{"id":232,"date":"2024-09-24T19:03:28","date_gmt":"2024-09-24T11:03:28","guid":{"rendered":"https:\/\/www.armfun.cn\/?p=232"},"modified":"2024-10-13T03:23:12","modified_gmt":"2024-10-12T19:23:12","slug":"%e5%a4%8d%e6%95%b0%e5%92%8c%e5%9b%9b%e5%85%83%e6%95%b0","status":"publish","type":"post","link":"https:\/\/www.armfun.cn\/?p=232","title":{"rendered":"\u590d\u6570\u548c\u56db\u5143\u6570"},"content":{"rendered":"

\u5f15\u8a00<\/strong><\/h3>\n

\u5728\u6570\u5b66\u548c\u5de5\u7a0b\u5b66\u4e2d\uff0c\u590d\u6570<\/strong>\u548c\u56db\u5143\u6570<\/strong>\u662f\u4e24\u79cd\u91cd\u8981\u7684\u6570\u7cfb\uff0c\u5b83\u4eec\u6269\u5c55\u4e86\u5b9e\u6570\u7684\u6982\u5ff5\uff0c\u80fd\u591f\u8868\u793a\u66f4\u590d\u6742\u7684\u6570\u91cf\u548c\u5173\u7cfb\u3002\u590d\u6570\u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\u63cf\u8ff0\u65cb\u8f6c\u548c\u632f\u8361\uff0c\u800c\u56db\u5143\u6570\u5219\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u63d0\u4f9b\u4e86\u4e00\u79cd\u9ad8\u6548\u4e14\u7a33\u5b9a\u7684\u65cb\u8f6c\u8868\u793a\u65b9\u6cd5\u3002\u672c\u6587\u5c06\u9996\u5148\u4ecb\u7ecd\u590d\u6570\uff0c\u968f\u540e\u6df1\u5165\u63a2\u8ba8\u56db\u5143\u6570\u7684\u5b9a\u4e49\u3001\u610f\u4e49\u53ca\u5176\u8fd0\u7b97\u89c4\u5219\u3002<\/p>\n

\n

\u4e00\u3001\u590d\u6570\uff08Complex Numbers\uff09<\/strong><\/h2>\n

1. \u4ec0\u4e48\u662f\u590d\u6570<\/strong><\/h3>\n

\u590d\u6570<\/strong>\u662f\u5b9e\u6570\u7684\u6269\u5c55\uff0c\u7531\u4e24\u90e8\u5206\u7ec4\u6210\uff1a\u5b9e\u90e8\u548c\u865a\u90e8\u3002\u4e00\u4e2a\u590d\u6570\u901a\u5e38\u8868\u793a\u4e3a\uff1a<\/p>\n

$$
\nz = a + bi
\n$$<\/p>\n

\u5176\u4e2d\uff1a<\/p>\n

    \n
  • $a$ \u662f\u590d\u6570\u7684\u5b9e\u90e8<\/strong>\uff08real part\uff09\u3002<\/li>\n
  • $b$ \u662f\u590d\u6570\u7684\u865a\u90e8<\/strong>\uff08imaginary part\uff09\u3002<\/li>\n
  • $i$ \u662f\u865a\u6570\u5355\u4f4d\uff0c\u6ee1\u8db3 $i^2 = -1$\u3002<\/li>\n<\/ul>\n

    2. \u590d\u6570\u7684\u8868\u793a\u65b9\u5f0f<\/strong><\/h3>\n

    \u590d\u6570\u53ef\u4ee5\u901a\u8fc7\u591a\u79cd\u65b9\u5f0f\u8868\u793a\uff1a<\/p>\n

      \n
    1. \u4ee3\u6570\u5f62\u5f0f<\/strong>\uff1a<\/li>\n<\/ol>\n

      $$
      \nz = a + bi
      \n$$<\/p>\n

        \n
      1. \u51e0\u4f55\u5f62\u5f0f\uff08\u590d\u5e73\u9762\uff09<\/strong>\uff1a\n
          \n
        • \u590d\u5e73\u9762<\/strong>\uff08\u6216\u79f0\u963f\u6839\u5ef7\u5e73\u9762\uff09\u5c06\u590d\u6570\u8868\u793a\u4e3a\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u70b9\u6216\u5411\u91cf\u3002<\/li>\n
        • \u5b9e\u90e8 $a$ \u5bf9\u5e94\u4e8e\u6c34\u5e73\u8f74\uff08\u5b9e\u8f74\uff09\u3002<\/li>\n
        • \u865a\u90e8 $b$ \u5bf9\u5e94\u4e8e\u5782\u76f4\u8f74\uff08\u865a\u8f74\uff09\u3002<\/li>\n
        • \u590d\u6570 $z = a + bi$ \u5728\u590d\u5e73\u9762\u4e0a\u7684\u4f4d\u7f6e\u4e3a\u70b9 $(a, b)$\u3002<\/li>\n<\/ul>\n<\/li>\n
        • \u6781\u5750\u6807\u5f62\u5f0f<\/strong>\uff1a<\/li>\n<\/ol>\n

          $$
          \nz = r (\\cos \\theta + i \\sin \\theta)
          \n$$<\/p>\n

          \u6216\u8005\u4f7f\u7528\u6b27\u62c9\u516c\u5f0f\uff1a<\/p>\n

          $$
          \nz = re^{i\\theta}
          \n$$<\/p>\n

          \u5176\u4e2d\uff1a<\/p>\n

            \n
          • $r = |z| = \\sqrt{a^2 + b^2}$ \u662f\u590d\u6570\u7684\u6a21<\/strong>\uff08magnitude\uff09\u3002<\/li>\n
          • $\\theta = \\arg(z) = \\tan^{-1}\\left(\\frac{b}{a}\\right)$ \u662f\u590d\u6570\u7684\u5e45\u89d2<\/strong>\uff08argument\uff09\uff0c\u8868\u793a\u590d\u6570\u4e0e\u5b9e\u8f74\u7684\u5939\u89d2\u3002<\/li>\n<\/ul>\n

            3. \u590d\u6570\u7684\u8fd0\u7b97<\/strong><\/h3>\n

            \u590d\u6570\u4e4b\u95f4\u53ef\u4ee5\u8fdb\u884c\u591a\u79cd\u8fd0\u7b97\uff0c\u5305\u62ec\u52a0\u6cd5\u3001\u51cf\u6cd5\u3001\u4e58\u6cd5\u3001\u9664\u6cd5\u4ee5\u53ca\u53d6\u6a21\u548c\u5171\u8f6d\u3002<\/p>\n

            3.1 \u52a0\u6cd5\u4e0e\u51cf\u6cd5<\/strong><\/h4>\n
              \n
            • \u52a0\u6cd5<\/strong>\uff1a<\/li>\n<\/ul>\n

              $$
              \n(a + bi) + (c + di) = (a + c) + (b + d)i
              \n$$<\/p>\n

                \n
              • \u51cf\u6cd5<\/strong>\uff1a<\/li>\n<\/ul>\n

                $$
                \n(a + bi) – (c + di) = (a – c) + (b – d)i
                \n$$<\/p>\n

                3.2 \u4e58\u6cd5<\/strong><\/h4>\n

                \u590d\u6570\u7684\u4e58\u6cd5\u57fa\u4e8e\u5206\u914d\u5f8b\u548c $i^2 = -1$ \u7684\u6027\u8d28\uff1a<\/p>\n

                $$
                \n(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac – bd) + (ad + bc)i
                \n$$<\/p>\n

                3.3 \u9664\u6cd5<\/strong><\/h4>\n

                \u590d\u6570\u7684\u9664\u6cd5\u6d89\u53ca\u5230\u4e58\u4ee5\u5171\u8f6d\u590d\u6570\u4ee5\u6d88\u9664\u5206\u6bcd\u4e2d\u7684\u865a\u90e8\uff1a<\/p>\n

                $$
                \n\\frac{a + bi}{c + di} = \\frac{(a + bi)(c – di)}{c^2 + d^2} = \\frac{(ac + bd) + (bc – ad)i}{c^2 + d^2}
                \n$$<\/p>\n

                3.4 \u6a21\u4e0e\u5171\u8f6d<\/strong><\/h4>\n
                  \n
                • \u6a21<\/strong>\uff08Magnitude\uff09\uff1a<\/li>\n<\/ul>\n

                  $$
                  \n|z| = \\sqrt{a^2 + b^2}
                  \n$$<\/p>\n

                    \n
                  • \u5171\u8f6d<\/strong>\uff08Conjugate\uff09\uff1a<\/li>\n<\/ul>\n

                    $$
                    \n\\overline{z} = a – bi
                    \n$$<\/p>\n

                    4. \u590d\u6570\u7684\u51e0\u4f55\u610f\u4e49<\/strong><\/h3>\n

                    \u590d\u6570\u5728\u590d\u5e73\u9762\u4e2d\u7684\u8868\u793a\u8d4b\u4e88\u4e86\u5b83\u4eec\u4e30\u5bcc\u7684\u51e0\u4f55\u610f\u4e49\uff1a<\/p>\n

                      \n
                    • \u52a0\u6cd5<\/strong>\u5bf9\u5e94\u4e8e\u5411\u91cf\u52a0\u6cd5\u3002<\/li>\n
                    • \u4e58\u6cd5<\/strong>\u6d89\u53ca\u6a21\u7684\u76f8\u4e58\u548c\u5e45\u89d2\u7684\u76f8\u52a0\u3002<\/li>\n
                    • \u590d\u6570\u7684\u4e58\u6cd5\u4e0e\u65cb\u8f6c<\/strong>\uff1a\n
                        \n
                      • \u4e58\u4ee5 $e^{i\\theta}$ \u76f8\u5f53\u4e8e\u5728\u590d\u5e73\u9762\u4e0a\u5c06\u590d\u6570\u65cb\u8f6c $\\theta$ \u89d2\u5ea6\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                        5. \u590d\u6570\u7684\u5e94\u7528<\/strong><\/h3>\n

                        \u590d\u6570\u5728\u591a\u4e2a\u9886\u57df\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u5305\u62ec\uff1a<\/p>\n

                          \n
                        • \u5de5\u7a0b\u5b66<\/strong>\uff1a\u4ea4\u6d41\u7535\u8def\u5206\u6790\u3001\u7535\u78c1\u573a\u7406\u8bba\u3002<\/li>\n
                        • \u7269\u7406\u5b66<\/strong>\uff1a\u91cf\u5b50\u529b\u5b66\u3001\u6ce2\u52a8\u7406\u8bba\u3002<\/li>\n
                        • \u6570\u5b66<\/strong>\uff1a\u590d\u5206\u6790\u3001\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/li>\n
                        • \u8ba1\u7b97\u673a\u79d1\u5b66<\/strong>\uff1a\u4fe1\u53f7\u5904\u7406\u3001\u56fe\u5f62\u5b66\u3002<\/li>\n<\/ul>\n
                          \n

                          \u4e8c\u3001\u56db\u5143\u6570\uff08Quaternions\uff09<\/strong><\/h2>\n

                          1. \u4ec0\u4e48\u662f\u56db\u5143\u6570<\/strong><\/h3>\n

                          \u56db\u5143\u6570<\/strong>\u662f\u7531\u5a01\u5ec9\u00b7\u54c8\u5bc6\u5c14\u987f\uff08William Rowan Hamilton\uff09\u572819\u4e16\u7eaa\u4e2d\u53f6\u5f15\u5165\u7684\u4e00\u79cd\u6570\u7cfb\uff0c\u7528\u4e8e\u6269\u5c55\u590d\u6570\u7684\u6982\u5ff5\u3002\u56db\u5143\u6570\u7531\u4e00\u4e2a\u5b9e\u6570\u90e8\u5206\u548c\u4e09\u4e2a\u865a\u6570\u90e8\u5206\u7ec4\u6210\uff0c\u901a\u5e38\u8868\u793a\u4e3a\uff1a<\/p>\n

                          $$
                          \nq = a + bi + cj + dk
                          \n$$<\/p>\n

                          \u5176\u4e2d\uff1a<\/p>\n

                            \n
                          • $a$ \u662f\u5b9e\u90e8\u3002<\/li>\n
                          • $b, c, d$ \u662f\u4e09\u4e2a\u865a\u90e8\uff0c\u5206\u522b\u4e0e\u865a\u6570\u5355\u4f4d $i, j, k$ \u5173\u8054\u3002<\/li>\n
                          • $i, j, k$ \u6ee1\u8db3\u7279\u5b9a\u7684\u4e58\u6cd5\u89c4\u5219\u3002<\/li>\n<\/ul>\n

                            2. \u56db\u5143\u6570\u7684\u5b9a\u4e49\u4e0e\u4e58\u6cd5\u89c4\u5219<\/strong><\/h3>\n

                            \u56db\u5143\u6570\u7684\u4e58\u6cd5\u4e0d\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u5373 $pq \\neq qp$\uff08\u975e\u4ea4\u6362\u4ee3\u6570\uff09\u3002\u5176\u4e58\u6cd5\u89c4\u5219\u57fa\u4e8e\u4ee5\u4e0b\u5173\u7cfb\uff1a<\/p>\n

                            $$
                            \ni^2 = j^2 = k^2 = ijk = -1
                            \n$$<\/p>\n

                            \u7531\u6b64\u53ef\u4ee5\u63a8\u5bfc\u51fa\uff1a<\/p>\n

                            $$
                            \nij = k,\\quad ji = -k
                            \n$$<\/p>\n

                            $$
                            \njk = i,\\quad kj = -i
                            \n$$<\/p>\n

                            $$
                            \nki = j,\\quad ik = -j
                            \n$$<\/p>\n

                            \u8fd9\u4e9b\u4e58\u6cd5\u89c4\u5219\u5b9a\u4e49\u4e86\u56db\u5143\u6570\u7684\u4e58\u6cd5\u8868\u3002<\/p>\n

                            3. \u56db\u5143\u6570\u7684\u8fd0\u7b97<\/strong><\/h3>\n

                            \u56db\u5143\u6570\u4e4b\u95f4\u53ef\u4ee5\u8fdb\u884c\u52a0\u6cd5\u3001\u51cf\u6cd5\u3001\u4e58\u6cd5\u3001\u53d6\u6a21\u3001\u53d6\u5171\u8f6d\u7b49\u8fd0\u7b97\u3002<\/p>\n

                            3.1 \u52a0\u6cd5\u4e0e\u51cf\u6cd5<\/strong><\/h4>\n
                              \n
                            • \u52a0\u6cd5<\/strong>\uff1a<\/li>\n<\/ul>\n

                              $$
                              \n(a + bi + cj + dk) + (e + fi + gj + hk) = (a + e) + (b + f)i + (c + g)j + (d + h)k
                              \n$$<\/p>\n

                                \n
                              • \u51cf\u6cd5<\/strong>\uff1a<\/li>\n<\/ul>\n

                                $$
                                \n(a + bi + cj + dk) – (e + fi + gj + hk) = (a – e) + (b – f)i + (c – g)j + (d – h)k
                                \n$$<\/p>\n

                                3.2 \u4e58\u6cd5<\/strong><\/h4>\n

                                \u56db\u5143\u6570\u7684\u4e58\u6cd5\u57fa\u4e8e\u5206\u914d\u5f8b\u548c\u4e58\u6cd5\u89c4\u5219\uff1a<\/p>\n

                                $$
                                \n(a + bi + cj + dk)(e + fi + gj + hk) = (ae – bf – cg – dh) + (af + be + ch – dg)i + (ag – bh + ce + df)j + (ah + bg – cf + de)k
                                \n$$<\/p>\n

                                3.3 \u6a21\u4e0e\u5171\u8f6d<\/strong><\/h4>\n
                                  \n
                                • \u6a21<\/strong>\uff08Magnitude\uff09\uff1a<\/li>\n<\/ul>\n

                                  $$
                                  \n|q| = \\sqrt{a^2 + b^2 + c^2 + d^2}
                                  \n$$<\/p>\n

                                    \n
                                  • \u5171\u8f6d<\/strong>\uff08Conjugate\uff09\uff1a<\/li>\n<\/ul>\n

                                    $$
                                    \n\\overline{q} = a – bi – cj – dk
                                    \n$$<\/p>\n

                                      \n
                                    • \u9006<\/strong>\uff08Inverse\uff09\uff1a<\/li>\n<\/ul>\n

                                      $$
                                      \nq^{-1} = \\frac{\\overline{q}}{|q|^2}
                                      \n$$<\/p>\n

                                      3.4 \u6807\u51c6\u56db\u5143\u6570<\/strong><\/h4>\n
                                        \n
                                      • \u5355\u4f4d\u56db\u5143\u6570<\/strong>\uff08Unit Quaternion\uff09\uff1a\u6a21\u4e3a1\u7684\u56db\u5143\u6570\uff0c\u7528\u4e8e\u8868\u793a\u65cb\u8f6c\u3002<\/li>\n<\/ul>\n

                                        $$
                                        \nq = a + bi + cj + dk,\\quad |q| = 1
                                        \n$$<\/p>\n

                                        4. \u56db\u5143\u6570\u7684\u51e0\u4f55\u610f\u4e49<\/strong><\/h3>\n

                                        \u56db\u5143\u6570\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u6709\u7740\u91cd\u8981\u7684\u51e0\u4f55\u610f\u4e49\uff0c\u7279\u522b\u662f\u5728\u63cf\u8ff0\u65cb\u8f6c\u65f6\u7684\u5e94\u7528\uff1a<\/p>\n

                                          \n
                                        • \u65cb\u8f6c\u8868\u793a<\/strong>\uff1a\n
                                            \n
                                          • \u5355\u4f4d\u56db\u5143\u6570\u53ef\u7528\u6765\u8868\u793a\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u65cb\u8f6c\uff0c\u907f\u514d\u4e86\u6b27\u62c9\u89d2\u7684\u4e07\u5411\u9501\u95ee\u9898\u548c\u65cb\u8f6c\u77e9\u9635\u7684\u5197\u4f59\u3002<\/li>\n
                                          • \u7ed9\u5b9a\u4e00\u4e2a\u5355\u4f4d\u56db\u5143\u6570 $q = \\cos(\\theta\/2) + \\sin(\\theta\/2)(xi + yj + zk)$\uff0c\u5176\u4e2d $\\theta$ \u662f\u65cb\u8f6c\u89d2\u5ea6\uff0c$\\vec{v} = (x, y, z)$ \u662f\u65cb\u8f6c\u8f74\uff08\u5355\u4f4d\u5411\u91cf\uff09\u3002<\/li>\n<\/ul>\n<\/li>\n
                                          • \u65cb\u8f6c\u516c\u5f0f<\/strong>\uff1a<\/li>\n<\/ul>\n

                                            $$
                                            \n\\vec{v}’ = q \\vec{v} q^{-1}
                                            \n$$<\/p>\n

                                            \u5176\u4e2d $\\vec{v}$ \u4f5c\u4e3a\u7eaf\u865a\u56db\u5143\u6570 $0 + xi + yj + zk$ \u8fdb\u884c\u8fd0\u7b97\u3002<\/p>\n

                                            5. \u56db\u5143\u6570\u7684\u5e94\u7528<\/strong><\/h3>\n

                                            \u56db\u5143\u6570\u5728\u591a\u4e2a\u9886\u57df\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u5305\u62ec\uff1a<\/p>\n

                                              \n
                                            • \u8ba1\u7b97\u673a\u56fe\u5f62\u5b66<\/strong>\uff1a3D\u6a21\u578b\u7684\u65cb\u8f6c\u3001\u52a8\u753b\u3001\u865a\u62df\u73b0\u5b9e\u3002<\/li>\n
                                            • \u673a\u5668\u4eba\u5b66<\/strong>\uff1a\u673a\u5668\u4eba\u8fd0\u52a8\u63a7\u5236\u3001\u5bfc\u822a\u3002<\/li>\n
                                            • \u822a\u7a7a\u822a\u5929<\/strong>\uff1a\u536b\u661f\u59ff\u6001\u63a7\u5236\u3001\u98de\u884c\u5668\u5bfc\u822a\u3002<\/li>\n
                                            • \u6e38\u620f\u5f00\u53d1<\/strong>\uff1a\u89d2\u8272\u548c\u6444\u50cf\u673a\u7684\u5e73\u6ed1\u65cb\u8f6c\u3002<\/li>\n
                                            • \u7269\u7406\u5b66<\/strong>\uff1a\u63cf\u8ff0\u7a7a\u95f4\u4e2d\u7684\u65cb\u8f6c\u548c\u89d2\u52a8\u91cf\u3002<\/li>\n<\/ul>\n

                                              6. \u56db\u5143\u6570\u4e0e\u65cb\u8f6c\u77e9\u9635\u3001\u6b27\u62c9\u89d2\u7684\u6bd4\u8f83<\/strong><\/h3>\n

                                              \u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u65cb\u8f6c\u53ef\u4ee5\u901a\u8fc7\u65cb\u8f6c\u77e9\u9635\u3001\u6b27\u62c9\u89d2\u6216\u56db\u5143\u6570\u6765\u8868\u793a\u3002\u56db\u5143\u6570\u76f8\u8f83\u4e8e\u5176\u4ed6\u65b9\u6cd5\u5177\u6709\u4ee5\u4e0b\u4f18\u70b9\uff1a<\/p>\n

                                                \n
                                              • \u907f\u514d\u4e07\u5411\u9501<\/strong>\uff1a\u6b27\u62c9\u89d2\u5728\u7279\u5b9a\u6761\u4ef6\u4e0b\u4f1a\u51fa\u73b0\u4e07\u5411\u9501\u73b0\u8c61\uff0c\u56db\u5143\u6570\u4e0d\u4f1a\u3002<\/li>\n
                                              • \u6548\u7387\u9ad8<\/strong>\uff1a\u56db\u5143\u6570\u7684\u63d2\u503c\uff08\u5982\u7403\u9762\u7ebf\u6027\u63d2\u503c\uff0cSLERP\uff09\u6bd4\u65cb\u8f6c\u77e9\u9635\u66f4\u9ad8\u6548\uff0c\u9002\u5408\u52a8\u753b\u548c\u5e73\u6ed1\u8fc7\u6e21\u3002<\/li>\n
                                              • \u5185\u5b58\u5360\u7528\u5c11<\/strong>\uff1a\u56db\u5143\u6570\u53ea\u97004\u4e2a\u6570\uff08\u4e00\u4e2a\u5b9e\u90e8\u548c\u4e09\u4e2a\u865a\u90e8\uff09\uff0c\u800c\u65cb\u8f6c\u77e9\u9635\u9700\u89819\u4e2a\u6570\u3002<\/li>\n
                                              • \u7a33\u5b9a\u6027\u597d<\/strong>\uff1a\u56db\u5143\u6570\u5728\u8fde\u7eed\u65cb\u8f6c\u4e2d\u66f4\u7a33\u5b9a\uff0c\u51cf\u5c11\u7d2f\u79ef\u8bef\u5dee\u3002<\/li>\n<\/ul>\n
                                                \n

                                                \u4e09\u3001\u56db\u5143\u6570\u7684\u8fd0\u7b97\u89c4\u5219\u4e0e\u6027\u8d28<\/strong><\/h2>\n

                                                1. \u56db\u5143\u6570\u7684\u52a0\u6cd5\u4e0e\u51cf\u6cd5<\/strong><\/h3>\n

                                                \u56db\u5143\u6570\u7684\u52a0\u6cd5\u548c\u51cf\u6cd5\u4e0e\u590d\u6570\u7c7b\u4f3c\uff0c\u6309\u5bf9\u5e94\u90e8\u5206\u8fdb\u884c\u64cd\u4f5c\uff1a<\/p>\n

                                                  \n
                                                • \u52a0\u6cd5<\/strong>\uff1a<\/li>\n<\/ul>\n

                                                  $$
                                                  \nq_1 + q_2 = (a_1 + a_2) + (b_1 + b_2)i + (c_1 + c_2)j + (d_1 + d_2)k
                                                  \n$$<\/p>\n

                                                    \n
                                                  • \u51cf\u6cd5<\/strong>\uff1a<\/li>\n<\/ul>\n

                                                    $$
                                                    \nq_1 – q_2 = (a_1 – a_2) + (b_1 – b_2)i + (c_1 – c_2)j + (d_1 – d_2)k
                                                    \n$$<\/p>\n

                                                    2. \u56db\u5143\u6570\u7684\u4e58\u6cd5<\/strong><\/h3>\n

                                                    \u56db\u5143\u6570\u4e58\u6cd5\u57fa\u4e8e\u865a\u6570\u5355\u4f4d\u7684\u4e58\u6cd5\u89c4\u5219\uff0c\u6ee1\u8db3\u5206\u914d\u5f8b\u4f46\u4e0d\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff1a<\/p>\n

                                                    $$
                                                    \nq_1 q_2 = (a_1 a_2 – b_1 b_2 – c_1 c_2 – d_1 d_2) + (a_1 b_2 + b_1 a_2 + c_1 d_2 – d_1 c_2)i + (a_1 c_2 – b_1 d_2 + c_1 a_2 + d_1 b_2)j + (a_1 d_2 + b_1 c_2 – c_1 b_2 + d_1 a_2)k
                                                    \n$$<\/p>\n

                                                    3. \u56db\u5143\u6570\u7684\u5171\u8f6d\u4e0e\u9006<\/strong><\/h3>\n
                                                      \n
                                                    • \u5171\u8f6d<\/strong>\uff1a<\/li>\n<\/ul>\n

                                                      $$
                                                      \n\\overline{q} = a – bi – cj – dk
                                                      \n$$<\/p>\n

                                                        \n
                                                      • \u9006<\/strong>\uff1a<\/li>\n<\/ul>\n

                                                        $$
                                                        \nq^{-1} = \\frac{\\overline{q}}{|q|^2} = \\frac{a – bi – cj – dk}{a^2 + b^2 + c^2 + d^2}
                                                        \n$$<\/p>\n

                                                        4. \u56db\u5143\u6570\u7684\u6a21<\/strong><\/h3>\n

                                                        \u56db\u5143\u6570\u7684\u6a21\u4e0e\u5176\u5b9e\u6570\u548c\u865a\u6570\u90e8\u5206\u76f8\u5173\uff1a<\/p>\n

                                                        $$
                                                        \n|q| = \\sqrt{a^2 + b^2 + c^2 + d^2}
                                                        \n$$<\/p>\n

                                                        \u5355\u4f4d\u56db\u5143\u6570\u6ee1\u8db3 $|q| = 1$\u3002<\/p>\n

                                                        5. \u56db\u5143\u6570\u7684\u65cb\u8f6c\u8fd0\u7b97<\/strong><\/h3>\n

                                                        \u56db\u5143\u6570\u7528\u4e8e\u8868\u793a\u65cb\u8f6c\u65f6\uff0c\u901a\u5e38\u4f7f\u7528\u5355\u4f4d\u56db\u5143\u6570\u3002\u7ed9\u5b9a\u4e00\u4e2a\u65cb\u8f6c\u56db\u5143\u6570 $q$ \u548c\u4e00\u4e2a\u7eaf\u865a\u56db\u5143\u6570 $v$\uff08\u8868\u793a\u5411\u91cf\uff09\uff0c\u65cb\u8f6c\u540e\u7684\u5411\u91cf $v’$ \u7531\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\uff1a<\/p>\n

                                                        $$
                                                        \nv’ = qvq^{-1}
                                                        \n$$<\/p>\n

                                                        \u5176\u4e2d\uff1a<\/p>\n

                                                          \n
                                                        • $q = \\cos(\\theta\/2) + \\sin(\\theta\/2)(xi + yj + zk)$<\/li>\n
                                                        • $v = 0 + xi + yj + zk$<\/li>\n<\/ul>\n

                                                          6. \u56db\u5143\u6570\u7684\u6027\u8d28<\/strong><\/h3>\n
                                                            \n
                                                          • \u975e\u4ea4\u6362\u6027<\/strong>\uff1a$q_1 q_2 \\neq q_2 q_1$ \u4e00\u822c\u60c5\u51b5\u4e0b\u3002<\/li>\n
                                                          • \u7ed3\u5408\u5f8b<\/strong>\uff1a$q_1 (q_2 q_3) = (q_1 q_2) q_3$\u3002<\/li>\n
                                                          • \u5206\u914d\u5f8b<\/strong>\uff1a$q_1 (q_2 + q_3) = q_1 q_2 + q_1 q_3$\u3002<\/li>\n
                                                          • \u4e0e\u6807\u91cf\u7684\u4e58\u6cd5<\/strong>\uff1a\u56db\u5143\u6570\u53ef\u4ee5\u4e0e\u5b9e\u6570\uff08\u6807\u91cf\uff09\u8fdb\u884c\u4e58\u6cd5\uff0c\u4f5c\u7528\u5728\u6240\u6709\u90e8\u5206\u4e0a\u3002<\/li>\n<\/ul>\n
                                                            \n

                                                            \u56db\u3001\u56db\u5143\u6570\u7684\u5b9e\u9645\u5e94\u7528<\/strong><\/h2>\n

                                                            1. \u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e0e\u52a8\u753b<\/strong><\/h3>\n

                                                            \u5728\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e2d\uff0c\u56db\u5143\u6570\u88ab\u5e7f\u6cdb\u7528\u4e8e\u8868\u793a\u548c\u63d2\u503c\u4e09\u7ef4\u65cb\u8f6c\u3002\u4f7f\u7528\u56db\u5143\u6570\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u76f8\u6bd4\u4e8e\u6b27\u62c9\u89d2\u548c\u65cb\u8f6c\u77e9\u9635\uff0c\u5177\u6709\u66f4\u9ad8\u7684\u6548\u7387\u548c\u66f4\u597d\u7684\u6570\u503c\u7a33\u5b9a\u6027\uff0c\u5c24\u5176\u5728\u8fde\u7eed\u65cb\u8f6c\u548c\u52a8\u753b\u8fc7\u6e21\u4e2d\u3002<\/p>\n

                                                            2. \u673a\u5668\u4eba\u5b66\u4e0e\u822a\u7a7a\u822a\u5929<\/strong><\/h3>\n

                                                            \u673a\u5668\u4eba\u548c\u822a\u7a7a\u822a\u5929\u9886\u57df\u4e2d\uff0c\u56db\u5143\u6570\u7528\u4e8e\u59ff\u6001\u63a7\u5236\u548c\u5bfc\u822a\u3002\u5b83\u4eec\u80fd\u591f\u6709\u6548\u5730\u8868\u793a\u548c\u8ba1\u7b97\u98de\u884c\u5668\u6216\u673a\u5668\u4eba\u5728\u7a7a\u95f4\u4e2d\u7684\u65b9\u5411\uff0c\u907f\u514d\u4e86\u4e07\u5411\u9501\u95ee\u9898\uff0c\u786e\u4fdd\u8fd0\u52a8\u63a7\u5236\u7684\u5e73\u6ed1\u6027\u548c\u51c6\u786e\u6027\u3002<\/p>\n

                                                            3. \u865a\u62df\u73b0\u5b9e\u4e0e\u589e\u5f3a\u73b0\u5b9e<\/strong><\/h3>\n

                                                            \u5728\u865a\u62df\u73b0\u5b9e\uff08VR\uff09\u548c\u589e\u5f3a\u73b0\u5b9e\uff08AR\uff09\u7cfb\u7edf\u4e2d\uff0c\u56db\u5143\u6570\u7528\u4e8e\u8ddf\u8e2a\u548c\u8868\u793a\u7528\u6237\u7684\u5934\u90e8\u548c\u8bbe\u5907\u7684\u65b9\u5411\u3002\u5b83\u4eec\u786e\u4fdd\u4e86\u7528\u6237\u89c6\u89d2\u7684\u81ea\u7136\u548c\u8fde\u7eed\u53d8\u5316\uff0c\u63d0\u5347\u4e86\u6c89\u6d78\u4f53\u9a8c\u3002<\/p>\n

                                                            4. \u7269\u7406\u6a21\u62df<\/strong><\/h3>\n

                                                            \u56db\u5143\u6570\u5728\u7269\u7406\u6a21\u62df\u4e2d\u7528\u4e8e\u63cf\u8ff0\u521a\u4f53\u7684\u65cb\u8f6c\u72b6\u6001\uff0c\u786e\u4fdd\u4e86\u6a21\u62df\u8fc7\u7a0b\u4e2d\u7684\u7269\u7406\u4e00\u81f4\u6027\u548c\u8ba1\u7b97\u6548\u7387\u3002<\/p>\n

                                                            \n

                                                            \u4e94\u3001\u603b\u7ed3<\/strong><\/h2>\n

                                                            \u590d\u6570<\/strong>\u548c\u56db\u5143\u6570<\/strong>\u90fd\u662f\u5b9e\u6570\u4f53\u7cfb\u7684\u91cd\u8981\u6269\u5c55\uff0c\u80fd\u591f\u63cf\u8ff0\u66f4\u9ad8\u7ef4\u5ea6\u7684\u6570\u91cf\u548c\u5173\u7cfb\u3002\u590d\u6570\u4e3b\u8981\u7528\u4e8e\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\u7684\u65cb\u8f6c\u548c\u632f\u8361\u73b0\u8c61\uff0c\u800c\u56db\u5143\u6570\u5219\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u63d0\u4f9b\u4e86\u4e00\u79cd\u9ad8\u6548\u4e14\u7a33\u5b9a\u7684\u65cb\u8f6c\u8868\u793a\u65b9\u6cd5\u3002\u56db\u5143\u6570\u7684\u8fd0\u7b97\u89c4\u5219\u867d\u7136\u6bd4\u590d\u6570\u590d\u6742\uff0c\u4f46\u5176\u5728\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u3001\u673a\u5668\u4eba\u5b66\u3001\u822a\u7a7a\u822a\u5929\u7b49\u9886\u57df\u7684\u5e7f\u6cdb\u5e94\u7528\u8bc1\u660e\u4e86\u5176\u5f3a\u5927\u7684\u5b9e\u7528\u4ef7\u503c\u3002<\/p>\n

                                                            \u901a\u8fc7\u7406\u89e3\u590d\u6570\u7684\u57fa\u7840\uff0c\u8fdb\u4e00\u6b65\u638c\u63e1\u56db\u5143\u6570\u7684\u5b9a\u4e49\u3001\u8fd0\u7b97\u89c4\u5219\u53ca\u5176\u51e0\u4f55\u610f\u4e49\uff0c\u53ef\u4ee5\u4e3a\u89e3\u51b3\u590d\u6742\u7684\u7a7a\u95f4\u65cb\u8f6c\u548c\u65b9\u5411\u63a7\u5236\u95ee\u9898\u63d0\u4f9b\u6709\u529b\u7684\u5de5\u5177\u548c\u65b9\u6cd5\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

                                                            \u5f15\u8a00 \u5728\u6570\u5b66\u548c\u5de5\u7a0b\u5b66\u4e2d\uff0c\u590d\u6570\u548c\u56db\u5143\u6570\u662f\u4e24\u79cd\u91cd\u8981\u7684\u6570\u7cfb\uff0c\u5b83\u4eec\u6269\u5c55\u4e86\u5b9e\u6570\u7684\u6982...<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-232","post","type-post","status-publish","format-standard","hentry","category-8"],"views":1933,"_links":{"self":[{"href":"https:\/\/www.armfun.cn\/index.php?rest_route=\/wp\/v2\/posts\/232","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.armfun.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.armfun.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.armfun.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.armfun.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=232"}],"version-history":[{"count":3,"href":"https:\/\/www.armfun.cn\/index.php?rest_route=\/wp\/v2\/posts\/232\/revisions"}],"predecessor-version":[{"id":235,"href":"https:\/\/www.armfun.cn\/index.php?rest_route=\/wp\/v2\/posts\/232\/revisions\/235"}],"wp:attachment":[{"href":"https:\/\/www.armfun.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=232"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.armfun.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=232"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.armfun.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=232"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}