\u5b9e\u9645\u4e0a\u5750\u6807\u6d4b\u91cf\u91cc\u7814\u7a76\u7684\u53d8\u6362\u53ea\u6709\u4e24\u79cd\uff0c\u4e00\u79cd\u662f\u65cb\u8f6c\uff0c\u4e0a\u4e00\u7bc7\u6587\u7ae0\u90a3\u6837\u7684\uff0c\u53e6\u5916\u4e00\u79cd\u5c31\u662f\u5e73\u79fb\u3002<\/p>\n
\u57fa\u672c\u5750\u6807\u7cfb\uff08\u81ea\u7136\u57fa\uff09<\/p>\n
\\(O=\\begin{bmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{bmatrix}\\)<\/p>\n
\u70b9P<\/p>\n
\\(\\begin{bmatrix}4\\\\1\\\\0\\\\1\\end{bmatrix}\\)<\/p>\n
\u4ee3\u8868\u5750\u6807\u7cfb\u7684\u77e9\u9635\u7531\\(3*3\\)\u53d8\u6210\u4e86\\(4*4\\)\uff0c\u5750\u6807\u7684\u5217\u5411\u91cf\u4e5f\u75313\u4e2a\u5143\u7d20\u52a0\u4e86\u4e00\u4e2a1\u3002<\/p>\n
\u867d\u7136\u4ecb\u7ecd\u4eff\u5c04\u53d8\u6362\u65b9\u6cd5\u7684\u6587\u7ae0\u5f88\u591a\uff0c\u4f46\u662f\u539f\u7406\u6027\u7684\u4ecb\u7ecd\u6ca1\u627e\u5230\uff0c\u8fd9\u79cd\u8868\u8fbe\u5f62\u5f0f\u610f\u5473\u7740\u7a7a\u95f4\u75313\u7ef4\u53d8\u6210\u4e864\u7ef4\uff0c\u867d\u7136\u7b2c\u56db\u7ef4\u7684\u5750\u6807\u6c38\u8fdc\u662f1\uff0c\u5982\u679c\u6709\u54ea\u4f4d\u670b\u53cb\u6709\u76f8\u5173\u6587\u7ae0\u63d0\u4f9b\u9605\u8bfb\uff0c\u611f\u6fc0\u4e0d\u5c3d\u3002<\/p>\n
\u5148\u5c06\u5750\u6807\u7cfb\u5ef6\u57fa\u672c\u5750\u6807\u7cfb\u5e73\u79fb\\([0,-3,0]\\)\uff0c\u518d\u5ef6Z\u8f74\u65cb\u8f6c45\u5ea6\uff0c\u5728\u65b0\u5750\u6807\u7cfb<\/p>\n
\\(A=\\begin{bmatrix}\\frac{\\sqrt{2}}{2}&-\\frac{\\sqrt{2}}{2}&0&0\\\\\\frac{\\sqrt{2}}{2}&\\frac{\\sqrt{2}}{2}&0&-3\\\\0&0&1&0\\\\0&0&0&1\\end{bmatrix}\\)<\/p>\n
\u4e0b\u7684\u5750\u6807\u4e3a\u70b9<\/p>\n
\\(P_{A}=\\begin{bmatrix}4\\sqrt{2}\\\\0\\\\0\\\\1\\end{bmatrix}\\)<\/p>\n
\u6ee1\u8db3<\/p>\n
\\[OP=AP_{A}\\]<\/p>\n
\u65b0\u5750\u6807\u7cfb\u7684\u8fc7\u6e21\u77e9\u9635\uff08\u76f8\u5bf9\u4e8e\u81ea\u7136\u57fa\uff09\u548c\u4e4b\u524d\u7684\u76f8\u6bd4\uff0c\u52a0\u4e86\u4e00\u5217\uff08\u57fa\u7684\u4e2a\u6570+1\uff09\uff0c\u662f\u65b0\u5750\u6807\u7cfb0\u70b9\u5728\u57fa\u672c\u5750\u6807\u7cfb\u7684\u5750\u6807\u503c\u3002<\/p>\n
\u5c06\u57fa\u672c\u5750\u6807\u7cfb\u5e73\u79fb\\([-10,-10,-10]\\)\u518d\u5ef6Y\u8f74\u65cb\u8f6c90\u5ea6\u3002<\/p>\n
\\(B=\\begin{bmatrix}0&0&1&-10\\\\0&1&0&-10\\\\-1&0&0&-10\\\\0&0&0&1\\end{bmatrix}\\)<\/p>\n
\u5728B\u5750\u6807\u7cfb\u4e0bP\u70b9\u5750\u6807<\/p>\n
\\(P_{B}=B^{-1}P=\\begin{bmatrix}-10\\\\11\\\\14\\\\1\\end{bmatrix}\\)<\/p>\n
\u6309\u5411\u91cf\u7a7a\u95f4\u7684\u5b9a\u4e49\uff0c\u7531A\u5230B\u7684\u8fc7\u6e21\u77e9\u9635\u4e3a\\(A^{-1}B\\)<\/p>\n
\\(\\begin{bmatrix}0&\\frac{\\sqrt{2}}{2}&\\frac{\\sqrt{2}}{2}&-12.0208\\\\0&\\frac{\\sqrt{2}}{2}&-\\frac{\\sqrt{2}}{2}&2.12132\\\\-1&0&0&-10\\\\0&0&0&1\\end{bmatrix}\\)<\/p>\n
<\/p>\n
B\u5230A\u7684\u8fc7\u6e21\u77e9\u9635\\(Q^{-1}\\)\u4e3a<\/p>\n
\\(\\begin{bmatrix}0&0&-1&-10\\\\\\frac{\\sqrt{2}}{2}&\\frac{\\sqrt{2}}{2}&0&7\\\\\\frac{\\sqrt{2}}{2}&-\\frac{\\sqrt{2}}{2}&0&10\\\\0&0&0&1\\end{bmatrix}\\)<\/p>\n
\u8fc7\u6e21\u77e9\u9635\u7684\u6700\u540e\u4e00\u884c\u4f9d\u65e7\u662f\\(\\begin{bmatrix}0,0,0,1\\end{bmatrix}\\)<\/p>\n
\u8fd9\u6837\u70b9\u7684\u53d8\u6362\u5728\u5e73\u79fb\u548c\u65cb\u8f6c\u5750\u6807\u7cfb\u7684\u6570\u5b66\u5173\u7cfb\u5c31\u6574\u7406\u597d\u4e86\u3002<\/p>\n
\u5728\u5411\u91cf\u7a7a\u95f4\u91cc\uff0c\u6240\u6709\u7684\u5411\u91cf\u8868\u793a\u7684\u90fd\u662f\u5411\u91cf\uff08\u65e2\u6709\u65b9\u5411\uff0c\u53c8\u6709\u957f\u5ea6-\u5927\u5c0f\uff09\uff0c\u8fd9\u548c\u7a7a\u95f4\u4e0a\u7684\u70b9\u5750\u6807\u662f\u7c7b\u4f3c\u7684\uff0c\u4f46\u662f\u5728\u6d4b\u91cf\u6216\u8005\u5efa\u6a21\u65f6\uff0c\u70b9\u8fd8\u6709\u4e00\u4e2a\u7279\u6027\u5c31\u662f\u70b9\u7684\u6cd5\u5411\uff08\u77e2\u91cf\uff09\uff0c\u8fd9\u4e2a\u53c2\u6570\u53ea\u6709\u65b9\u5411\uff0c\u6ca1\u6709\u5927\u5c0f\uff08\u603b\u662f\u5355\u4f4d\u957f\u5ea61\uff09\u3002<\/p>\n
\u5f53\u5750\u6807\u7cfb\u5e73\u79fb\u65f6\uff0c\u8fd9\u4e2a\u6cd5\u5411\u662f\u4e0d\u4f1a\u968f\u4e4b\u5e73\u79fb\u7684\uff0c\u53ea\u80fd\u5f53\u5750\u6807\u7cfb\u65cb\u8f6c\u65f6\uff0c\u6cd5\u5411\u624d\u4f1a\u968f\u4e4b\u65cb\u8f6c\u3002<\/p>\n
\u6240\u4ee5\u5bf9\u4e8e\u6cd5\u5411\uff0c\u5750\u6807\u7684\u6700\u540e\u4e00\u4e2a\u503c\u6c38\u8fdc\u4e3a0\uff0c\u8fd9\u6837\u5728\u77e9\u9635\u53d8\u6362\u65f6\uff0c\u5c31\u4e0d\u4f1a\u53d7\u6700\u540e\u57fa\u7684\u6700\u540e\u4e00\u4e2a\u5217\u5411\u91cf\u5f71\u54cd\u4e86\u3002<\/p>\n
\u7b97\u662f\u5199\u5b8c\uff0c\u6d88\u8017\u65f6\u95f4\u5927\u7ea6\u662f2\u4e2a\u665a\u4e0a\u52a0\u5468\u516d\u5468\u65e5\u7684\u4e1a\u4f59\u65f6\u95f4\uff0c\u5f53\u7136\u5468\u516d\u5468\u65e5\u6211\u90fd\u6709\u65e9\u4e0a\u6253\u7bee\u7403\uff0c\u5468\u516d\u8fd8\u53bb\u7406\u4e86\u53d1\u3002\u4e4b\u6240\u4ee5\u8981\u628a\u8fd9\u4e9b\u5f88\u7b80\u5355\u7684\u4e1c\u897f\u6574\u7406\u4e00\u904d\uff0c\u800c\u4e0d\u662f\u628a\u7f51\u4e0a\u73b0\u6210\u7684\u6750\u6599\u62f7\u8d1d\u4fdd\u7559\u4e0b\u6765\uff0c\u539f\u56e0\u6709\u4e24\u4e2a\u3002<\/p>\n
1.\u81ea\u5df1\u6574\u7406\u4e00\u904d\uff0c\u628a\u81ea\u4ee5\u4e3a\u7406\u89e3\u7684\u4e1c\u897f\u8bb2\u4e00\u904d\uff0c\u53ef\u4ee5\u6392\u67e5\u4e00\u4e0b\u81ea\u5df1\u662f\u5426\u771f\u6b63\u7406\u89e3\u4e86\uff0c\u800c\u4e14\u7406\u89e3\u7684\u6bd4\u8f83\u5168\u9762\u4e86\u3002<\/p>\n
2.2013\u5e74\u6211\u7b2c\u4e00\u6b21\u5c1d\u8bd5\u7528C#\u89e3\u51b3\u5b9e\u9645\u7684\u95ee\u9898\uff0c\u5728\u7528\u4ee3\u7801\u5b9e\u73b0\u7684\u8fc7\u7a0b\u4e2d\uff0c\u53d1\u73b0\u590d\u6742\u5ea6\u6bd4\u6211\u4f30\u8ba1\u5f97\u8981\u591a\uff08\u7136\u540e\u6211\u5e76\u6ca1\u6709\u5438\u53d6\u6559\u8bad\uff0c\u76f4\u5230\u73b0\u5728\u6211\u9884\u4f30\u4f9d\u7136\u5f88\u5dee\uff09\u3002\u800c\u4e14\uff0c\u867d\u7136\u77e9\u9635\u7684\u7406\u8bba\u5f88\u5b8c\u5907\uff0c\u5e76\u4e0d\u610f\u5473\u7740\u7b97\u6cd5\u662f\u7edf\u4e00\u7684\uff0c\u6bd4\u5982DirectX\u7528\u884c\u5411\u91cf\uff0cOpenGL\u7528\u5217\u5411\u91cf\uff0c\u518d\u6bd4\u5982Math.Net\u91cc\u7684DenseMatrix.OfArray\u6784\u9020\u7684\u77e9\u9635\uff0c\u5c06\u6570\u7ec4\u91cc\u7684\u5143\u7d20\u6309\u5217\u4f18\u5148\u7684\u987a\u5e8f\u5b58\u5165\u6570\u7ec4\u3002\u8fd9\u5c31\u5bfc\u81f4\u6211\u5728\u6ca1\u6709\u8db3\u591f\u7406\u8bba\u50a8\u5907\u7684\u60c5\u51b5\u4e0b\u53ea\u80fd\u7528\u5b9e\u9645\u6570\u636e\u7684\u6d4b\u8bd5\u7684\u65b9\u6cd5\u5b9e\u73b0\u4e86\u9700\u8981\u7684\u529f\u80fd\u3002\u7b49\u52302015\u5e74\u6211\u56e0\u4e3a\u5176\u4ed6\u4e8b\u60c5\u8981\u6269\u5c55\u8fd9\u90e8\u5206\u4ee3\u7801\u65f6\uff0c\u6211\u53d1\u73b0\u6211\u5df2\u7ecf\u65e0\u6cd5\u4fee\u6539\u4e86\u3002<\/p>\n
\u6240\u4ee5\u8fd9\u6b21\u501f\u5199\u65b0\u5e93\u7684\u673a\u4f1a\uff0c\u5c06\u7406\u8bba\u6574\u7406\u6e05\u695a\uff0c\u672a\u6765\u5199\u4ee3\u7801\u65f6\u5c31\u4e0d\u4f1a\u73b0\u4e00\u4f1a\u77e9\u9635\u4e58\u5411\u91cf\uff0c\u4e00\u4f1a\u5411\u91cf\u4e58\u77e9\u9635\uff0c\u6700\u540e\u4e71\u6210\u4e00\u56e2\u7684\u60c5\u51b5\u4e86\u3002<\/p>\n
using System;\nusing MathNet.Numerics.LinearAlgebra.Double;\nusing System.Diagnostics;\n\nnamespace CoordinateTest\n{\n class Program\n {\n static void Main(string[] args)\n {\n dimension3();\n dimension4();\n }\n static void dimension3 ()\n {\n DenseMatrix O = DenseMatrix.OfArray(new double[,]\n{\n {1,0,0 },\n {0,1,0 },\n {0,0,1 }\n});\n double SqrtTwo = Math.Sqrt(2) \/ 2.0;\n DenseMatrix A = DenseMatrix.OfArray(new double[,]\n {\n {SqrtTwo,-SqrtTwo,0 },\n {SqrtTwo,SqrtTwo,0 },\n {0,0,1 }\n });\n DenseMatrix B = DenseMatrix.OfArray(new double[,]\n {\n {0,0,1 },\n {0,1,0 },\n {-1,0,0 }\n });\n\n DenseVector P = DenseVector.OfArray(new double[] { 1, 1, 0 });\n DenseVector PA = A.Inverse() * P as DenseVector;\n DenseVector PB = B.Inverse() * P as DenseVector;\n\n Debug.WriteLine(PA);\n Debug.WriteLine(PB);\n }\n static void dimension4()\n {\n DenseMatrix O = DenseMatrix.OfArray(new double[,]\n {\n {1,0,0,0 },\n {0,1,0,0 },\n {0,0,1,0 },\n {0,0,0,1 }\n });\n DenseVector P = DenseVector.OfArray(new double[] { 4, 1, 0, 1 });\n double SqrtTwo = Math.Sqrt(2) \/ 2.0;\n\n DenseMatrix A = DenseMatrix.OfArray(new double[,]\n {\n {SqrtTwo,-SqrtTwo,0,0 },\n {SqrtTwo,SqrtTwo,0,-3 },\n {0,0,1,0 },\n {0,0,0,1 }\n });\n DenseVector PA = DenseVector.OfArray(new double[] { 8* SqrtTwo, 0, 0, 1 });\n\n Debug.WriteLine(A * PA);\n DenseMatrix B = DenseMatrix.OfArray(new double[,]\n {\n {0,0,1,-10 },\n {0,1,0,-10 },\n {-1,0,0,-10 },\n {0,0,0,1 }\n });\n Debug.WriteLine(B.Inverse());\n\n DenseVector PB = B.Inverse() * P as DenseVector;\n Debug.WriteLine(PB);\n\n DenseMatrix Q = A.Inverse() * B as DenseMatrix;\n Debug.WriteLine(Q);\n\n Debug.WriteLine(Q * PB);\n Debug.WriteLine(A * Q);\n\n Debug.WriteLine(Q.Inverse());\n\n \/\/for vector test\n DenseVector v1 = DenseVector.OfArray(new double[] { SqrtTwo, SqrtTwo, 0, 0 });\n\n Debug.WriteLine(A.Inverse() * v1);\n }\n }\n}\n<\/pre>\n <\/p>\n
\u8f93\u51fa\u4e3a\uff1a<\/p>\n
DenseVector 3-Double\n1.41421\n 0\n 0\n\nDenseVector 3-Double\n0\n1\n1\n\nDenseVector 4-Double\n4\n1\n0\n1\n\nDenseMatrix 4x4-Double\n0 0 -1 -10\n0 1 0 10\n1 0 0 10\n0 0 0 1\n\nDenseVector 4-Double\n-10\n 11\n 14\n 1\n\nDenseMatrix 4x4-Double\n 0 0.707107 0.707107 -12.0208\n 0 0.707107 -0.707107 2.12132\n-1 0 0 -10\n 0 0 0 1\n\nDenseVector 4-Double\n 5.65685\n4.44089E-16\n 0\n 1\n\nDenseMatrix 4x4-Double\n 0 0 1 -10\n 0 1 0 -10\n-1 0 0 -10\n 0 0 0 1\n\nDenseMatrix 4x4-Double\n 0 0 -1 -10\n0.707107 0.707107 0 7\n0.707107 -0.707107 0 10\n 0 0 0 1\n\nDenseVector 4-Double\n1\n0\n0\n0\n<\/pre>\n <\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
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