{"id":51646,"date":"2014-01-25T22:51:43","date_gmt":"2014-01-25T14:51:43","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=51646"},"modified":"2021-10-06T16:19:10","modified_gmt":"2021-10-06T08:19:10","slug":"%e5%ba%a7%e6%a8%99%e5%b9%b3%e9%9d%a2%e4%b8%8a%e7%9a%84%e6%97%8b%e8%bd%89%e8%ae%8a%e6%8f%9b","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%ba%a7%e6%a8%99%e5%b9%b3%e9%9d%a2%e4%b8%8a%e7%9a%84%e6%97%8b%e8%bd%89%e8%ae%8a%e6%8f%9b\/","title":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u4e0a\u7684\u65cb\u8f49\u8b8a\u63db"},"content":{"rendered":"

\u5ea7\u6a19\u5e73\u9762\u4e0a\u7684\u65cb\u8f49\u8b8a\u63db<\/span>
\n\u81fa\u5317\u5e02\u7acb\u897f\u677e\u9ad8\u4e2d \u8607\u60e0\u7389\u6559\u5e2b<\/span><\/strong><\/p>\n

\u4e8c\u968e\u65b9\u9663\u6240\u5c0d\u61c9\u7684\u65cb\u8f49\u8b8a\u63db<\/b><\/span><\/p>\n

\u5c07\u5e73\u9762\u4e0a\u7684\u9ede \\(P(x,y)\\)\uff0c\u4ee5\u5750\u6a19\u8ef8\u539f\u9ede \\(O\\)\u00a0\u70ba\u65cb\u8f49\u4e2d\u5fc3\uff0c\u9006\u6642\u91dd\u65cb\u8f49 \\(\\theta\\)\u00a0\u89d2\uff08\u7576 \\(\\theta<0\\)\u00a0\u6642\u53ef\u8003\u616e\u70ba\u9806\u6642\u91dd\u65cb\u8f49\uff09\uff0c\u5f97\u5230\u9ede \\(P\\)\u00a0\u7d93\u65cb\u8f49\u4e4b\u5f8c\u7684\u50cf\u70ba \\(P’=(x’,y’)\\)\uff0c\u9019\u6a23\u7684\u8b8a\u63db\u7a31\u70ba\u65cb\u8f49\u8b8a\u63db\u3002<\/p>\n

\u6211\u5011\u5148\u4ee5\u6975\u5750\u6a19\u4f86\u8868\u793a \\(P\\)\u00a0\u9ede\u5750\u6a19\uff1a
\n\u5728\u5750\u6a19\u5e73\u9762\u4e0a\uff0c\u82e5 \\(P\\)\u00a0\u9ede\u5230\u539f\u9ede \\(O\\)\u00a0\u7684\u8ddd\u96e2\u70ba \\(r\\)\uff0c\u4ee5 \\(x\\)\u00a0\u8ef8\u6b63\u5411\u70ba\u59cb\u908a\uff0c
\n\u9006\u6642\u91dd\u65cb\u8f49\u5230 \\(\\overrightarrow{OP}\\)\u00a0\u7684\u89d2\u5ea6\u70ba \\(\\alpha\\)\uff0c\u90a3\u9ebc\u9ede \\(P(x,y)\\)\u00a0\u7684\u6975\u5750\u6a19\u70ba \\(P[r,\\alpha]\\)\uff0c
\n\u4e14 \\(x=r\\cos \\alpha,y=r\\sin \\alpha\\)\uff0c
\n\u4e26\u53ef\u5f97\u9ede \\(P\\)\u00a0\u7d93\u9006\u6642\u91dd\u65cb\u8f49 \\(\\theta\\)\u00a0\u89d2\u7684\u50cf\u00a0\\(P’=(x’,y’)\\) \u7684\u6975\u5750\u6a19\u70ba\u00a0\\(P'[r,\\alpha+\\theta]\\)\uff0c
\n\u5176\u4e2d\u00a0\\(x’=r\\cos {(\\alpha+\\theta)},y=r\\sin {(\\alpha+\\theta)}\\)\u3002<\/p>\n

\"51646_p1\"<\/a><\/p>\n

\u82e5\u6211\u5011\u5c07\u9ede\u5750\u6a19\u4ee5\u884c\u77e9\u9663\u4f86\u8868\u793a\uff0c\u5373
\n\\(\\left[ \\begin{array}{l} {x’}\\\\ {y’} \\end{array} \\right] = \\left[ \\begin{array}{l} r\\cos (\\alpha + \\theta )\\\\ r\\sin (\\alpha + \\theta ) \\end{array} \\right]=\\left[ \\begin{array}{l} r\\cos \\alpha \\cos \\theta – r\\sin \\alpha \\sin \\theta \\\\ r\\sin \\alpha \\cos \\theta + r\\cos \\alpha \\sin \\theta \\end{array} \\right]=\\left[ \\begin{array}{l} \\cos \\theta \\cdot x – \\sin \\theta \\cdot y\\\\ \\sin \\theta \\cdot x + \\cos \\theta \\cdot y \\end{array} \\right]=\\left[ \\begin{array}{rr} \\cos \\theta& -\\sin \\theta \\\\ \\sin \\theta &\\cos \\theta \\end{array} \\right]\\left[ \\begin{array}{l} x\\\\ y \\end{array} \\right]\\)<\/p>\n

\u5176\u4e2d \\(\\left[ \\begin{array}{rr} \\cos \\theta& -\\sin \\theta \\\\ \\sin \\theta &\\cos \\theta \\end{array} \\right]\\)\u00a0\u7a31\u70ba\u6b64\u65cb\u8f49\u8b8a\u63db\u6240\u5c0d\u61c9\u7684\u65cb\u8f49\u77e9\u9663\uff0c\u8a18\u4f5c \\(R(\\theta)\\)\u3002<\/p>\n

\u7576\u6211\u5011\u5229\u7528\u4e8c\u968e\u65b9\u9663\u8868\u793a\u65cb\u8f49\u8b8a\u63db\uff0c
\n\u90a3\u9ebc\u63a5\u4e0b\u4f86\u5c31\u53ef\u4ee5\u7528\u77e9\u9663\u7684\u904b\u7b97\u4f86\u8868\u5fb5\u9019\u500b\u8b8a\u63db\u7684\u67d0\u4e9b\u6027\u8cea\u3002<\/p>\n

\u8003\u616e\u5169\u500b\u65cb\u8f49\u77e9\u9663 \\(R(\\theta_1)\\)\u00a0\u8207\u00a0\\(R(\\theta_2)\\)\u00a0\u76f8\u4e58\uff0c<\/p>\n

\\(R({\\theta _1})R({\\theta _2}) = \\left[ \\begin{array}{rr} \\cos {\\theta _1} &- \\sin {\\theta _1}\\\\ \\sin {\\theta _1} &\\cos {\\theta _1} \\end{array} \\right]\\left[ \\begin{array}{rr} \\cos {\\theta _2}& – \\sin {\\theta _2}\\\\ \\sin {\\theta _2}&\\cos {\\theta _2} \\end{array} \\right] = \\left[ \\begin{array}{rr} \\cos\\theta_1\\cos\\theta_2-\\sin\\theta_1\\sin\\theta_2 & -\\cos\\theta_1\\sin\\theta_2- \\sin\\theta_1\\cos\\theta_2\\\\ \\sin\\theta_1\\cos\\theta_2+\\cos\\theta_1\\sin\\theta_2&-\\sin\\theta_1\\sin\\theta_2+\\cos\\theta_1\\cos \\theta_2\\end{array}\\right] =\\left[ \\begin{array}{rr} \\cos{(\\theta_1+\\theta_2)} & -\\sin{(\\theta_1+\\theta_2)}\\\\ \\sin{(\\theta_1+\\theta_2)}&\\cos{(\\theta_1+\\theta_2)} \\end{array} \\right]\\)<\/p>\n

\u4e5f\u5c31\u662f\u8aaa\uff0c\u5169\u500b\u65cb\u8f49\u77e9\u9663\u76f8\u4e58\u7684\u7d50\u679c\uff0c\u5c31\u662f\u300c\u89d2\u5ea6\u300d\u76f8\u52a0\u3002\u4ee5\u65cb\u8f49\u7684\u89d2\u5ea6\u4f86\u8003\u616e\uff0c\u4ea6\u5373 \\(P\\)\u00a0\u9ede\u5148\u9006\u6642\u91dd\u65cb\u8f49 \\(\\theta_2\\)\uff0c\u518d\u65cb\u8f49\u00a0\\(\\theta_1\\) \u7684\u89d2\u5ea6\uff0c\u4e5f\u5c31\u662f \\(P\\)\u00a0\u9ede\u5171\u65cb\u8f49\u4e86\u00a0\\(\\theta_1+\\theta_2\\)\u7684\u89d2\u5ea6\u3002<\/p>\n

\u7531\u6b64\u63a8\u5ee3\uff0c\u53ef\u5f97 \\([R(\\theta)]^n=R(n\\theta)\\)\u3002\u540c\u6642\uff0c\u8003\u616e \\(R(\\theta)\\)\u00a0\u7684\u53cd\u65b9\u9663<\/p>\n

\\(R^{-1}(\\theta) = \\frac{1}{1} \\cdot \\left[ \\begin{array}{rr} \\cos\\theta & \\sin \\theta \\\\ -\\sin\\theta & \\cos\\theta \\end{array} \\right] =\\left[ \\begin{array}{rr} \\cos{(-\\theta)} & -\\sin{(-\\theta)}\\\\ \\sin{(-\\theta)}&\\cos{(-\\theta)}\\end{array}\\right]= R( – \\theta )\\)<\/p>\n

\u4e5f\u5c31\u662f\u8aaa\uff0c\u7576 \\(P\\)\u00a0\u9ede\u9006\u6642\u91dd\u65cb\u8f49 \\(\\theta\\)\u00a0\u89d2\u5ea6\u4e4b\u5f8c\uff0c\u5728\u9806\u6642\u91dd\u65cb\u8f49 \\(\\theta\\)\u00a0\u89d2\u5ea6\u5373\u53ef\u56de\u5230\u539f\u4f86\u7684 \\(P\\)\u00a0\u9ede\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u7576\u9006\u6642\u91dd\u65cb\u8f49 \\(\\theta\\)\u00a0\u89d2\u5ea6\u7684\u65cb\u8f49\u77e9\u9663\u70ba \\(R(\\theta)\\)\uff0c\u90a3\u9ebc\u9806\u6642\u91dd\u65cb\u8f49 \\(\\theta\\)\u00a0\u89d2\u5ea6\u7684\u65cb\u8f49\u77e9\u9663\u5373\u70ba \\(R(-\\theta)\\)\u3002<\/p>\n

\u8907\u6578\u76f8\u4e58\u7684\u5e7e\u4f55\u610f\u7fa9<\/b><\/span><\/p>\n

\u5c0d\u4efb\u4e00\u8907\u6578 \\(z=x+yi\\)\uff0c\\(x,y\\in \\Re\\)\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u5176\u8003\u616e\u6210\u6578\u5c0d \\((x,y)\\)\u00a0\u4e26\u756b\u5728\u8907\u6578\u5e73\u9762\u4e0a\uff0c\u4ee5\u9ede \\(P(x,y)\\)\u00a0\u4f86\u8868\u793a\u9019\u500b\u8907\u6578\uff0c\u90a3\u9ebc\u5c31\u53ef\u5b9a\u7fa9 \\(|z|=\\sqrt{x^2+y^2}\\)\u00a0\u70ba\u9ede \\(P\\) \u5230\u539f\u9ede\u7684\u8ddd\u96e2\uff0c\u7c21\u55ae\u8aaa\uff0c\u4e5f\u5c31\u662f \\(z\\)\u00a0\u9019\u500b\u8907\u6578\u7684\u300c\u9577\u5ea6\u300d\uff1b\u82e5\u4ee5\u5be6\u8ef8\uff08\u6a6b\u8ef8\uff09\u7684\u6b63\u5411\u70ba\u59cb\u908a\uff0c\u9006\u6642\u91dd\u65cb\u8f49\u5230 \\(\\overline{OP}\\)\u00a0\u7684\u89d2\u5ea6\u70ba \\(\\theta\\)\uff0c\u6539\u4ee5\u6975\u5750\u6a19\u4f86\u8868\u793a\u70ba \\(P[r,\\alpha]\\)\uff0c\u5176\u4e2d \\(x=r\\cos\\theta,y=r\\sin\\theta\\)\u3002<\/p>\n

\u56e0\u6b64\uff0c\u53ef\u5c07\u8907\u6578 \\(z\\)\u00a0\u8868\u793a\u6210\u6975\u5f0f \\(z=r(\\cos\\theta,i\\sin\\theta)\\)\uff0c\u5176\u4e2d \\(r=|z|=\\sqrt{x^2+y^2}\\)\uff0c\\(\\theta\\)\u00a0\u7a31\u70ba \\(z\\)\u00a0\u7684\u5e45\u89d2 (argument)\u3002<\/p>\n

\u7576\u8907\u6578\u4ee5\u6975\u5f0f\u4f86\u8868\u793a\u7684\u6642\u5019\uff0c\u85c9\u7531\u6b64\uff0c\u6211\u5011\u5c31\u53ef\u5f9e\u5e7e\u4f55\u7684\u89d2\u5ea6\u4f86\u63a2\u8a0e\u8907\u6578\u4e58\u9664\u904b\u7b97\u7684\u5e7e\u4f55\u610f\u7fa9\u3002<\/p>\n

\u5047\u8a2d\u5169\u500b\u8907\u6578 \\(z_1,z_2\\)\uff0c\\(z_1=r_1(\\cos{\\theta_1}+i\\sin\\theta_1),z_2=r_2(\\cos{\\theta_2}+i\\sin\\theta_2)\\)\u00a0\uff0c
\n\u5247\u5b83\u5011\u7684\u4e58\u7a4d\u53ef\u8a08\u7b97\u70ba
\n\\(\\begin{array}{ll}z_1\\cdot z_2 &=r_1(\\cos {\\theta _1} + i\\sin {\\theta _1}) \\cdot {r_2}(\\cos {\\theta _2} + i\\sin {\\theta _2}) \\\\&= {r_1}{r_2}[(\\cos {\\theta _1}\\cos {\\theta _2} – \\sin {\\theta _1}\\sin {\\theta _2}{\\rm{) + }}i(\\cos {\\theta _1}\\sin {\\theta _2} + \\sin {\\theta _1}\\cos {\\theta _2})] \\\\&= {r_1}{r_2}[\\cos ({\\theta _1}{\\rm{ + }}{\\theta _{\\rm{2}}}{\\rm{) + }}i\\sin ({\\theta _1}{\\rm{ + }}{\\theta _{\\rm{2}}}{\\rm{)]}}\\end{array}\\)<\/p>\n

\u9019\u5169\u500b\u8907\u6578\u7684\u6975\u5f0f\u76f8\u4e58\u544a\u8a34\u6211\u5011\uff0c\u7576\u5169\u500b\u8907\u6578\u76f8\u4e58\u6642\uff0c\u5176\u300c\u9577\u5ea6\u300d\u76f8\u4e58\uff0c\u300c\u89d2\u5ea6\u300d\u76f8\u52a0\u3002<\/p>\n

\u63db\u500b\u89d2\u5ea6\u8aaa\u660e\uff0c\u7576 \\(z_1\\)\u00a0\u9019\u500b\u8907\u6578\u4e58\u4ee5\u53e6\u4e00\u500b\u8907\u6578 \\(z_2\\)\uff0c\u53ef\u4ee5\u8003\u616e\u6210 \\(z_1\\)\u00a0\u6240\u4ee3\u8868\u7684\u9ede \\(P\\)\uff0c\u5c07\u5b83\u5230\u539f\u9ede\u7684\u8ddd\u96e2 \\(r_1\\)\uff0c\u4f5c\u4f38\u7e2e\u8b8a\u63db\u8b8a\u6210 \\(r_1r_2\\)\uff1b\u5c07\u5b83\u8207\u6a6b\u8ef8\u6b63\u5411\u6240\u6210\u7684\u6709\u5411\u89d2 \\(\\theta_1\\)\uff0c\u7e5e\u539f\u9ede\u9006\u6642\u91dd\u65cb\u8f49\u4e86 \\(\\theta_2\\)\u00a0\u7684\u89d2\u5ea6\uff0c\u5982\u4e0b\u5716\u6240\u793a\uff1a<\/p>\n

\"51646_p2\"<\/a><\/p>\n

\u82e5\u55ae\u7d14\u53ea\u8003\u616e\u4e00\u500b\u8907\u6578 \\(z\\)\uff0c\u5c07\u5b83\u4e58\u4ee5 \\(\\cos\\theta+i\\sin\\theta\\)\uff0c\u5e7e\u4f55\u5448\u73fe\u4e0a\u7684\u610f\u7fa9\u5373\u662f\u5c07\u6b64\u8907\u6578\u4ee5\u539f\u9ede\u70ba\u65cb\u8f49\u4e2d\u5fc3\uff0c\u9006\u6642\u91dd\u65cb\u8f49 \\(\\theta\\)\u00a0\u89d2\u5ea6\uff08\u7576 \\(\\theta<0\\)\u00a0\u6642\u70ba\u9806\u6642\u91dd\u65cb\u8f49\uff09\u3002\u4f46\u662f\u9019\u6a23\u7684\u65cb\u8f49\u8207\u4ee5\u4e8c\u968e\u65b9\u9663\u6240\u8868\u5fb5\u7684\u65cb\u8f49\u8b8a\u63db\u662f\u5426\u76f8\u540c\u5462\uff1f<\/p>\n

\u8a2d\u6b64\u8907\u6578 \\(z=x+yi\\)\uff0c\u5728\u8907\u6578\u5e73\u9762\u4e0a\u4ee5\u9ede \\(P(x,y)\\)\u00a0\u8868\u793a\u6b64\u8907\u6578\uff0c\u7d93\u7e5e\u539f\u9ede\u9006\u6642\u91dd\u65cb\u8f49 \\(\\theta\\)\u00a0\u89d2\u5ea6\u4e4b\u5f8c\uff0c\u6240\u5f97\u5230\u65b0\u7684\u9ede\u70ba \\(P’=(x’,y’)\\)\uff0c\u7531\u65cb\u8f49\u8b8a\u63db\u53ef\u77e5<\/p>\n

\\(\\left[ \\begin{array}{l} {x’}\\\\ {y’} \\end{array} \\right] = \\left[ \\begin{array}{rr} \\cos \\theta&-\\sin\\theta\\\\ \\sin\\theta & \\cos \\theta \\end{array} \\right]\\left[ \\begin{array}{l} x\\\\ y \\end{array} \\right] = \\left[ \\begin{array}{l} x\\cos \\theta – y\\sin \\theta \\\\ x\\sin \\theta + y\\cos \\theta \\end{array} \\right]\\)<\/p>\n

\u4ea6\u5373 \\(x’=x\\cos \\theta-y\\sin \\theta,y’=x\\sin \\theta+y\\cos\\theta\\)\uff0c\u56e0\u6b64<\/p>\n

\\(\\begin{array}{ll} z’ &= (x\\cos\\theta-y\\sin\\theta)+i(x\\sin\\theta+y\\cos\\theta)\\\\ &=x(\\cos\\theta+i\\sin\\theta)+yi\\cdot(\\cos \\theta+i\\sin\\theta)\\\\&= (x +yi)\\cdot(\\cos\\theta+i\\sin\\theta)=z\\cdot(\\cos\\theta+i\\sin\\theta)\\end{array}\\)<\/p>\n

\u7531\u6b64\u53ef\u77e5\uff0c\u7576\u6211\u5011\u5c07\u5e73\u9762\u4e0a\u7684\u9ede\u8003\u616e\u6210\u8907\u6578\u6642\uff0c\u4f9d\u8907\u6578\u6975\u5f0f\u4e58\u6cd5\u904b\u7b97\u6240\u8655\u7406\u7684\u65cb\u8f49\u8b8a\u63db\uff0c\u8207\u4e8c\u968e\u65b9\u9663\u6240\u4ee3\u8868\u7684\u65cb\u8f49\u8b8a\u63db\u662f\u4e00\u81f4\u7684\u3002\u5c0d\u5e73\u9762\u4e0a\u7684\u65cb\u8f49\u8b8a\u63db\u800c\u8a00\uff0c\u6211\u5011\u5e38\u4f9d\u9ede\u6240\u8868\u5fb5\u7684\u610f\u7fa9\u4e0d\u540c\uff0c\u800c\u9078\u64c7\u4e0d\u540c\u7684\u5448\u73fe\u65b9\u5f0f\u4f86\u8655\u7406\u5e7e\u4f55\u4e0a\u7684\u65cb\u8f49\u8b8a\u63db\uff0c\u4e0d\u904e\u8868\u9762\u4e0a\u4e0d\u540c\uff0c\u88e1\u5b50\u537b\u662f\u4e00\u6a23\u7684\u3002\u800c\u6709\u6642\u5019\u901a\u904e\u8907\u6578\u7684\u6975\u5f0f\u4f86\u4f5c\u904b\u7b97\uff0c\u53cd\u800c\u53ef\u4ee5\u8b93\u904b\u7b97\u8b8a\u5f97\u66f4\u52a0\u7684\u7c21\u6f54\u6613\u61c2\u3002<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u5ea7\u6a19\u5e73\u9762\u4e0a\u7684\u65cb\u8f49\u8b8a\u63db \u81fa\u5317\u5e02\u7acb\u897f\u677e\u9ad8\u4e2d \u8607\u60e0\u7389\u6559\u5e2b \u4e8c\u968e\u65b9\u9663\u6240\u5c0d\u61c9\u7684\u65cb\u8f49\u8b8a\u63db \u5c07\u5e73\u9762\u4e0a\u7684\u9ede \\(P(x,y)\\…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111],"tags":[6085,6084,5575,6086,3818],"class_list":["post-51646","post","type-post","status-publish","format-standard","hentry","category-mathematics00","tag-6085","tag-6084","tag-5575","tag-6086","tag-3818","loop-entry","cat-111","no-thumbnail"],"views":52649,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/51646","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=51646"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/51646\/revisions"}],"predecessor-version":[{"id":87180,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/51646\/revisions\/87180"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=51646"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=51646"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=51646"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}