{"id":47407,"date":"2013-10-11T09:29:55","date_gmt":"2013-10-11T01:29:55","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=47407"},"modified":"2021-10-06T16:20:08","modified_gmt":"2021-10-06T08:20:08","slug":"%e9%bb%9e%e5%88%b0%e7%9b%b4%e7%b7%9a%e7%9a%84%e8%b7%9d%e9%9b%a2%e5%85%ac%e5%bc%8fthe-formula-of-the-distance-from-a-point-to-a-line","status":"publish","type":"post","link":"http:\/\/localhost\/%e9%bb%9e%e5%88%b0%e7%9b%b4%e7%b7%9a%e7%9a%84%e8%b7%9d%e9%9b%a2%e5%85%ac%e5%bc%8fthe-formula-of-the-distance-from-a-point-to-a-line\/","title":{"rendered":"\u9ede\u5230\u76f4\u7dda\u7684\u8ddd\u96e2\u516c\u5f0f"},"content":{"rendered":"

\u9ede\u5230\u76f4\u7dda\u7684\u8ddd\u96e2\u516c\u5f0f (The Formula of the distance from a point to a line)<\/strong><\/span>
\n \u81fa\u5317\u5e02\u7acb\u7b2c\u4e00\u5973\u5b50\u4e2d\u5b78\u6578\u5b78\u79d1\u8607\u4fca\u9d3b\u8001\u5e2b<\/strong><\/span><\/p>\n

99\u8ab2\u7db1\u4e2d\u5c07\u5713\u8207\u76f4\u7dda\u7684\u55ae\u5143\u653e\u5728\u5e73\u9762\u5411\u91cf\u4e4b\u524d\uff0c\u9032\u800c\u8a0e\u8ad6\u5713\u8207\u76f4\u7dda\u7684\u95dc\u4fc2\u6642\uff0c\u7279\u5225\u5f37\u8abf\u904b\u7528\u5713\u8207\u76f4\u7dda\u7684\u806f\u7acb\u65b9\u7a0b\u5f0f\u4e4b\u89e3\u7684\u5f62\u614b(\u76f8\u7b49\u5be6\u6839\u3001\u5169\u76f8\u7570\u5be6\u6839\uff0c\u53ca\u6c92\u6709\u5be6\u6839) \u7684\u300c\u4ee3\u6578\u5224\u5b9a\u300d\u65b9\u6cd5\u3002\u5118\u7ba1\u6b64\u6cd5\u7684\u4f7f\u7528\u5177\u6709\u4e00\u822c\u6027\uff0c\u4f46\u8a08\u7b97\u901a\u5e38\u8f03\u70ba\u7e41\u96dc\u3002\u56e0\u6b64\uff0c\u8001\u5e2b\u901a\u5e38\u9084\u6703\u4ecb\u7d39\u9ede\u5230\u76f4\u7dda\u7684\u8ddd\u96e2\u516c\u5f0f\uff0c\u5229\u7528\u5713\u5fc3\u8207\u76f4\u7dda\u7684\u8ddd\u96e2\u4f86\u5224\u65b7\u5169\u8005\u7684\u95dc\u4fc2\u3002<\/p>\n

\u7136\u800c\uff0c\u6b64\u8ddd\u96e2\u516c\u5f0f\u7684\u4ecb\u7d39\u5e38\u501f\u52a9\u5411\u91cf\u65b9\u6cd5\u9032\u884c\u8b49\u660e\uff0c\u4f7f\u5f97\u8a31\u591a\u8001\u5e2b\u5021\u8b70\u5c07\u5713\u8207\u76f4\u7dda\u8207\u5e73\u9762\u5411\u91cf\u5169\u500b\u55ae\u5143\u4e92\u63db\u3002\u4e0d\u904e\uff0c\u50c5\u50c5\u70ba\u4e86\u4e00\u500b\u516c\u5f0f\u7684\u8b49\u660e\u5927\u8cbb\u5468\u7ae0\uff0c\u4e26\u4e14\uff0c\u66f4\u52d5\u5f8c\u4e5f\u6d89\u53ca\u5e73\u9762\u4e0a\u76f4\u7dda\u7684\u5411\u91cf\u8868\u793a\u548c\u76f4\u7dda\u65b9\u7a0b\u5f0f\u4e4b\u9593\u7684\u8abf\u6574\u554f\u984c\u3002\u672c\u6587\u4e2d\u63d0\u51fa\u5e7e\u500b\u572899\u8ab2\u7db1\u4e2d\u7121\u9808\u8abf\u52d5\u6b21\u5e8f\uff0c\u4e5f\u53ef\u9054\u5230\u8b49\u660e\u9ede\u5230\u76f4\u7dda\u7684\u8ddd\u96e2\u516c\u5f0f\u4e4b\u76ee\u6a19\u7684\u8b49\u6cd5\u3002<\/p>\n

\"47407_p1\"<\/a><\/p>\n

\u8b49\u6cd5\u4e00\uff1a<\/strong>(\u904e \\(P\\) \u9ede\u4f5c\u76f4\u7dda \\(L\\) \u7684\u5782\u7dda\uff0c\u627e\u5782\u8db3 \\(H\\))<\/span><\/span><\/p>\n

\"47407_p2\"<\/a><\/p>\n

\u5982\u53f3\u5716\uff0c\u4f5c \\(\\overline{PH}\\)\u00a0\u5782\u76f4 \\(L\\)\u00a0\u65bc \\(H\\)\uff0c<\/p>\n

\u5247\u76f4\u7dda \\(PH\\)\u00a0\u65b9\u7a0b\u5f0f\u70ba \\(bx-ay=bx_0-ay_0\\)<\/p>\n

\u89e3\u806f\u7acb\u65b9\u7a0b\u5f0f \\(\\left\\{ \\begin{array}{l} ax + by + c = 0\\\\ bx – ay = b{x_0} – a{y_0} \\end{array} \\right.\\)\uff0c<\/p>\n

\u5373\u5f97 \\(\\displaystyle H(\\frac{{{b^2}{x_0} – ab{y_0} – ac}}{{{a^2} + {b^2}}},\\frac{{ – ab{x_0} + {a^2}{y_0} – bc}}{{{a^2} + {b^2}}})\\)<\/p>\n

\u9032\u800c \\(\\begin{array}{ll} d(P,L) &=\\displaystyle \\overline {PH}= \\sqrt {{{({x_0} – \\frac{{{b^2}{x_0} – ab{y_0} – ac}}{{{a^2} + {b^2}}})}^2} + {{({y_0} – \\frac{{ – ab{x_0} + {a^2}{y_0} – bc}}{{{a^2} + {b^2}}})}^2}}\\\\&=\\displaystyle\\sqrt {\\frac{{{{(a{x_0} + b{y_0} + c)}^2}}}{{{a^2} + {b^2}}}}= \\frac{{\\left| {a{x_0} + b{y_0} + c} \\right|}}{{\\sqrt {{a^2} + {b^2}} }}\\end{array}\\)<\/p>\n

\u6b64\u6cd5\u6982\u5ff5\u7c21\u55ae\uff0c\u76f4\u63a5\u786c\u7b97\uff0c\u4f46\u6574\u9ad4\u904e\u7a0b\u4e0d\u7b97\u8907\u96dc\uff0c\u53ef\u9f13\u52f5\u5b78\u751f\u5617\u8a66\u3002<\/p>\n

 <\/p>\n

\u8b49\u6cd5\u4e8c\uff1a<\/strong>(\u82e5 \\(A\\)\u00a0\u70ba\u76f4\u7dda \\(L\\)\u00a0\u4e0a\u4efb\u4e00\u9ede\uff0c\u5247 \\(\\overline{AP}\\)\u00a0\u7684\u6700\u5c0f\u503c\\(=\\)\u9ede \\(P\\)\u00a0\u5230\u76f4\u7dda \\(L\\)\u00a0\u7684\u8ddd\u96e2)<\/span><\/span><\/p>\n

\u4ee4 \\(A(x,y)\\)\u00a0\u70ba\u76f4\u7dda \\(L\\)\u00a0\u4e0a\u4efb\u4e00\u9ede\uff0c\u5247\u00a0\\(ax + by + c = 0 \\Rightarrow y = \\frac{{ – 1}}{b}(ax + c)\\)<\/p>\n

\\(\\begin{array}{ll}\\displaystyle{\\overline {AP} ^2} &=\\displaystyle {(x – {x_0})^2} + {(y – {y_0})^2} = {(x – {x_0})^2} + \\frac{1}{{{b^2}}}{(ax + c – {y_0})^2} \\\\&=\\displaystyle\\frac{1}{{{b^2}}}[({a^2} + {b^2}){x^2} + 2( – {b^2}{x_0} + ab{y_0} + ac)x + {b^2}{x_0}^2 + {(b{y_0} + c)^2}]\\end{array}\\)<\/p>\n

\u5229\u7528\u4e8c\u6b21\u51fd\u6578\u7684\u6027\u8cea\uff0c\u7576\u00a0\\(\\displaystyle x =\\frac{{{b^2}{x_0} – ab{y_0} – ac}}{{{a^2} + {b^2}}}\\)\u00a0\u6642\uff0c\u6709\u6700\u5c0f\u503c\u3002<\/p>\n

\u6b64\u6642\uff0c\\(\\displaystyle x = \\frac{{{b^2}{x_0} – ab{y_0} – ac}}{{{a^2} + {b^2}}}\\)<\/p>\n

\u6545\u00a0\\(\\begin{array}{ll} d(P,L) &=\\displaystyle \\sqrt {{{(\\frac{{{b^2}{x_0} – ab{y_0} – ac}}{{{a^2} + {b^2}}} – {x_0})}^2} + {{(\\frac{{ – ab{x_0} + {a^2}{y_0} – bc}}{{{a^2} + {b^2}}} – {y_0})}^2}} \\\\&=\\displaystyle \\sqrt {\\frac{{{{(a{x_0} + b{y_0} + c)}^2}}}{{{a^2} + {b^2}}}}=\\frac{{\\left| {a{x_0} + b{y_0} + c} \\right|}}{{\\sqrt {{a^2} + {b^2}} }}\\end{array}\\)<\/p>\n

\u4e8b\u5be6\u4e0a\uff0c\u767c\u751f\u6700\u5c0f\u503c\u7684 \\(A\\) \u9ede\u5373\u70ba\u8b49\u6cd5\u4e00\u6240\u6c42\u51fa\u7684\u5782\u8db3 \\(H\\)\u3002\u5229\u7528\u5169\u9ede\u8ddd\u96e2\u7684\u6700\u5c0f\u503c\u4f86\u6c42\u9ede\u5230\u76f4\u7dda\u8ddd\u96e2\u7684\u4f5c\u6cd5\u5177\u6709\u4e00\u822c\u6027\uff0c\u4e5f\u662f\u7a7a\u9593\u4e2d\u89e3\u6c7a\u9ede\u5230\u76f4\u7dda\u8ddd\u96e2\u7684\u4e3b\u8981\u65b9\u6cd5\u3002<\/p>\n

\u8b49\u6cd5\u4e09\uff1a(\u5229\u7528\u4e09\u89d2\u5f62\u9762\u7a4d)<\/strong><\/span><\/p>\n

\"47407_p3\"<\/a><\/p>\n

\u5982\u53f3\u5716\uff0c\u904e\u9ede \\(P(x_0,y_0)\\)\u00a0\u5206\u5225\u4f5c\u925b\u76f4\u7dda\u548c\u6c34\u5e73\u7dda\u4ea4\u76f4\u7dda \\(L\\)<\/p>\n

\u65bc \\(A({x_0}, – \\frac{{a{x_0} + c}}{b})\\)\u3001\\(B(-\\frac{{b{y_0}+c}}{a},{y_0})\\)<\/p>\n

\\(\\Delta APB\\text{\u7684\u9762\u7a4d} = \\frac{1}{2}\\overline {PA}\\times \\overline {PB}= \\frac{1}{2}\\overline {AB}\\times\\overline{PH}\\)<\/p>\n

\u53c8<\/p>\n

\\(\\overline {PA}\\displaystyle= \\left| {{y_0} – ( – \\frac{{a{x_0} + c}}{b})} \\right| = \\left| {\\frac{{a{x_0} + b{y_0} + c}}{b}} \\right|\\\\\\overline {PB}\\displaystyle= \\left| {{x_0} – ( – \\frac{{b{y_0} + c}}{a})} \\right| = \\left| {\\frac{{a{x_0} + b{y_0} + c}}{a}} \\right|\\\\\\overline {AB}=\\displaystyle\\sqrt {{{({x_0} + \\frac{{b{y_0} + c}}{a})}^2} + {{({y_0} + \\frac{{a{x_0} + c}}{b})}^2}}= \\sqrt {{a^2} + {b^2}} \\left| {\\frac{{a{x_0} + b{y_0} + c}}{{ab}}} \\right|\\)<\/p>\n

\u56e0\u6b64\uff0c\\(d(P,L)=\\displaystyle\\overline{PH}=\\frac{\\overline{PA}\\times\\overline{PB}}{\\overline{AB}}=\\frac{{\\left| {a{x_0} + b{y_0} + c} \\right|}}{{\\sqrt {{a^2} + {b^2}} }}\\)<\/p>\n

\u5229\u7528\u4e09\u89d2\u5f62\u9762\u7a4d\u7684\u9ad8\u4f86\u6c42\u8ddd\u96e2\uff0c\u4e5f\u662f\u53e6\u4e00\u7a2e\u5e38\u898b\u7684\u4f5c\u6cd5\u3002<\/p>\n

\u8b49\u6cd5\u56db\uff1a(\u5229\u7528\u4e09\u89d2\u51fd\u6578)<\/strong><\/span><\/p>\n

\"47407_p4\"<\/a><\/p>\n

\u5982\u53f3\u5716\uff0c\u904e\u9ede \\(P(x_0,y_0)\\) \u4f5c\u6c34\u5e73\u7dda\u4ea4\u76f4\u7dda \\(L\\) \u65bc<\/p>\n

\\(\\displaystyle B( – \\frac{{b{y_0} + c}}{a},{y_0})\\) \u4e14 \\(L\\) \u8207 \\(x\\) \u8ef8\u7684\u6b63\u5411\u593e\u89d2\u70ba \\(\\theta\\)\u3002<\/p>\n

\u76f4\u7dda \\(L:ax+by+c=0\\) \u7684\u659c\u7387 \\(m=-\\frac{a}{b}=\\tan\\theta\\)<\/p>\n

\u5247 \\(\\displaystyle\\sin \\theta= \\sin (\\pi-\\theta)=\\frac{{\\left| a \\right|}}{{\\sqrt {{a^2} + {b^2}} }}\\)<\/p>\n

\u5247 \\(\\begin{array}{ll}\\displaystyle\\overline {PH}&=\\displaystyle\\overline {PB} \\sin \\theta\\\\&=\\displaystyle \\left| {\\frac{{a{x_0} + b{y_0} + c}}{a}} \\right|\\frac{{\\left| a \\right|}}{{\\sqrt {{a^2} + {b^2}} }}\\\\&=\\displaystyle \\frac{{\\left| {a{x_0} + b{y_0} + c} \\right|}}{{\\sqrt {{a^2} + {b^2}} }}\\end{array}\\)<\/p>\n

\u6b64\u4e00\u8b49\u6cd5\u904b\u7528\u5b78\u751f\u525b\u5b78\u904e\u7684\u4e09\u89d2\u51fd\u6578\uff0c\u4e5f\u8aaa\u660e\u4e86\u6b63\u5207\u51fd\u6578 \\((\\tan\\theta)\\) \u8207\u659c\u7387\u4e4b\u9593\u7684\u95dc\u4fc2\uff0c\u7528\u4f86\u8b49\u660e\u9ede\u5230\u76f4\u7dda\u7684\u8ddd\u96e2\u6070\u5230\u80fd\u547c\u61c999\u8ab2\u7db1\u7684\u5b89\u6392\uff0c\u6b64\u6cd5\u6700\u65e9\u898b\u65bc\u5167\u58e2\u9ad8\u4e2d\u6797\u5354\u5104\u8001\u5e2b\u5beb\u65bc\u6578\u5b78\u5b78\u79d1\u4e2d\u5fc3\u96fb\u5b50\u5831\u7b2c60\u671f\u3008\u4e5d\u4e5d\u8ab2\u7db1\u4e2d\uff0c\u9ede\u5230\u76f4\u7dda\u8ddd\u96e2\u516c\u5f0f\u7684\u8a6e\u91cb\u3009\u4e00\u6587\u3002<\/p>\n

\u4e0a\u8ff0\u6240\u63d0\u56db\u7a2e\u8b49\u6cd5\u90fd\u80fd\u907f\u958b\u5411\u91cf\u7684\u4f7f\u7528\uff0c\u540c\u6642\uff0c\u6240\u904b\u7528\u7684\u8b49\u660e\u6982\u5ff5\u4e5f\u5177\u6709\u4e00\u822c\u6027\uff0c\u8001\u5e2b\u5011\u5728\u4ecb\u7d39\u9ede\u5230\u76f4\u7dda\u7684\u8ddd\u96e2\u516c\u5f0f\u6642\u6216\u53ef\u4e00\u8a66\uff01<\/p>\n

\n

\u53c3\u8003\u8cc7\u6599\uff1a<\/strong><\/p>\n

    \n
  1. \u6797\u5354\u5104\uff0c\u3008\u4e5d\u4e5d\u8ab2\u7db1\u4e2d\uff0c\u9ede\u5230\u76f4\u7dda\u8ddd\u96e2\u516c\u5f0f\u7684\u8a6e\u91cb\u3009\uff0c\u6578\u5b78\u5b78\u79d1\u4e2d\u5fc3\u96fb\u5b50\u5831\u7b2c60\u671f(http:\/\/mathcenter.ck.tp.edu.tw\/Resources\/ePaper\/Default.aspx?id=60<\/a>)\u3002<\/li>\n
  2. \u90ed\u6176\u7ae0\uff0c\u3008\u4e5f\u8ac7\u9ede\u8207\u76f4\u7dda\u7684\u8ddd\u96e2\u516c\u5f0f<\/a>\u3009\u3002<\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u9ede\u5230\u76f4\u7dda\u7684\u8ddd\u96e2\u516c\u5f0f (The Formula of the distance from a point to …<\/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":[5694,5695],"class_list":["post-47407","post","type-post","status-publish","format-standard","hentry","category-mathematics00","tag-5694","tag-5695","loop-entry","cat-111","no-thumbnail"],"views":259775,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47407","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=47407"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47407\/revisions"}],"predecessor-version":[{"id":87455,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/47407\/revisions\/87455"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=47407"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=47407"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=47407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}