{"id":840,"date":"2021-12-18T20:08:29","date_gmt":"2021-12-18T12:08:29","guid":{"rendered":"https:\/\/blog.cauchyschwarz.com\/?p=840"},"modified":"2021-12-18T20:08:33","modified_gmt":"2021-12-18T12:08:33","slug":"4x4%e7%9f%a9%e9%98%b5%e6%b1%82%e9%80%86","status":"publish","type":"post","link":"https:\/\/blog.cauchyschwarz.com\/?p=840","title":{"rendered":"4&#215;4\u77e9\u9635\u6c42\u9006"},"content":{"rendered":"\n<p>\u5bf94&#215;4\u77e9\u9635\u6c42\u9006\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u516c\u5f0f\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"\">bool gluInvertMatrix(const double m[16], double invOut[16])\n{\n    double inv[16], det;\n    int i;\n\n    inv[0] = m[5]  * m[10] * m[15] - \n             m[5]  * m[11] * m[14] - \n             m[9]  * m[6]  * m[15] + \n             m[9]  * m[7]  * m[14] +\n             m[13] * m[6]  * m[11] - \n             m[13] * m[7]  * m[10];\n\n    inv[4] = -m[4]  * m[10] * m[15] + \n              m[4]  * m[11] * m[14] + \n              m[8]  * m[6]  * m[15] - \n              m[8]  * m[7]  * m[14] - \n              m[12] * m[6]  * m[11] + \n              m[12] * m[7]  * m[10];\n\n    inv[8] = m[4]  * m[9] * m[15] - \n             m[4]  * m[11] * m[13] - \n             m[8]  * m[5] * m[15] + \n             m[8]  * m[7] * m[13] + \n             m[12] * m[5] * m[11] - \n             m[12] * m[7] * m[9];\n\n    inv[12] = -m[4]  * m[9] * m[14] + \n               m[4]  * m[10] * m[13] +\n               m[8]  * m[5] * m[14] - \n               m[8]  * m[6] * m[13] - \n               m[12] * m[5] * m[10] + \n               m[12] * m[6] * m[9];\n\n    inv[1] = -m[1]  * m[10] * m[15] + \n              m[1]  * m[11] * m[14] + \n              m[9]  * m[2] * m[15] - \n              m[9]  * m[3] * m[14] - \n              m[13] * m[2] * m[11] + \n              m[13] * m[3] * m[10];\n\n    inv[5] = m[0]  * m[10] * m[15] - \n             m[0]  * m[11] * m[14] - \n             m[8]  * m[2] * m[15] + \n             m[8]  * m[3] * m[14] + \n             m[12] * m[2] * m[11] - \n             m[12] * m[3] * m[10];\n\n    inv[9] = -m[0]  * m[9] * m[15] + \n              m[0]  * m[11] * m[13] + \n              m[8]  * m[1] * m[15] - \n              m[8]  * m[3] * m[13] - \n              m[12] * m[1] * m[11] + \n              m[12] * m[3] * m[9];\n\n    inv[13] = m[0]  * m[9] * m[14] - \n              m[0]  * m[10] * m[13] - \n              m[8]  * m[1] * m[14] + \n              m[8]  * m[2] * m[13] + \n              m[12] * m[1] * m[10] - \n              m[12] * m[2] * m[9];\n\n    inv[2] = m[1]  * m[6] * m[15] - \n             m[1]  * m[7] * m[14] - \n             m[5]  * m[2] * m[15] + \n             m[5]  * m[3] * m[14] + \n             m[13] * m[2] * m[7] - \n             m[13] * m[3] * m[6];\n\n    inv[6] = -m[0]  * m[6] * m[15] + \n              m[0]  * m[7] * m[14] + \n              m[4]  * m[2] * m[15] - \n              m[4]  * m[3] * m[14] - \n              m[12] * m[2] * m[7] + \n              m[12] * m[3] * m[6];\n\n    inv[10] = m[0]  * m[5] * m[15] - \n              m[0]  * m[7] * m[13] - \n              m[4]  * m[1] * m[15] + \n              m[4]  * m[3] * m[13] + \n              m[12] * m[1] * m[7] - \n              m[12] * m[3] * m[5];\n\n    inv[14] = -m[0]  * m[5] * m[14] + \n               m[0]  * m[6] * m[13] + \n               m[4]  * m[1] * m[14] - \n               m[4]  * m[2] * m[13] - \n               m[12] * m[1] * m[6] + \n               m[12] * m[2] * m[5];\n\n    inv[3] = -m[1] * m[6] * m[11] + \n              m[1] * m[7] * m[10] + \n              m[5] * m[2] * m[11] - \n              m[5] * m[3] * m[10] - \n              m[9] * m[2] * m[7] + \n              m[9] * m[3] * m[6];\n\n    inv[7] = m[0] * m[6] * m[11] - \n             m[0] * m[7] * m[10] - \n             m[4] * m[2] * m[11] + \n             m[4] * m[3] * m[10] + \n             m[8] * m[2] * m[7] - \n             m[8] * m[3] * m[6];\n\n    inv[11] = -m[0] * m[5] * m[11] + \n               m[0] * m[7] * m[9] + \n               m[4] * m[1] * m[11] - \n               m[4] * m[3] * m[9] - \n               m[8] * m[1] * m[7] + \n               m[8] * m[3] * m[5];\n\n    inv[15] = m[0] * m[5] * m[10] - \n              m[0] * m[6] * m[9] - \n              m[4] * m[1] * m[10] + \n              m[4] * m[2] * m[9] + \n              m[8] * m[1] * m[6] - \n              m[8] * m[2] * m[5];\n\n    det = m[0] * inv[0] + m[1] * inv[4] + m[2] * inv[8] + m[3] * inv[12];\n\n    if (det == 0)\n        return false;\n\n    det = 1.0 \/ det;\n\n    for (i = 0; i &lt; 16; i++)\n        invOut[i] = inv[i] * det;\n\n    return true;\n}\n<\/code><\/pre>\n\n\n\n<p>\u4f46\u662f\u6839\u636e\u8fd9\u7bc7<a href=\"http:\/\/www.freevec.org\/function\/inverse_matrix_4x4_using_partitioning\">\u535a\u5ba2<\/a>\uff0c\u53ef\u4ee5\u6709\u66f4\u5feb\u7684\u65b9\u6cd5\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code class=\"\">#define SUBP(i,j) input[i][j]\n#define SUBQ(i,j) input[i][2+j]\n#define SUBR(i,j) input[2+i][j]\n#define SUBS(i,j) input[2+i][2+j]\n\n#define OUTP(i,j) output[i][j]\n#define OUTQ(i,j) output[i][2+j]\n#define OUTR(i,j) output[2+i][j]\n#define OUTS(i,j) output[2+i][2+j]\n\n#define INVP(i,j) invP[i][j]\n#define INVPQ(i,j) invPQ[i][j]\n#define RINVP(i,j) RinvP[i][j]\n#define INVPQ(i,j) invPQ[i][j]\n#define RINVPQ(i,j) RinvPQ[i][j]\n#define INVPQR(i,j) invPQR[i][j]\n#define INVS(i,j) invS[i][j]\n\n#define MULTI(MAT1, MAT2, MAT3) \\\n\tMAT3(0,0)=MAT1(0,0)*MAT2(0,0) + MAT1(0,1)*MAT2(1,0); \\\nMAT3(0,1)=MAT1(0,0)*MAT2(0,1) + MAT1(0,1)*MAT2(1,1); \\\nMAT3(1,0)=MAT1(1,0)*MAT2(0,0) + MAT1(1,1)*MAT2(1,0); \\\nMAT3(1,1)=MAT1(1,0)*MAT2(0,1) + MAT1(1,1)*MAT2(1,1);\n\n#define INV(MAT1, MAT2) \\\n\t_det = 1.0 \/ (MAT1(0,0) * MAT1(1,1) - MAT1(0,1) * MAT1(1,0)); \\\nMAT2(0,0) = MAT1(1,1) * _det; \\\nMAT2(1,1) = MAT1(0,0) * _det; \\\nMAT2(0,1) = -MAT1(0,1) * _det; \\\nMAT2(1,0) = -MAT1(1,0) * _det; \\\n\n#define SUBTRACT(MAT1, MAT2, MAT3) \\\n\tMAT3(0,0)=MAT1(0,0) - MAT2(0,0); \\\nMAT3(0,1)=MAT1(0,1) - MAT2(0,1); \\\nMAT3(1,0)=MAT1(1,0) - MAT2(1,0); \\\nMAT3(1,1)=MAT1(1,1) - MAT2(1,1);\n\n#define NEGATIVE(MAT) \\\n\tMAT(0,0)=-MAT(0,0); \\\nMAT(0,1)=-MAT(0,1); \\\nMAT(1,0)=-MAT(1,0); \\\nMAT(1,1)=-MAT(1,1);\n\n\nvoid getInvertMatrix(complex&lt;double&gt; input[4][4], complex&lt;double&gt; output[4][4]) {\n\tcomplex&lt;double&gt; _det;\n\tcomplex&lt;double&gt; invP[2][2];\n\tcomplex&lt;double&gt; invPQ[2][2];\n\tcomplex&lt;double&gt; RinvP[2][2];\n\tcomplex&lt;double&gt; RinvPQ[2][2];\n\tcomplex&lt;double&gt; invPQR[2][2];\n\tcomplex&lt;double&gt; invS[2][2];\n\n\n\tINV(SUBP, INVP);\n\tMULTI(SUBR, INVP, RINVP);\n\tMULTI(INVP, SUBQ, INVPQ);\n\tMULTI(RINVP, SUBQ, RINVPQ);\n\tSUBTRACT(SUBS, RINVPQ, INVS);\n\tINV(INVS, OUTS);\n\tNEGATIVE(OUTS);\n\tMULTI(OUTS, RINVP, OUTR);\n\tMULTI(INVPQ, OUTS, OUTQ);\n\tMULTI(INVPQ, OUTR, INVPQR);\n\tSUBTRACT(INVP, INVPQR, OUTP);\n}\n<\/code><\/pre>\n\n\n\n<p>\u7531\u4e8e\u8fd9\u4e2a\u65b9\u6cd5\u9700\u8981\u5b50\u77e9\u9635<strong>P<\/strong>\u4e3a\u975e\u9006\u77e9\u9635\uff0c\u6240\u4ee5\u8fd9\u5e76\u4e0d\u662f\u4e00\u4e2a\u901a\u7528\u7684\u53ef\u4ee5\u5e94\u5bf9\u6240\u67094&#215;4\u77e9\u9635\u6c42\u9006\u7684\u65b9\u6cd5\uff0c\u4f46\u662f\u901a\u8fc7\u548c\u524d\u9762\u7684\u6765\u81ea\u4e8eMESA\u7684\u5b9e\u73b0\u7ed3\u5408\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u83b7\u5f97\u4e00\u4e2a\u66f4\u5feb\u7684\u77e9\u9635\u6c42\u9006\u65b9\u6cd5\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5bf94&#215;4\u77e9\u9635\u6c42\u9006\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u516c\u5f0f\uff1a \u4f46\u662f\u6839\u636e\u8fd9\u7bc7\u535a\u5ba2\uff0c\u53ef\u4ee5\u6709\u66f4\u5feb\u7684\u65b9\u6cd5\uff1a \u7531\u4e8e\u8fd9\u4e2a\u65b9\u6cd5\u9700\u8981\u5b50\u77e9\u9635P\u4e3a\u975e\u9006\u77e9\u9635\uff0c\u6240\u4ee5\u8fd9\u5e76\u4e0d\u662f\u4e00\u4e2a\u901a\u7528\u7684\u53ef\u4ee5\u5e94\u5bf9\u6240\u67094&#215;4\u77e9\u9635\u6c42\u9006\u7684\u65b9\u6cd5\uff0c\u4f46\u662f\u901a\u8fc7\u548c\u524d&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[26],"class_list":["post-840","post","type-post","status-publish","format-standard","hentry","category-10","tag-algorithm"],"_links":{"self":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/840","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=840"}],"version-history":[{"count":1,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/840\/revisions"}],"predecessor-version":[{"id":841,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/840\/revisions\/841"}],"wp:attachment":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=840"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=840"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}