{"id":734,"date":"2014-01-28T17:22:16","date_gmt":"2014-01-28T16:22:16","guid":{"rendered":"http:\/\/darksun190.cn\/?p=734"},"modified":"2022-12-17T16:02:51","modified_gmt":"2022-12-17T08:02:51","slug":"%e6%9c%80%e5%b0%8f%e4%ba%8c%e4%b9%98%e6%b3%95%e6%8b%9f%e5%90%88%e4%b8%80%e4%ba%9b%e5%9f%ba%e6%9c%ac%e7%a9%ba%e9%97%b4%e5%85%83%e7%b4%a0%e7%9a%84c%e5%ae%9e%e7%8e%b0-%ef%bc%88-3","status":"publish","type":"post","link":"http:\/\/darksun190.cn\/blog\/?p=734","title":{"rendered":"\u6700\u5c0f\u4e8c\u4e58\u6cd5\u62df\u5408\u4e00\u4e9b\u57fa\u672c\u7a7a\u95f4\u5143\u7d20\u7684C#\u5b9e\u73b0 \u2014\u2014 \uff08\u4e09\uff09\u5706\u67f1"},"content":{"rendered":"\n

\u540c\u524d\u4e00\u7bc7\u6587\u7ae0\u76f8\u540c\uff0c\u7b97\u6cd5\u4e3b\u8981\u6839\u636e\u8fd9\u7bc7\u6587\u7ae0<\/a>\uff0c\u4fee\u6539\u4e86\u5176\u4e2d\u4e00\u4e9b\u7b14\u8bef\u5e76\u5fae\u8c03\u4ee5\u9002\u5408C#\u4ee3\u7801\u3002<\/p>\n\n\n\n

\u9996\u5148\u662f\u5c06\u5706\u67f1\u53c2\u6570\u5316\u5e76\u7b80\u5316\u8be5\u6a21\u578b\u3002<\/p>\n\n\n\n

\u5706\u67f1\u7684\u8f74\u7ebf\u4e3a\u4e00\u7a7a\u95f4\u76f4\u7ebf\uff0c\u7531\u4e00\u70b9\\(P\\)\u548c\u4e00\u7a7a\u95f4\u5411\u91cf\\(V\\)\u5b9a\u4e49\u3002<\/p>\n\n\n\n

\u5982\u679c\u7ecf\u8fc7\u5750\u6807\u53d8\u6362\uff0c\u65b0\u5750\u6807\u7cfb\u96f6\u70b9\u4e3a\\(P\\)\u4e14\u5750\u6807z\u8f74\u4e3a\\(V\\)\uff0c\u90a3\u4e48\u8be5\u8f74\u7ebf\u7684\u672a\u77e5\u91cf\u5c31\u7531\\(x_0,y_0,z_0,a,b,c\\)\u51cf\u5c11\u4e3a4\u4e2a\u3002<\/p>\n\n\n\n

\u53c2\u6570\u5316<\/h3>\n\n\n\n

\u5706\u67f1\u7684\u8f74\u7ebf\u8868\u793a\u4e3a\u7ecf\u8fc7\u70b9\\(P\\left(x_0,y_0,z_0\\right)\\)\u5411\u91cf\u4e3a\\(V\\left(a,b,c\\right)\\)\u7684\u76f4\u7ebf\uff0c\u534a\u5f84\u4e3aR<\/p>\n\n\n\n

\u6c42\u521d\u59cb\u503c<\/h3>\n\n\n\n

\u6211\u4e2a\u4eba\u89c9\u5f97\u8fd9\u79cd\u6c42\u6cd5\u504f\u7e41\u7410\uff0c\u4e0d\u8fc7\u7684\u786e\u76f8\u5f53\u63a5\u8fd1\u6700\u540e\u62df\u5408\u503c\uff0c\u8fed\u4ee3\u6b21\u6570\u76f8\u5f53\u5c11\u3002<\/p>\n\n\n\n

\\(a,b,c\\)\u7684\u521d\u59cb\u503c<\/h4>\n\n\n\n

\u5bf9\u4e8e\u5706\u67f1\u9762\u4e0a\u4efb\u4e00\u70b9\\(P_i\\left(x_i,y_i,z_i\\right)\\)\u6709\\[A^2+B^2+C^2=R^2\\]\u5176\u4e2d\\[A=c\\left(y_i-y_0\\right)-b\\left(z_i-z_0\\right)\\]\\[B=a\\left(z_i-z_0\\right)-c\\left(x_i-x_0\\right)\\]\\[C=b\\left(x_i-x_0\\right)-a\\left(y_i-y_0\\right)\\]<\/p>\n\n\n\n

\u5c06\u4e0a\u9762\u7684\u7b49\u5f0f\u4ee3\u5165\u5c55\u5f00\\[\\left[c\\left(y_i-y_0\\right)-b\\left(z_i-z_0\\right)\\right]^2+\\left[a\\left(z_i-z_0\\right)-c\\left(x_i-x_0\\right)\\right]^2+\\left[b\\left(x_i-x_0\\right)-a\\left(y_i-y_0\\right)\\right]^2=R^2\\]\u8fd9\u4e2a\u7b49\u5f0f\u5f88\u7b80\u5355\uff08\u4e0d\u5f97\u4e0d\u5410\u69fd\uff0c\u5f53\u65f6\u770b\u5230\u8fd9\u91cc\u6211\u90fd\u75af\u4e86\uff0c\u73b0\u5728\u624b\u6253\u8fd9\u4e9b\u516c\u5f0f\u66f4\u75af\u4e86\uff09\uff0c\u5b83\u7b49\u4ef7\u4e8e\\[Ax_i^2+By_i^2+Cz_i^2+Dx_iy_i+Ex_iz_i+Fy_iz_i+Gx_i+Hy_i+Iz_i+J=0\\]\u5176\u4e2d\\begin{array}{lclp{12}}
A&=&\\left(b^2+c^2\\right)\\\\
B&=&\\left(a^2+c^2\\right)\\\\
C&=&\\left(a^2+b^2\\right)\\\\
D&=&-2ab\\\\
E&=&-2ac\\\\
F&=&-2bc\\\\
G&=&-2\\left(b^2+c^2\\right)x_0+2aby_0+2acz_0\\\\
H&=&2abx_0-2\\left(a^2+c^2\\right)y_0+2bcz_0\\\\
I&=&2acx_0+2bcy_0-2\\left(a^2+b^2\\right)z_0\\\\
J&=&\\left(b^2+c^2\\right)x_0^2+\\left(a^2+c^2\\right)y_0^2+\\left(a^2+b^2\\right)z_0^2-2bcy_0z_0-2acz_0x_0-2abx_0y_0-R^2
\\end{array}<\/p>\n\n\n\n

\u4e24\u8fb9\u540c\u9664\u4ee5\\(A\\)\u5e76\u5c06\\(x\\)\u4e8c\u6b21\u9879\u79fb\u5230\u7b49\u5f0f\u53f3\u8fb9\u5f97
\\[\\frac{B}{A}y_i^2+\\frac{C}{A}z_i^2+\\frac{D}{A}x_iy_i+\\frac{E}{A}x_iz_i+\\frac{F}{A}y_iz_i+\\frac{G}{A}x_i+\\frac{G}{A}y_i+\\frac{I}{A}z_i+\\frac{J}{A}=-x^2\\]<\/p>\n\n\n\n

\u8fd9\u662f\u4e00\u4e2a\u7ebf\u6027\u65b9\u7a0b\u7ec4
\\[\\begin{bmatrix}y_1^2 &z_1^2 & x_1y_1 & x_1z_1 & y_1z_1 & x_1 & y_1 & z_1 & 1 \\\\y_2^2 & z_2^2 & x_2y_2 & x_2z_2 & y_2z_2 & x_2 & y_2 & z_2 & 1 \\\\\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots \\\\y_n^2 &z_n^2 & x_ny_n & x_nz_n & y_nz_n & x_n & y_n & z_n & 1 \\end{bmatrix}\\begin{bmatrix}B\/A\\\\C\/A\\\\D\/A\\\\E\/A\\\\F\/A\\\\G\/A\\\\H\/A\\\\I\/A\\\\J\/A\\end{bmatrix}=\\begin{bmatrix}-x_1^2\\\\-x_2^2\\\\\\vdots\\\\-x_n^2\\end{bmatrix}\\]<\/p>\n\n\n\n

\u6c42\u89e3\uff1a<\/p>\n\n\n\n

\u4ee4\\(B’=B\/A,C’=C\/A,D’=D\/A,E’=E\/A,F’=F\/A,G’=G\/A,H’=H\/A,I’=I\/A,J’=J\/A\\)
\u5982\u679c\\(|D’|\\)\uff0c\\(|E’|\\)\u548c\\(|F’|\\)\u8fd1\u4f3c\u7b49\u4e8e0\uff08\u610f\u5473\u7740abc\u4e2d\u6709\u4e24\u4e2a\u8fd1\u4f3c\u4e8e0\uff0c\u5411\u91cf\u4e0e\u67d0\u5750\u6807\u8f74\u5e73\u884c\uff09\uff0c\u5982\u679c\\(B’\\)\u8fd1\u4f3c\u4e8e1\u5411\u91cf\u4e3a\\(\\left(0,0,1\\right)\\)\uff0c\u5982\u679c\\(C’\\)\u8fd1\u4f3c\u4e8e1\u5411\u91cf\u4e3a\\(\\left(0,1,0\\right)\\)<\/p>\n\n\n\n

\u5426\u5219\uff0c\u4ee4\\[k=\\frac{2}{1+B’+C’}\\]\u5219\\[A=k,B=kB’,C=kC’,D=kD’,E=kE’,F=kF’,G=kG’,H=kH’,I=kI’,J=kJ’\\]\u5982\u679c\\(A,B\\)\u8fd1\u4f3c\u4e8e1\uff0c\u5219\\begin{array}{lcl}c’&=&\\left(1-C\\right)^{1\/2}\\\\a’&=&E\/-2c’\\\\b’&=&F\/-2c’\\end{array}
\u5982\u679cA\u8fd1\u4f3c1\uff0cB\u4e0d\u8fd1\u4f3c1\u5219\\begin{array}{lcl}b’&=&\\left(1-B\\right)^{1\/2}\\\\a’&=&D\/-2b’\\\\c’&=&F\/-2b’\\end{array}
\u5982\u679cA\u4e0d\u8fd1\u4f3c\u4e8e1\\begin{array}{lcl}a’&=&\\left(1-A\\right)^{1\/2}\\\\b’&=&D\/-2a’\\\\c’&=&E\/-2a’\\end{array}
\u5c06\u5411\u91cf\\(\\left(a’,b’,c’\\right)\\)\u6b63\u89c4\u5316\u5f97\u5230\u5355\u4f4d\u5411\u91cf\\[\\left(a,b,c\\right)\\]<\/p>\n\n\n\n

\u6ce8\uff1a\u8fd1\u4f3c\u4e8e\u8fd9\u4e2a\u6982\u5ff5\u5f88\u6a21\u7cca\uff0c\u7531\u4e8e\u6211\u4f7f\u7528\u5728\u673a\u68b0\u6d4b\u91cf\u4e2d\uff0c\u901a\u5e38\u8bbe\u5907\u91c7\u96c6\u539f\u59cb\u70b9\u7684\u7cbe\u5ea6\u4e3a0.001\u5de6\u53f3\uff0c\u800c\u70b9\u4e91\u672c\u8eab\u7684\u6807\u51c6\u5dee\u57280.001~1\uff0c\u6240\u4ee5\u6211\u5728\u7a0b\u5e8f\u4e2d\u5c06\u8fd1\u4f3c\u4e8e\u5b9a\u4e49\u4e3a\\(\\Delta<0.001\\)
\u4e0a\u6587\u4e24\u6b21\u63d0\u5230\u8fd1\u4f3c\u4e8e\uff0c\u7b2c\u4e00\u6b21\u662f\\(DEF\\)\uff0c\u7528\u6765\u5224\u65ad\u5411\u91cf\u662f\u5426\u5e73\u884c\u4e8e\u67d0\u4e00\u8f74\u3002\u7b2c\u4e8c\u6b21\u662f\\(A,B\\)\uff0c\u5b9e\u9645\u4e0a\u6b64\u65f6\u5df2\u7ecf\u6392\u9664\u4e86\u5411\u91cf\u5e73\u884c\u4e8e\u8f74\u7ebf\u7684\u53ef\u80fd\u6027\uff0c\u8fd9\u91cc\u662f\u7528\u6765\u5224\u65ad\u5411\u91cf\u66f4\u63a5\u8fd1\u4e8e\u54ea\u4e2a\u8f74\uff0c\u4ece\u800c\u907f\u514d\u8fd1\u4f3c0 \u7684\u5c0f\u6570\u5f00\u5e73\u65b9\u9020\u6210\u7684\u7cbe\u5ea6\u8bef\u5dee\uff0c\u6240\u4ee5\u5728\u7b97\u6cd5\u4e2d\u6211\u5c06\u8fd9\u90e8\u5206\u6539\u4e3a\u4e86\u6bd4\u5927\u5c0f\u3002
\u53e6\u5916\u5c31\u662f\\(ABC\\)\u4e09\u4e2a\u503c\u6709\u53ef\u80fd\u7565\u5fae\u5927\u4e8e1\uff0c\u7528\u539f\u6587\u7b97\u6cd5\u6709\u53ef\u80fd\u51fa\u73b0\u8d1f\u6570\u5f00\u65b9\u9020\u6210\u65e0\u89e3\u3002<\/p>\n\n\n\n

\\(x_0,y_0,z_0\\)\u7684\u521d\u59cb\u503c<\/h4>\n\n\n\n

\u6839\u636e\\(x_0,y_0,z_0\\)\u4e0e\\(G,H,I\\)\u7684\u5173\u7cfb\uff0c\u4ee5\u53ca\u7b49\u5f0f\\(ax_0+by_0+cz_0=0\\)\uff08\u6ce8\uff1a\u539f\u6587\u4e2d\u4f7f\u7528\u539f\u70b9\u5230\u8f74\u7ebf\u7684\u5782\u8db3\u4f5c\u4e3a\u8d77\u59cb\u70b9\uff0c\u56e0\u4e3a\u4e2a\u4eba\u9700\u8981\uff0c\u6211\u5c06\u4e4b\u66ff\u6362\u4e3a\u6570\u636e\u7b2c\u4e00\u70b9\u6295\u5f71\u5230\u8f74\u7ebf\u7684\u5782\u8db3\\(ax_0+by_0+cz_0=ax_1+by_1+cz_1\\)\u5f97\u5230\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/p>\n\n\n\n

\\[\\begin{bmatrix}-2\\left(b^2+c^2\\right)&2ab&2ac\\\\2ab&-2\\left(a^2+c^2\\right)&2bc\\\\2ac&2bc&-2\\left(a^2+b^2\\right)\\\\a&b&c\\end{bmatrix}\\begin{bmatrix}x_0\\\\y_0\\\\z_0\\end{bmatrix}=\\begin{bmatrix}G\\\\H\\\\I\\\\ax_1+by_1+cz_1\\end{bmatrix}\\]<\/p>\n\n\n\n

\u6ce8\uff1a\u5728\u5b9e\u9645\u4f7f\u7528\u4e2d\uff0c\u8fd9\u4e2a\u521d\u59cb\u503c\u7b97\u6cd5\u6709\u4e00\u4e9b\u95ee\u9898\uff0c\u5f53\u5706\u67f1\u7684\u8f74\u7ebf\u975e\u5e38\u63a5\u8fd1\u4e8e\\((0,0,1)\\)\u65f6\uff08\u5f88\u5e38\u89c1\uff0c\u901a\u5e38\u4e3a\u4e86\u4eea\u5668\u6d4b\u91cf\u65b9\u4fbf\u90fd\u4f1a\u5148\u627e\u6b63\uff09\uff0c\u8be5\u65b9\u7a0b\u7ec4\u53d8\u4e3a\u4e86\\[\\begin{bmatrix}-2&0&0\\\\0&-2&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}\\begin{bmatrix}x_0\\\\y_0\\\\z_0\\end{bmatrix}=\\begin{bmatrix}G\\\\H\\\\I\\\\z_1\\end{bmatrix}\\]\u8fd9\u4e2a\u4e00\u770b\u5c31\u77e5\u9053\\(x_0=-\\frac{G}{2},y_0=-\\frac{H}{2},z_0=z_1\\)\uff0c\u4f46\u662f\u7531\u4e8e\u7b2c\u4e09\u884c\u90fd\u662f0\u7684\u5b58\u5728\uff0c\u4f7f\u5f97\u77e9\u9635\u7684\u4efb\u4f55\u64cd\u4f5c\u90fd\u6709\u95ee\u9898\u3002<\/p>\n\n\n\n

\\(r\\)\u7684\u521d\u59cb\u503c<\/h4>\n\n\n\n

\u539f\u6587\u4f7f\u7528\u4e86\u524d\u9762\\(J\\)\u7684\u7b49\u5f0f\u3002\\[R^2=(b^2+c^2)x_0^2+(a^2+c^2)y_0^2+(a^2+b^2)z_0^2-2bcy_0z_0-2acx_0z_0-2abx_0y_0-J\\]<\/p>\n\n\n\n

\u6ce8\uff1a\u4f9d\u7136\u662f\u5728\u6211\u7684\u5b9e\u9645\u4f7f\u7528\u4e2d\uff0c\u53d1\u73b0J\u7684\u89e3\u5728\u7279\u5b9a\u60c5\u51b5\u4e0b\u4f1a\u662f\u6781\u5927\u7684\u6b63\u6570\uff08\u53ef\u80fd\u662f\u7531\u4e8e\u5706\u67f1\u7684\u8f74\u7ebf\u957f\u5ea6\u5c0f\u4e8e\u76f4\u5f84\u7684\u539f\u56e0\uff09\uff0c\u4e8e\u662f\u53c8\u51fa\u73b0\u4e86\u8d1f\u6570\u5f00\u65b9\u65e0\u89e3\u7684\u60c5\u51b5\uff0c\u4f8b\u5982\u8fd9\u7ec4\u6570\u636e<\/a>\uff0c\u4e8e\u662f\u6211\u7528\u4e86\u6240\u6709\u6d4b\u91cf\u70b9\u534a\u5f84\u5e73\u5747\u503c\u6765\u4ee3\u66ff\u3002<\/p>\n\n\n\n

\u8fed\u4ee3\u7b97\u6cd5<\/h3>\n\n\n\n
1.\u5750\u6807\u53d8\u6362<\/h6>\n\n\n\n

\u5728\\((x_0,y_0,z_0)\\)\u5904\u4ee5\\((a,b,c)\\)\u4e3aZ\u8f74\u5efa\u7acb\u4e00\u4e2a\u5750\u6807\u7cfbU\u5e76\u5c06\u6240\u6709\u5b9e\u6d4b\u70b9\u7684\u4e00\u4e2a\u526f\u672c\u53d8\u6362\u5230\u8be5\u5750\u6807\u7cfb\u4e0b\u3002<\/p>\n\n\n\n

\\[\\begin{bmatrix}x_i\\\\y_i\\\\z_i\\end{bmatrix}=\\left(\\begin{bmatrix}x_i\\\\y_i\\\\z_i\\end{bmatrix}-\\begin{bmatrix}x_0\\\\y_0\\\\z_0\\end{bmatrix}\\right)U\\]<\/p>\n\n\n\n

\u5176\u4e2d\\[U=\\begin{bmatrix}1&0&0\\\\0&c_1&s_1\\\\0&-s1&c1\\end{bmatrix}\\begin{bmatrix}c_2&0&s2\\\\0&1&0\\\\-s2&0&c2\\end{bmatrix}\\]<\/p>\n\n\n\n

\u5f53\\((a,b,c)\\)\u4e3a\\((1,0,0)\\)\u65f6\\(s_1=0,c_1=1,s_2=-1,c_2=0\\)\u5426\u5219\\[\\begin{array}{lcl}c_1&=&\\frac{c}{\\sqrt{b^2+c^2}}\\\\s_1&=&\\frac{-b}{\\sqrt{b^2+c^2}}\\\\c_2&=&\\frac{cc_1-b_s1}{\\sqrt{a^2+(cc_1-bs_1)^2}}\\\\s_2&=&\\frac{-a}{\\sqrt{a^2+(cc_1-bs_1)^2}}\\end{array}\\]<\/p>\n\n\n\n

2.\u6784\u9020Jacobian\u77e9\u9635<\/h6>\n\n\n\n

\u53c2\u8003\u76f4\u7ebf\u8ddd\u79bb\u516c\u5f0f\uff0c\u5e76\u4e14\u5c06\\(a=b=0,c=1,z=0\\)\u4ee3\u5165\u5f97\u5230\\[\\begin{bmatrix}-x_1\/r_1&-y_1\/r_1&-x_1z_1\/r_1&-y_1z_1\/r_1&-1\\\\-x_2\/r_2&-y_2\/r_2&-x_2z_2\/r_2&-y_2z_2\/r_2&-1\\\\\\vdots&\\vdots&\\vdots&\\vdots&\\vdots\\\\-x_m\/r_m&-y_m\/r_m&-x_mz_m\/r_m&-y_mz_m\/r_m&-1\\end{bmatrix}\\begin{bmatrix}P_{x0}\\\\P_{y0}\\\\P_a\\\\P_b\\\\P_r\\end{bmatrix}=-1\\begin{bmatrix}r_1-r\\\\r_2-r\\\\\\vdots\\\\r_m-r\\end{bmatrix}\\]<\/p>\n\n\n\n

3.\u66f4\u65b0\u53c2\u6570<\/h6>\n\n\n\n

\\[\\begin{bmatrix}x_0\\\\y_0\\\\z_0\\end{bmatrix}=\\begin{bmatrix}x_0\\\\y_0\\\\z_0\\end{bmatrix}+\\begin{bmatrix}P_{x_0}\\\\P_{y_0}\\\\-P_{x_0}P_a-P_{y_0}P_b\\end{bmatrix}U^T\\]<\/p>\n\n\n\n

\\[\\begin{bmatrix}a\\\\b\\\\c\\end{bmatrix}=\\begin{bmatrix}P_a\\\\P_b\\\\1\\end{bmatrix}U^T\\]<\/p>\n\n\n\n

\\[r=r+P_r\\]<\/p>\n\n\n\n

4.\u786e\u5b9a\u8fed\u4ee3\u7ed3\u675f\u6761\u4ef6<\/h6>\n\n\n\n

\u91cd\u590d1~3\uff0c\u76f4\u5230\u6ee1\u8db3\u7ec8\u6b62\u6761\u4ef6\u3002<\/p>\n\n\n\n

\u7ec8\u6b62\u6761\u4ef63\u90091\uff1a<\/p>\n\n\n\n