n^{16}(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^6a^8b^8+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+19200m^8n^8(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^{12}a^4b^4+480m^4n^4(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^{15}a^2b^2+25165824m^{20}n^{20}(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^3a^{10}b^{10}-16777216m^{22}n^{22}(a^2b^2r^2-a^2b^2-a^2n^2-b^2m^2)^3a^8b^8-262144m^{18}n^{18}(a^2b^2r^2-a^2b^2-a^2n^2-b^2m^2)^9+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+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+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^6-1572864m^{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+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+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+100663296m^{22}n^{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}+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-(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^{18}-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)^2-589824m^{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^{24}n^{24}a^{12}b^{12}=0

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当\(N=10\)时

2949120m^{16}n^{16}(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^6a^8b^8+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+35584m^8n^8(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^{12}a^4b^4+1152m^4n^4(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^{15}a^2b^2+10485760m^{20}n^{20}(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^3a^{10}b^{10}+20971520m^{22}n^{22}(a^2b^2r^2-a^2b^2-a^2n^2-b^2m^2)^3a^8b^8+262144m^{18}n^{18}(a^2b^2r^2-a^2b^2-a^2n^2-b^2m^2)^9+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+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+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+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^4-4194304m^{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+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+41943040m^{22}n^{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}+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+5(-R^2m^2n^2+a^2n^2+b^2m^2+m^2n^2)^{18}-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)^2-720896m^{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^{24}n^{24}a^{12}b^{12}=0

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数学星空 · 发表于 2014-5-3 13:18:03

由于椭圆与双曲线较接近，根据mathe提供的特征方程：

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

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

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

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\(r_1,r_2,r_3\)）

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

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

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

数学星空 · 发表于 2014-5-3 13:43:55

根据4#的递推式，我们可以得到特征方程

（即mathe公布的结果 http://bbs.emath.ac.cn/forum.php ... 92&fromuid=1455 ）

当\(N=3\)时

\(4u_0u_2-u_1^2=0\)

当\(N=4\)时

\(16u_0u_1u_2-4u_1^3-32u_0^2=0\)

当\(N=5\)时

\(-64u_0^3u_2^3+48u_0^2u_1^2u_2^2-12u_0u_1^4u_2+u_1^6+128u_0^3u_1u_2-32u_0^2u_1^3-256u_0^4=0\)

当\(N=6\)时

\(64u_0^3u_2^3+16u_0^2u_1^2u_2^2-20u_0u_1^4u_2+3u_1^6-384u_0^3u_1u_2+96u_0^2u_1^3+512u_0^4=0\)

当\(N=7\)时

\(-4096u_0^6u_2^6+6144u_0^5u_1^2u_2^5-3840u_0^4u_1^4u_2^4+1280u_0^3u_1^6u_2^3-240u_0^2u_1^8u_2^2+24u_0u_1^{10}u_2-u_1^{12}+8192u_0^6u_1u_2^4-16384u_0^5u_1^3u_2^3+9216u_0^4u_1^5u_2^2-2048u_0^3u_1^7u_2+160u_0^2u_1^9-16384u_0^7u_2^3+61440u_0^6u_1^2u_2^2-27648u_0^5u_1^4u_2+3328u_0^4u_1^6-98304u_0^7u_1u_2+24576u_0^6u_1^3+65536u_0^8=0\)

当\(N=8\)时

\(4096u_0^6u_2^6-4096u_0^5u_1^2u_2^5+1280u_0^4u_1^4u_2^4-80u_0^2u_1^8u_2^2+16u_0u_1^{10}u_2-u_1^{12}-20480u_0^6u_1u_2^4+20480u_0^5u_1^3u_2^3-7680u_0^4u_1^5u_2^2+1280u_0^3u_1^7u_2-80u_0^2u_1^9+32768u_0^7u_2^3-16384u_0^6u_1^2u_2^2+2048u_0^5u_1^4u_2-32768u_0^7u_1u_2+8192u_0^6u_1^3+32768u_0^8=0\)

当\(N=9\)时

\(262144u_0^9u_2^9-589824u_0^8u_1^2u_2^8+589824u_0^7u_1^4u_2^7-344064u_0^6u_1^6u_2^6+129024u_0^5u_1^8u_2^5-32256u_0^4u_1^{10}u_2^4+5376u_0^3u_1^{12}u_2^3-576u_0^2u_1^{14}u_2^2+36u_0u_1^{16}u_2-u_1^{18}-1572864u_0^9u_1u_2^7+2228224u_0^8u_1^3u_2^6-1802240u_0^7u_1^5u_2^5+1024000u_0^6u_1^7u_2^4-378880u_0^5u_1^9u_2^3+83456u_0^4u_1^{11}u_2^2-9856u_0^3u_1^{13}u_2+480u_0^2u_1^{15}+3145728u_0^{10}u_2^6+1572864u_0^9u_1^2u_2^5+1376256u_0^8u_1^4u_2^4-3342336u_0^7u_1^6u_2^3+1560576u_0^6u_1^8u_2^2-288768u_0^5u_1^{10}u_2+19200u_0^4u_1^{12}-18874368u_0^{10}u_1u_2^4-12582912u_0^9u_1^3u_2^3+16515072u_0^8u_1^5u_2^2-4718592u_0^7u_1^7u_2+417792u_0^6u_1^9+16777216u_0^{11}u_2^3+67108864u_0^{10}u_1^2u_2^2-36700160u_0^9u_1^4u_2+4718592u_0^8u_1^6-100663296u_0^{11}u_1u_2+25165824u_0^{10}u_1^3+50331648u_0^{12}=0\)

当\(N=10\)时

\(-262144u_0^9u_2^9-196608u_0^8u_1^2u_2^8+720896u_0^7u_1^4u_2^7-573440u_0^6u_1^6u_2^6+215040u_0^5u_1^8u_2^5-39424u_0^4u_1^{10}u_2^4+1792u_0^3u_1^{12}u_2^3+576u_0^2u_1^{14}u_2^2-100u_0u_1^{16}u_2+5u_1^{18}+4194304u_0^9u_1u_2^7-3670016u_0^8u_1^3u_2^6-524288u_0^7u_1^5u_2^5+1802240u_0^6u_1^7u_2^4-901120u_0^5u_1^9u_2^3+210944u_0^4u_1^{11}u_2^2-24576u_0^3u_1^{13}u_2+1152u_0^2u_1^{15}-5242880u_0^{10}u_2^6-8912896u_0^9u_1^2u_2^5+21299200u_0^8u_1^4u_2^4-14090240u_0^7u_1^6u_2^3+4280320u_0^6u_1^8u_2^2-624640u_0^5u_1^{10}u_2+35584u_0^4u_1^{12}+31457280u_0^{10}u_1u_2^4-52428800u_0^9u_1^3u_2^3+27525120u_0^8u_1^5u_2^2-5898240u_0^7u_1^7u_2+450560u_0^6u_1^9-20971520u_0^{11}u_2^3+57671680u_0^{10}u_1^2u_2^2-24903680u_0^9u_1^4u_2+2949120u_0^8u_1^6-41943040u_0^{11}u_1u_2+10485760u_0^{10}u_1^3+16777216u_0^{12}=0\)

当\(N=11\)时

1073741824u_0^{15}u_2^{15}-4026531840u_0^{14}u_1^2u_2^{14}+7046430720u_0^{13}u_1^4u_2^{13}-7633633280u_0^{12}u_1^6u_2^{12}+5725224960u_0^{11}u_1^8u_2^{11}-3148873728u_0^{10}u_1^{10}u_2^{10}+1312030720u_0^9u_1^{12}u_2^9-421724160u_0^8u_1^{14}u_2^8+105431040u_0^7u_1^{16}u_2^7-20500480u_0^6u_1^{18}u_2^6+3075072u_0^5u_1^{20}u_2^5-349440u_0^4u_1^{22}u_2^4+29120u_0^3u_1^{24}u_2^3-1680u_0^2u_1^{26}u_2^2+60u_0u_1^{28}u_2-u_1^{30}-6442450944u_0^{15}u_1u_2^{13}+29527900160u_0^{14}u_1^3u_2^{12}-50734301184u_0^{13}u_1^5u_2^{11}+48653926400u_0^{12}u_1^7u_2^{10}-30744248320u_0^{11}u_1^9u_2^9+14136901632u_0^{10}u_1^{11}u_2^8-5064622080u_0^9u_1^{13}u_2^7+1467482112u_0^8u_1^{15}u_2^6-343572480u_0^7u_1^{17}u_2^5+62791680u_0^6u_1^{19}u_2^4-8474624u_0^5u_1^{21}u_2^3+783360u_0^4u_1^{23}u_2^2-43904u_0^3u_1^{25}u_2+1120u_0^2u_1^{27}+12884901888u_0^{16}u_2^{12}-77309411328u_0^{15}u_1^2u_2^{11}+56371445760u_0^{14}u_1^4u_2^{10}+37580963840u_0^{13}u_1^6u_2^9-68702699520u_0^{12}u_1^8u_2^8+36842766336u_0^{11}u_1^{10}u_2^7-7839154176u_0^{10}u_1^{12}u_2^6-754974720u_0^9u_1^{14}u_2^5+875888640u_0^8u_1^{16}u_2^4-233963520u_0^7u_1^{18}u_2^3+32514048u_0^6u_1^{20}u_2^2-2408448u_0^5u_1^{22}u_2+75520u_0^4u_1^{24}+51539607552u_0^{16}u_1u_2^{10}+429496729600u_0^{15}u_1^3u_2^9-724775731200u_0^{14}u_1^5u_2^8+386547056640u_0^{13}u_1^7u_2^7-14092861440u_0^{12}u_1^9u_2^6-76101451776u_0^{11}u_1^{11}u_2^5+39636172800u_0^{10}u_1^{13}u_2^4-10066329600u_0^9u_1^{15}u_2^3+1450967040u_0^8u_1^{17}u_2^2-114032640u_0^7u_1^{19}u_2+3817472u_0^6u_1^{21}-17179869184u_0^{17}u_2^9-1352914698240u_0^{16}u_1^2u_2^8+854698491904u_0^{15}u_1^4u_2^7+877247070208u_0^{14}u_1^6u_2^6-1245406298112u_0^{13}u_1^8u_2^5+638977048576u_0^{12}u_1^{10}u_2^4-177251287040u_0^{11}u_1^{12}u_2^3+28273803264u_0^{10}u_1^{14}u_2^2-2451832832u_0^9u_1^{16}u_2+89980928u_0^8u_1^{18}+1649267441664u_0^{17}u_1u_2^7+2439541424128u_0^{16}u_1^3u_2^6-5617817223168u_0^{15}u_1^5u_2^5+3833258311680u_0^{14}u_1^7u_2^4-1309965025280u_0^{13}u_1^9u_2^3+246021095424u_0^{12}u_1^{11}u_2^2-24360517632u_0^{11}u_1^{13}u_2+998244352u_0^{10}u_1^{15}-755914244096u_0^{18}u_2^6-7524782702592u_0^{17}u_1^2u_2^5+9083855831040u_0^{16}u_1^4u_2^4-4144643440640u_0^{15}u_1^6u_2^3+922075791360u_0^{14}u_1^8u_2^2-100260642816u_0^{13}u_1^{10}u_2+4244635648u_0^{12}u_1^{12}+7559142440960u_0^{18}u_1u_2^4-4810363371520u_0^{17}u_1^3u_2^3+773094113280u_0^{16}u_1^5u_2^2+42949672960u_0^{15}u_1^7u_2-13421772800u_0^{14}u_1^9-2748779069440u_0^{19}u_2^3-2061584302080u_0^{18}u_1^2u_2^2+1546188226560u_0^{17}u_1^4u_2-214748364800u_0^{16}u_1^6+3298534883328u_0^{19}u_1u_2-824633720832u_0^{18}u_1^3-1099511627776u_0^{20}=0

复制代码

当\(N=12\)时

50331648u_0^{12}u_2^{12}-134217728u_0^{11}u_1^2u_2^{11}+161480704u_0^{10}u_1^4u_2^{10}-115343360u_0^9u_1^6u_2^9+54067200u_0^8u_1^8u_2^8-17301504u_0^7u_1^{10}u_2^7+3784704u_0^6u_1^{12}u_2^6-540672u_0^5u_1^{14}u_2^5+42240u_0^4u_1^{16}u_2^4-352u_0^2u_1^{20}u_2^2+32u_0u_1^{22}u_2-u_1^{24}-369098752u_0^{12}u_1u_2^{10}+922746880u_0^{11}u_1^3u_2^9-1038090240u_0^{10}u_1^5u_2^8+692060160u_0^9u_1^7u_2^7-302776320u_0^8u_1^9u_2^6+90832896u_0^7u_1^{11}u_2^5-18923520u_0^6u_1^{13}u_2^4+2703360u_0^5u_1^{15}u_2^3-253440u_0^4u_1^{17}u_2^2+14080u_0^3u_1^{19}u_2-352u_0^2u_1^{21}+671088640u_0^{13}u_2^9-838860800u_0^{12}u_1^2u_2^8+100663296u_0^{11}u_1^4u_2^7+478150656u_0^{10}u_1^6u_2^6-445644800u_0^9u_1^8u_2^5+200540160u_0^8u_1^{10}u_2^4-53084160u_0^7u_1^{12}u_2^3+8421376u_0^6u_1^{14}u_2^2-743424u_0^5u_1^{16}u_2+28160u_0^4u_1^{18}-2684354560u_0^{13}u_1u_2^7+4563402752u_0^{12}u_1^3u_2^6-4127195136u_0^{11}u_1^5u_2^5+2348810240u_0^{10}u_1^7u_2^4-828375040u_0^9u_1^9u_2^3+173015040u_0^8u_1^{11}u_2^2-19529728u_0^7u_1^{13}u_2+917504u_0^6u_1^{15}+2684354560u_0^{14}u_2^6-1610612736u_0^{13}u_1^2u_2^5+3791650816u_0^{12}u_1^4u_2^4-3623878656u_0^{11}u_1^6u_2^3+1390411776u_0^{10}u_1^8u_2^2-236978176u_0^9u_1^{10}u_2+15073280u_0^8u_1^{12}-5905580032u_0^{14}u_1u_2^4-6979321856u_0^{13}u_1^3u_2^3+7449083904u_0^{12}u_1^5u_2^2-2046820352u_0^{11}u_1^7u_2+178257920u_0^{10}u_1^9+4294967296u_0^{15}u_2^3+19327352832u_0^{14}u_1^2u_2^2-10468982784u_0^{13}u_1^4u_2+1342177280u_0^{12}u_1^6-21474836480u_0^{15}u_1u_2+5368709120u_0^{14}u_1^3+8589934592u_0^{16}=0

复制代码

当\(N=13\)时

-4398046511104u_0^{21}u_2^{21}+23089744183296u_0^{20}u_1^2u_2^{20}-57724360458240u_0^{19}u_1^4u_2^{19}+91396904058880u_0^{18}u_1^6u_2^{18}-102821517066240u_0^{17}u_1^8u_2^{17}+87398289506304u_0^{16}u_1^{10}u_2^{16}-58265526337536u_0^{15}u_1^{12}u_2^{15}+31213674823680u_0^{14}u_1^{14}u_2^{14}-13655982735360u_0^{13}u_1^{16}u_2^{13}+4931327098880u_0^{12}u_1^{18}u_2^{12}-1479398129664u_0^{11}u_1^{20}u_2^{11}+369849532416u_0^{10}u_1^{22}u_2^{10}-77051985920u_0^9u_1^{24}u_2^9+13335920640u_0^8u_1^{26}u_2^8-1905131520u_0^7u_1^{28}u_2^7+222265344u_0^6u_1^{30}u_2^6-20837376u_0^5u_1^{32}u_2^5+1532160u_0^4u_1^{34}u_2^4-85120u_0^3u_1^{36}u_2^3+3360u_0^2u_1^38u_2^2-84u_0u_1^40u_2+u_1^42+52776558133248u_0^{21}u_1u_2^{19}-215504279044096u_0^{20}u_1^3u_2^{18}+484884627849216u_0^{19}u_1^5u_2^{17}-755089610375168u_0^{18}u_1^7u_2^{16}+856244680130560u_0^{17}u_1^9u_2^{15}-722997614739456u_0^{16}u_1^{11}u_2^{14}+464079806267392u_0^{15}u_1^{13}u_2^{13}-231099305295872u_0^{14}u_1^{15}u_2^{12}+91017336324096u_0^{13}u_1^{17}u_2^{11}-28885668331520u_0^{12}u_1^{19}u_2^{10}+7528507244544u_0^{11}u_1^{21}u_2^9-1641994518528u_0^{10}u_1^{23}u_2^8+304200286208u_0^9u_1^{25}u_2^7-48055189504u_0^8u_1^{27}u_2^6+6381895680u_0^7u_1^{29}u_2^5-688586752u_0^6u_1^{31}u_2^4+57257984u_0^5u_1^{33}u_2^3-3400704u_0^4u_1^{35}u_2^2+127232u_0^3u_1^37u_2-2240u_0^2u_1^39-105553116266496u_0^{22}u_2^{18}-46179488366592u_0^{20}u_1^4u_2^{16}+492581209243648u_0^{19}u_1^6u_2^{15}-503026569707520u_0^{18}u_1^8u_2^{14}-184717953466368u_0^{17}u_1^{10}u_2^{13}+766050366914560u_0^{16}u_1^{12}u_2^{12}-761136924327936u_0^{15}u_1^{14}u_2^{11}+441749197553664u_0^{14}u_1^{16}u_2^{10}-171970490531840u_0^{13}u_1^{18}u_2^9+46552949194752u_0^{12}u_1^{20}u_2^8-8584565882880u_0^{11}u_1^{22}u_2^7+948860616704u_0^{10}u_1^{24}u_2^6-19730006016u_0^9u_1^{26}u_2^5-12983992320u_0^8u_1^{28}u_2^4+2436890624u_0^7u_1^{30}u_2^3-222781440u_0^6u_1^{32}u_2^2+11010048u_0^5u_1^{34}u_2-236032u_0^4u_1^{36}+1477743627730944u_0^{22}u_1u_2^{16}+70368744177664u_0^{21}u_1^3u_2^{15}-1847179534663680u_0^{20}u_1^5u_2^{14}-3232564185661440u_0^{19}u_1^7u_2^{13}+10005555812761600u_0^{18}u_1^9u_2^{12}-10430791934803968u_0^{17}u_1^{11}u_2^{11}+5984573070508032u_0^{16}u_1^{13}u_2^{10}-1977660641116160u_0^{15}u_1^{15}u_2^9+262562088222720u_0^{14}u_1^{17}u_2^8+80611167436800u_0^{13}u_1^{19}u_2^7-54546889965568u_0^{12}u_1^{21}u_2^6+15590932611072u_0^{11}u_1^{23}u_2^5-2793909780480u_0^{10}u_1^{25}u_2^4+333090652160u_0^9u_1^{27}u_2^3-25873612800u_0^8u_1^{29}u_2^2+1193017344u_0^7u_1^{31}u_2-24887296u_0^6u_1^{33}-1337006139375616u_0^{23}u_2^{15}-11874725579980800u_0^{22}u_1^2u_2^{14}+4244114883215360u_0^{21}u_1^4u_2^{13}+34257483786616832u_0^{20}u_1^6u_2^{12}-49218263627857920u_0^{19}u_1^8u_2^{11}+25495544344346624u_0^{18}u_1^{10}u_2^{10}+684875485020160u_0^{17}u_1^{12}u_2^9-8699434783211520u_0^{16}u_1^{14}u_2^8+5843808738607104u_0^{15}u_1^{16}u_2^7-2212838286295040u_0^{14}u_1^{18}u_2^6+556525688979456u_0^{13}u_1^{20}u_2^5-96839181271040u_0^{12}u_1^{22}u_2^4+11610483589120u_0^{11}u_1^{24}u_2^3-920285872128u_0^{10}u_1^{26}u_2^2+43567022080u_0^9u_1^{28}u_2-935133184u_0^8u_1^{30}+23221685578629120u_0^{23}u_1u_2^{13}+54711698598133760u_0^{22}u_1^3u_2^{12}-138432911983509504u_0^{21}u_1^5u_2^{11}+58410455713972224u_0^{20}u_1^7u_2^{10}+75500714700308480u_0^{19}u_1^9u_2^9-112234710989537280u_0^{18}u_1^{11}u_2^8+71477189337415680u_0^{17}u_1^{13}u_2^7-28001159545356288u_0^{16}u_1^{15}u_2^6+7371819589828608u_0^{15}u_1^{17}u_2^5-1336719987179520u_0^{14}u_1^{19}u_2^4+165376373555200u_0^{13}u_1^{21}u_2^3-13361039278080u_0^{12}u_1^{23}u_2^2+635663548416u_0^{11}u_1^{25}u_2-13490978816u_0^{10}u_1^{27}-12384898975268864u_0^{24}u_2^{12}-178173660257845248u_0^{23}u_1^2u_2^{11}+117163959055810560u_0^{22}u_1^4u_2^{10}+308654904149278720u_0^{21}u_1^6u_2^9-528029464123146240u_0^{20}u_1^8u_2^8+381158899908083712u_0^{19}u_1^{10}u_2^7-161016331062214656u_0^{18}u_1^{12}u_2^6+43225238061711360u_0^{17}u_1^{14}u_2^5-7445669404999680u_0^{16}u_1^{16}u_2^4+783552358645760u_0^{15}u_1^{18}u_2^3-42574937063424u_0^{14}u_1^{20}u_2^2+417954004992u_0^{13}u_1^{22}u_2+43620761600u_0^{12}u_1^{24}+193654783976931328u_0^{24}u_1u_2^{10}+346214221354106880u_0^{23}u_1^3u_2^9-1095641346846228480u_0^{22}u_1^5u_2^8+981643981278412800u_0^{21}u_1^7u_2^7-413768215764664320u_0^{20}u_1^9u_2^6+72871232642482176u_0^{19}u_1^{11}u_2^5+7042371975905280u_0^{18}u_1^{13}u_2^4-6094043196948480u_0^{17}u_1^{15}u_2^3+1252412463513600u_0^{16}u_1^{17}u_2^2-120130235269120u_0^{15}u_1^{19}u_2+4619237326848u_0^{14}u_1^{21}-73183493944770560u_0^{25}u_2^9-848647054782627840u_0^{24}u_1^2u_2^8+1253971021245972480u_0^{23}u_1^4u_2^7-376824625071390720u_0^{22}u_1^6u_2^6-302013853917511680u_0^{21}u_1^8u_2^5+286081930431037440u_0^{20}u_1^{10}u_2^4-103095707778416640u_0^{19}u_1^{12}u_2^3+19556188689530880u_0^{18}u_1^{14}u_2^2-1939693130219520u_0^{17}u_1^{16}u_2+79650168504320u_0^{16}u_1^{18}+702561541869797376u_0^{25}u_1u_2^7-191402984163246080u_0^{24}u_1^3u_2^6-1202461100507922432u_0^{23}u_1^5u_2^5+1298303330077900800u_0^{22}u_1^7u_2^4-583005045511946240u_0^{21}u_1^9u_2^3+135081600542048256u_0^{20}u_1^{11}u_2^2-15964908835307520u_0^{19}u_1^{13}u_2+764710337118208u_0^{18}u_1^{15}-211669182486413312u_0^{26}u_2^6-790381734603522048u_0^{25}u_1^2u_2^5+2132172948583219200u_0^{24}u_1^4u_2^4-1572037744929013760u_0^{23}u_1^6u_2^3+515363090171166720u_0^{22}u_1^8u_2^2-79507884827738112u_0^{21}u_1^{10}u_2+4724601464553472u_0^{20}u_1^{12}+675539944105574400u_0^{26}u_1u_2^4-1756403854674493440u_0^{25}u_1^3u_2^3+1063975411966279680u_0^{24}u_1^5u_2^2-244883229738270720u_0^{23}u_1^7u_2+19527326509301760u_0^{22}u_1^9-180143985094819840u_0^{27}u_2^3+945755921747804160u_0^{26}u_1^2u_2^2-439100963668623360u_0^{25}u_1^4u_2+53480245575024640u_0^{24}u_1^6-360287970189639680u_0^{27}u_1u_2+90071992547409920u_0^{26}u_1^3+72057594037927936u_0^{28}=0

复制代码

当\(N=14\)时

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复制代码

当\(N=15\)时

-281474976710656u_0^{24}u_2^{24}+1688849860263936u_0^{23}u_1^2u_2^{23}-4855443348258816u_0^{22}u_1^4u_2^{22}+8901646138474496u_0^{21}u_1^6u_2^{21}-11683410556747776u_0^{20}u_1^8u_2^{20}+11683410556747776u_0^{19}u_1^{10}u_2^{19}-9249366690758656u_0^{18}u_1^{12}u_2^{18}+5946021444059136u_0^{17}u_1^{14}u_2^{17}-3158823892156416u_0^{16}u_1^{16}u_2^{16}+1403921729847296u_0^{15}u_1^{18}u_2^{15}-526470648692736u_0^{14}u_1^{20}u_2^{14}+167513388220416u_0^{13}u_1^{22}u_2^{13}-45368209309696u_0^{12}u_1^{24}u_2^{12}+10469586763776u_0^{11}u_1^{26}u_2^{11}-2056525971456u_0^{10}u_1^{28}u_2^{10}+342754328576u_0^9u_1^{30}u_2^9-48199827456u_0^8u_1^{32}u_2^8+5670567936u_0^7u_1^{34}u_2^7-551305216u_0^6u_1^{36}u_2^6+43524096u_0^5u_1^38u_2^5-2720256u_0^4u_1^40u_2^4+129536u_0^3u_1^42u_2^3-4416u_0^2u_1^44u_2^2+96u_0u_1^46u_2-u_1^48+2814749767106560u_0^{24}u_1u_2^{22}-21110623253299200u_0^{23}u_1^3u_2^{21}+59496773202214912u_0^{22}u_1^5u_2^{20}-98269951244107776u_0^{21}u_1^7u_2^{19}+114593300870070272u_0^{20}u_1^9u_2^{18}-104602038708469760u_0^{19}u_1^{11}u_2^{17}+78843917243514880u_0^{18}u_1^{13}u_2^{16}-49897486935916544u_0^{17}u_1^{15}u_2^{15}+26416333793001472u_0^{16}u_1^{17}u_2^{14}-11596300030050304u_0^{15}u_1^{19}u_2^{13}+4194223469363200u_0^{14}u_1^{21}u_2^{12}-1246995436011520u_0^{13}u_1^{23}u_2^{11}+305208362008576u_0^{12}u_1^{25}u_2^{10}-61826691760128u_0^{11}u_1^{27}u_2^9+10471369342976u_0^{10}u_1^{29}u_2^8-1505209876480u_0^9u_1^{31}u_2^7+186513489920u_0^8u_1^{33}u_2^6-20005715968u_0^7u_1^{35}u_2^5+1817370624u_0^6u_1^37u_2^4-132456448u_0^5u_1^39u_2^3+7104000u_0^4u_1^41u_2^2-244480u_0^3u_1^43u_2+4000u_0^2u_1^45-5629499534213120u_0^{25}u_2^{21}+52072870691471360u_0^{24}u_1^2u_2^{20}-28851185112842240u_0^{23}u_1^4u_2^{19}-152664990493442048u_0^{22}u_1^6u_2^{18}+310867121544364032u_0^{21}u_1^8u_2^{17}-282388670873337856u_0^{20}u_1^{10}u_2^{16}+155525919748915200u_0^{19}u_1^{12}u_2^{15}-68634264584847360u_0^{18}u_1^{14}u_2^{14}+39976421719801856u_0^{17}u_1^{16}u_2^{13}-29348774253953024u_0^{16}u_1^{18}u_2^{12}+17870916537024512u_0^{15}u_1^{20}u_2^{11}-7965430241034240u_0^{14}u_1^{22}u_2^{10}+2592599965696000u_0^{13}u_1^{24}u_2^9-620135127711744u_0^{12}u_1^{26}u_2^8+107828274528256u_0^{11}u_1^{28}u_2^7-12997820940288u_0^{10}u_1^{30}u_2^6+923589672960u_0^9u_1^{32}u_2^5-5436538880u_0^8u_1^{34}u_2^4-6608027648u_0^7u_1^{36}u_2^3+727949312u_0^6u_1^38u_2^2-36545536u_0^5u_1^40u_2+759040u_0^4u_1^42-22517998136852480u_0^{25}u_1u_2^{19}-703687441776640000u_0^{24}u_1^3u_2^{18}+1536008947910049792u_0^{23}u_1^5u_2^{17}-838443586876866560u_0^{22}u_1^7u_2^{16}-623537442158280704u_0^{21}u_1^9u_2^{15}+1007240611973038080u_0^{20}u_1^{11}u_2^{14}-342286765819691008u_0^{19}u_1^{13}u_2^{13}-246008030133485568u_0^{18}u_1^{15}u_2^{12}+330971279341387776u_0^{17}u_1^{17}u_2^{11}-181920211769753600u_0^{16}u_1^{19}u_2^{10}+59581633166049280u_0^{15}u_1^{21}u_2^9-11393854599069696u_0^{14}u_1^{23}u_2^8+602161931091968u_0^{13}u_1^{25}u_2^7+346568158871552u_0^{12}u_1^{27}u_2^6-123000280055808u_0^{11}u_1^{29}u_2^5+22062794014720u_0^{10}u_1^{31}u_2^4-2507368562688u_0^9u_1^{33}u_2^3+182789603328u_0^8u_1^{35}u_2^2-7872118784u_0^7u_1^37u_2+153305088u_0^6u_1^39+1936547839769313280u_0^{25}u_1^2u_2^{17}+596726950626590720u_0^{24}u_1^4u_2^{16}-9025213653250473984u_0^{23}u_1^6u_2^{15}+11081106883145105408u_0^{22}u_1^8u_2^{14}-3217962671244574720u_0^{21}u_1^{10}u_2^{13}-4283591748699029504u_0^{20}u_1^{12}u_2^{12}+5240606669515259904u_0^{19}u_1^{14}u_2^{11}-2688297133819297792u_0^{18}u_1^{16}u_2^{10}+647361934986838016u_0^{17}u_1^{18}u_2^9+44330590845009920u_0^{16}u_1^{20}u_2^8-95855973465325568u_0^{15}u_1^{22}u_2^7+38974972785328128u_0^{14}u_1^{24}u_2^6-9447186441961472u_0^{13}u_1^{26}u_2^5+1544045574750208u_0^{12}u_1^{28}u_2^4-172982492200960u_0^{11}u_1^{30}u_2^3+12834167586816u_0^{10}u_1^{32}u_2^2-571165638656u_0^9u_1^{34}u_2+11585191936u_0^8u_1^{36}-2206763817411543040u_0^{26}u_1u_2^{16}-15132094747964866560u_0^{25}u_1^3u_2^{15}+29394994767847227392u_0^{24}u_1^5u_2^{14}+364791569817010176u_0^{23}u_1^7u_2^{13}-41885587596870942720u_0^{22}u_1^9u_2^{12}+45972744996198023168u_0^{21}u_1^{11}u_2^{11}-21525147933063774208u_0^{20}u_1^{13}u_2^{10}+1306026299751399424u_0^{19}u_1^{15}u_2^9+4434262499977592832u_0^{18}u_1^{17}u_2^8-3041436079405137920u_0^{17}u_1^{19}u_2^7+1133547078933282816u_0^{16}u_1^{21}u_2^6-280699889732550656u_0^{15}u_1^{23}u_2^5+48357056113016832u_0^{14}u_1^{25}u_2^4-5774858407378944u_0^{13}u_1^{27}u_2^3+458507891507200u_0^{12}u_1^{29}u_2^2-21852525166592u_0^{11}u_1^{31}u_2+474314964992u_0^{10}u_1^{33}+972777519512027136u_0^{27}u_2^{15}+33966148389628280832u_0^{26}u_1^2u_2^{14}+1952310438465110016u_0^{25}u_1^4u_2^{13}-134818069595102904320u_0^{24}u_1^6u_2^{12}+181813553919179096064u_0^{23}u_1^8u_2^{11}-81695613899849596928u_0^{22}u_1^{10}u_2^{10}-30355818396942073856u_0^{21}u_1^{12}u_2^9+62276582688814006272u_0^{20}u_1^{14}u_2^8-41197389525150597120u_0^{19}u_1^{16}u_2^7+16573803119046033408u_0^{18}u_1^{18}u_2^6-4526338318106886144u_0^{17}u_1^{20}u_2^5+86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复制代码

数学星空 · 发表于 2014-5-3 19:05:12

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 · 发表于 2014-5-3 19:18:23

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 · 发表于 2014-5-3 20:20:16

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
\(u_2=\frac{a^2b^2-a^2n^2+a^2y_0^2+b^2m^2-b^2x_0^2}{a^2b^2}, u_1=\frac{-a^2n^2+b^2m^2-m^2n^2+m^2y_0^2+n^2x_0^2}{a^2b^2}, u_0=-\frac{m^2n^2}{a^2b^2}\)

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