求外接圆的半径

zuijianqiugen

mathe 发表于 2014-4-21 12:33
横向量，就是这个向量横放，这样才可和矩阵左乘

特征向量本身就是横向量，何来还有一个横向量？

补充内容 (2014-4-29 12:07):
是本人理解错了，特征向量是列向量。

数学星空

根据61#mathe的结论，我们很容易得到\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)，与\(x^2+y^2=r^2\),有一个n边形外切内切的条件：




\(n=3\)时

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


\(n=4\)时

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


\(n=5\)时

\((a^3-a^2b-a^2r-ab^2-2abr-ar^2+b^3-b^2r-br^2+r^3)^2(a^3+a^2b-a^2r-ab^2+2abr-ar^2-b^3-b^2r+br^2+r^3)^2(a^3+a^2b+a^2r-ab^2-2abr-ar^2-b^3+b^2r+br^2-r^3)^2(a^3-a^2b+a^2r-ab^2+2abr-ar^2+b^3+b^2r-br^2-r^3)^2=0\)


\(n=6\)时

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



\(n=7\)时

\((a^6+2a^5b-2a^5r-a^4b^2+2a^4br-a^4r^2-4a^3b^3+4a^3r^3-a^2b^4+2a^2b^2r^2-a^2r^4+2ab^5+2ab^4r-2abr^4-2ar^5+b^6-2b^5r-b^4r^2+4b^3r^3-b^2r^4-2br^5+r^6)^2(a^6-2a^5b-2a^5r-a^4b^2-2a^4br-a^4r^2+4a^3b^3+4a^3r^3-a^2b^4+2a^2b^2r^2-a^2r^4-2ab^5+2ab^4r+2abr^4-2ar^5+b^6+2b^5r-b^4r^2-4b^3r^3-b^2r^4+2br^5+r^6)^2(a^6-2a^5b+2a^5r-a^4b^2+2a^4br-a^4r^2+4a^3b^3-4a^3r^3-a^2b^4+2a^2b^2r^2-a^2r^4-2ab^5-2ab^4r+2abr^4+2ar^5+b^6-2b^5r-b^4r^2+4b^3r^3-b^2r^4-2br^5+r^6)^2(a^6+2a^5b+2a^5r-a^4b^2-2a^4br-a^4r^2-4a^3b^3-4a^3r^3-a^2b^4+2a^2b^2r^2-a^2r^4+2ab^5-2ab^4r-2abr^4+2ar^5+b^6+2b^5r-b^4r^2-4b^3r^3-b^2r^4+2br^5+r^6)^2=0\)


\(n=8\)时

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


\(n=9\)时
\((r+a+b)^2(-r+a+b)^2(a-b-r)^2(-b+r+a)^2(a^9+3a^8b-3a^8r-8a^6b^3-4a^6b^2r+4a^6br^2+8a^6r^3-6a^5b^4+12a^5b^2r^2-6a^5r^4+6a^4b^5+14a^4b^4r+8a^4b^3r^2-8a^4b^2r^3-14a^4br^4-6a^4r^5+8a^3b^6-8a^3b^4r^2-8a^3b^2r^4+8a^3r^6-4a^2b^6r-12a^2b^5r^2-8a^2b^4r^3+8a^2b^3r^4+12a^2b^2r^5+4a^2br^6-3ab^8-4ab^6r^2+14ab^4r^4-4ab^2r^6-3ar^8-b^9-3b^8r+8b^6r^3+6b^5r^4-6b^4r^5-8b^3r^6+3br^8+r^9)^2(a^9-3a^8b-3a^8r+8a^6b^3-4a^6b^2r-4a^6br^2+8a^6r^3-6a^5b^4+12a^5b^2r^2-6a^5r^4-6a^4b^5+14a^4b^4r-8a^4b^3r^2-8a^4b^2r^3+14a^4br^4-6a^4r^5+8a^3b^6-8a^3b^4r^2-8a^3b^2r^4+8a^3r^6-4a^2b^6r+12a^2b^5r^2-8a^2b^4r^3-8a^2b^3r^4+12a^2b^2r^5-4a^2br^6-3ab^8-4ab^6r^2+14ab^4r^4-4ab^2r^6-3ar^8+b^9-3b^8r+8b^6r^3-6b^5r^4-6b^4r^5+8b^3r^6-3br^8+r^9)^2(a^9-3a^8b+3a^8r+8a^6b^3+4a^6b^2r-4a^6br^2-8a^6r^3-6a^5b^4+12a^5b^2r^2-6a^5r^4-6a^4b^5-14a^4b^4r-8a^4b^3r^2+8a^4b^2r^3+14a^4br^4+6a^4r^5+8a^3b^6-8a^3b^4r^2-8a^3b^2r^4+8a^3r^6+4a^2b^6r+12a^2b^5r^2+8a^2b^4r^3-8a^2b^3r^4-12a^2b^2r^5-4a^2br^6-3ab^8-4ab^6r^2+14ab^4r^4-4ab^2r^6-3ar^8+b^9+3b^8r-8b^6r^3-6b^5r^4+6b^4r^5+8b^3r^6-3br^8-r^9)^2(a^9+3a^8b+3a^8r-8a^6b^3+4a^6b^2r+4a^6br^2-8a^6r^3-6a^5b^4+12a^5b^2r^2-6a^5r^4+6a^4b^5-14a^4b^4r+8a^4b^3r^2+8a^4b^2r^3-14a^4br^4+6a^4r^5+8a^3b^6-8a^3b^4r^2-8a^3b^2r^4+8a^3r^6+4a^2b^6r-12a^2b^5r^2+8a^2b^4r^3+8a^2b^3r^4-12a^2b^2r^5+4a^2br^6-3ab^8-4ab^6r^2+14ab^4r^4-4ab^2r^6-3ar^8-b^9+3b^8r-8b^6r^3+6b^5r^4+6b^4r^5-8b^3r^6+3br^8-r^9)^2=0\)


\(n=10\)时

\((a^3-a^2b-a^2r-ab^2-2abr-ar^2+b^3-b^2r-br^2+r^3)^2(a^3+a^2b-a^2r-ab^2+2abr-ar^2-b^3-b^2r+br^2+r^3)^2(a^3+a^2b+a^2r-ab^2-2abr-ar^2-b^3+b^2r+br^2-r^3)^2(a^3-a^2b+a^2r-ab^2+2abr-ar^2+b^3+b^2r-br^2-r^3)^2(a^{12}-6a^{10}b^2+6a^{10}r^2+15a^8b^4+14a^8b^2r^2-29a^8r^4-20a^6b^6-20a^6b^4r^2+4a^6b^2r^4+36a^6r^6+15a^4b^8-20a^4b^6r^2+50a^4b^4r^4-36a^4b^2r^6-9a^4r^8-6a^2b^{10}+14a^2b^8r^2+4a^2b^6r^4-36a^2b^4r^6+34a^2b^2r^8-10a^2r^{10}+b^{12}+6b^{10}r^2-29b^8r^4+36b^6r^6-9b^4r^8-10b^2r^{10}+5r^{12})^2(5a^{12}-10a^{10}b^2-10a^{10}r^2-9a^8b^4+34a^8b^2r^2-9a^8r^4+36a^6b^6-36a^6b^4r^2-36a^6b^2r^4+36a^6r^6-29a^4b^8+4a^4b^6r^2+50a^4b^4r^4+4a^4b^2r^6-29a^4r^8+6a^2b^{10}+14a^2b^8r^2-20a^2b^6r^4-20a^2b^4r^6+14a^2b^2r^8+6a^2r^{10}+b^{12}-6b^{10}r^2+15b^8r^4-20b^6r^6+15b^4r^8-6b^2r^{10}+r^{12})^2(a^{12}+6a^{10}b^2-6a^{10}r^2-29a^8b^4+14a^8b^2r^2+15a^8r^4+36a^6b^6+4a^6b^4r^2-20a^6b^2r^4-20a^6r^6-9a^4b^8-36a^4b^6r^2+50a^4b^4r^4-20a^4b^2r^6+15a^4r^8-10a^2b^{10}+34a^2b^8r^2-36a^2b^6r^4+4a^2b^4r^6+14a^2b^2r^8-6a^2r^{10}+5b^{12}-10b^{10}r^2-9b^8r^4+36b^6r^6-29b^4r^8+6b^2r^{10}+r^{12})^2=0\)

mathe

你还可以做些优化：重因子可以去除，还有比如n=6对应的结果中可以去除n=3和n=2对应的因式，此外，所有因式中，只需要保留关于a,b对称的

mathe

于是n=6变成$a^4-2a^2b^2+2a^2r^2+b^4+2b^2r^2-3r^4=0$

mathe

如楼上所说，由于结果是关于a,b对称，我们可以写成a+b和ab的函数，然后再使用54#的方法将它们用$1/u$替换，得到$1/u$的另外一个多项式，然后再利用54#的多项式消去u就得到参数的约束条件,另外54#关于$1/u$的多项式是首1的，消去这个变量相对容易。

zuijianqiugen

数学星空 发表于 2014-4-21 20:12
根据61#mathe的结论，我们很容易得到\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)，与\(x^2+y^2=r^2\),有一个n边 ...

这次数学星空办了一件有实用价值的事，特表支持。

zuijianqiugen

mathe 发表于 2014-4-21 20:48
你还可以做些优化：重因子可以去除，还有比如n=6对应的结果中可以去除n=3和n=2对应的因式，此外，所有因式 ...

mathe大师说的有道理。那到底哪个因式才是弦切凸n边形的因式呢？

数学星空

由于n为奇数时，消元出来的结果为关于\(u_1,u_2,u_3\)为对称式，因此较易算出

\(n=3\)时

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


\(n=5\)时

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


\(n=7\)时

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


\(n=9\)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\)

mathe

n为偶数时结果应该更加简单。它总会有三组轮换对称的表达式，而我们只要使用其中一种即可

mathe

偶数结果能贴出来吗？另外两圆的情况是不是可以简化很多？