{"id":14544,"date":"2010-11-10T17:05:10","date_gmt":"2010-11-10T09:05:10","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=14544"},"modified":"2021-10-06T16:30:51","modified_gmt":"2021-10-06T08:30:51","slug":"%e5%a4%a7%e5%9c%93%e6%98%af%e7%90%83%e4%b8%8a%e7%9a%84%e6%8d%b7%e5%be%91%ef%bc%88great-circles%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%a4%a7%e5%9c%93%e6%98%af%e7%90%83%e4%b8%8a%e7%9a%84%e6%8d%b7%e5%be%91%ef%bc%88great-circles%ef%bc%89\/","title":{"rendered":"\u5927\u5713\u662f\u7403\u4e0a\u7684\u6377\u5f91\uff08Great Circles\uff09"},"content":{"rendered":"

\u5927\u5713\u662f\u7403\u4e0a\u7684\u6377\u5f91\uff08Great Circles\uff09
\n<\/span>\u570b\u7acb\u81fa\u7063\u5927\u5b78\u6578\u5b78\u7cfb\u66f9\u4eae\u5409\u6559\u6388\/\u570b\u7acb\u81fa\u7063\u5927\u5b78\u6578\u5b78\u7cfb\u66f9\u4eae\u5409\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n

\u5730\u7406\u8ab2\u672c\u8aaa\u5927\u5713\u662f\u5730\u7403\u4e0a\u7684\u6377\u5f91\uff0c\u9019\u5728\u4e86\u89e3\u5730\u7403\u4e0a\u7684\u5e7e\u4f55\u662f\u5f88\u91cd\u8981\u7684\u3002\u53ea\u5177\u6709\u9ad8\u4e2d\u6578\u5b78\u7684\u7a0b\u5ea6\uff0c\u6709\u8fa6\u6cd5\u8b49\u660e\u9019\u500b\u4e8b\u5be6\u55ce\uff1f<\/span><\/p>\n

\u82e5 $$K$$\u3001$$L$$ \u70ba\u5730\u7403\u4e0a\u5169\u9ede\uff0c\u5047\u8a2d\u5728\u5730\u7403\u4e0a\u806f\u7d50 $$K$$\u3001$$L$$ \u5169\u9ede\u7684\u67d0\u4e00\u66f2\u7dda\uff0c\u662f\u9019\u5169\u9ede\u4e4b\u9593\u7684\u6377\u5f91\uff0c\u6211\u5011\u53ef\u4ee5\u505a\u500b\u5408\u7406\u7684\u5047\u8a2d\uff1a\u9019\u66f2\u7dda\u6703\u843d\u5728\u67d0\u4e00\u500b\u5e73\u9762\u4e0a\u3002\u6709\u6b64\u5047\u8a2d\uff0c\u5c31\u53ef\u4ee5\u8b49\u660e\u6b64\u66f2\u7dda\u662f\u6b64\u5e73\u9762\u8207\u7403\u9762\u76f8\u4ea4\u4e4b\u5927\u5713\u7684\u4e00\u90e8\u5206\u3002
\n<\/span><\/p>\n

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

\u4e0a\u8ff0\u7684\u5e73\u9762\u8207\u7403\u9762\u4ea4\u6210\u4e00\u500b\u5713\uff0c\u5982\u679c\u5b83\u4e0d\u662f\u500b\u5927\u5713\uff0c\u90a3\u9ebc\u5c31\u662f\u500b\u5c0f\u5713\u3002\u6211\u5011\u53ef\u4ee5\u91cd\u65b0\u5b9a\u7fa9\u5730\u8ef8\uff0c\u4f7f\u5f97\u9019\u500b\u5c0f\u5713\u662f\u500b\u7def\u7dda\u5708\uff0c\u5176\u7def\u5ea6\u8a2d\u70ba $$\\phi$$\u3002\u5982\u679c\u5927\u5713\u4e0a\u7684 $$KL$$ \u5f27\u5728\u7403\u5fc3 $$O$$ \u7684\u5f35\u89d2\u70ba $$\\alpha$$\uff0c\u800c\u5c0f\u5713\u4e0a\u7684 $$KL$$ \u5f27\u5728\u5c0f\u5713\u5713\u5fc3 $$M$$ \u6240\u5f35\u7684\u89d2\u70ba $$\\beta$$\uff0c\u5982\u5716\u4e00\u6240\u793a\u3002\u5247\u5f26 $$KL$$ \u505a\u70ba\u7b49\u8170\u4e09\u89d2\u5f62 $$\\triangle KOL$$ \u53ca $$\\triangle KML$$ \u7684\u5e95\u908a\uff0c\u5176\u9577\u5ea6\u8981\u70ba\uff08\u8a2d\u7403\u534a\u5f91\u70ba $${r}$$\uff09<\/span><\/p>\n

$$\\begin{array}{ll}KL&=\\displaystyle 2OK\\sin{\\frac{\\alpha}{2}}=2{r}\\sin{\\frac{\\alpha}{2}}\\\\&=\\displaystyle 2{MK}{\\sin}{\\frac{\\beta}{2}}=2r\\cos{\\phi}\\sin\\frac{\\beta}{2}\\end{array}$$<\/span><\/p>\n

\u6240\u4ee5\u5f97 $$(1)$$ \u5f0f $$\\displaystyle\\sin\\frac{\\alpha}{2}=\\cos{\\phi}\\sin\\frac{\\beta}{2}$$\uff0c\u800c\u6709 $$\\alpha < \\beta$$ \u3002<\/span><\/p>\n

\u7136\u800c\u5728\u5927\u5713\u4e0a\u7684 $$KL$$ \u5713\u5f27\uff0c\u5176\u9577\u70ba $$r\\alpha$$\uff0c\u800c\u5728\u5c0f\u5713\u4e0a\u7684 $$KL$$ \u5713\u5f27\uff0c\u5176\u9577\u70ba $${r}\\cos{\\phi}\\beta$$\u3002\u9019\u6a23\u5c31\u5f97\u5230\u6211\u5011\u60f3\u8b49\u660e\u7684\u3002$$(2)$$ \u5f0f\u4e4b\u6210\u7acb\u662f\u8207 $$(2)$$ \u5f0f\u9664\u4ee5 $$(1)$$ \u5f0f\u800c\u5f97\u4e4b $$(3)$$ \u5f0f $$\\frac{\\frac{\\alpha}{2}}{\\sin\\frac{\\alpha}{2}}<\\frac{\\frac{\\beta}{2}}{\\sin\\frac{\\beta}{2}}$$\u00a0\u80fd\u6210\u7acb\u662f\u76f8\u7576\u7684\u3002\u800c $$(3)$$ \u662f\u5c0d\u7684\uff0c\u56e0\u70ba\u5982\u5716\u4e8c\uff0c$$\\displaystyle\\frac{\\frac{\\theta}{2}}{\\sin\\frac{\\theta}{2}}=AB$$ \u5f27\u8207 $$AB$$ \u5f26\u4e4b\u6bd4\uff0c\u800c\u6b64\u6bd4\u5927\u65bc $$1$$\uff08\u5f27\u5927\u65bc\u5f26\uff09\uff0c\u4e14\u96a8\u8457 $$\\theta$$ \u8b8a\u5c0f\u8da8\u8fd1\u65bc $$0$$\uff0c\u6b64\u6bd4\u6703\u8b8a\u5c0f\u800c\u8da8\u8fd1\u65bc $$1$$\uff0c\u6240\u4ee5\u7576 $$\\alpha\/2 < \\beta\/2$$ \u6642\uff0c$$(3)$$ \u5f0f\u662f\u5c0d\u7684\u3002\"\"<\/a><\/span><\/p>\n

\u70ba\u4e86\u78ba\u5b9a\u9060\u6d0b\u822a\u6d77\u4e0d\u8d70\u51a4\u6789\u8def\uff0c\u9700\u8981\u4e00\u7a2e\u5730\u5716\uff0c\u4e0a\u9762\u7684\u76f4\u7dda\u6b63\u662f\u5169\u9ede\u9593\u7684\u6700\u77ed\u8ddd\u96e2\uff08\u5927\u5713\uff09\u3002<\/span><\/p>\n

\u9019\u7a2e\u5730\u5716\u7a31\u70ba\u9ea5\u5361\u6258\u6295\u5f71\u5716\uff0c\u662f\u5341\u516d\u4e16\u7d00\u5730\u5716\u88fd\u4f5c\u5bb6\u9ea5\u5361\u6258\u767c\u660e\u7684\u3002\u82e5 $$(\\theta,\\phi)$$ \u8868\u5730\u7403\uff08\u8a2d\u534a\u5f91\u70ba $$1$$\uff09\u4e0a\u7d93\u5ea6\u70ba $${\\theta}$$\u3001\u7def\u5ea6\u70ba $$\\phi$$ \u7684\u90a3\u4e00\u9ede\uff0c\u5247\u9019\u7a2e\u6295\u5f71\u6cd5\u628a $$(\\theta,\\phi)$$<\/span>\u5c0d\u61c9\u5230\u5730\u5716\u4e0a\u7684\u9ede $$({x},{y})$$\uff0c$${x=\\theta}$$\uff0c$$y=\\int_0^{\\theta}\\sec\\theta {d\\theta}$$\u3002\u5927\u5b78\u7684\u5fae\u7a4d\u5206\u6703\u628a $${y}$$ \u7684\u7a4d\u5206\u5c0e\u6210 $$y=\\ln(\\sec\\theta+\\tan\\theta)$$\uff0c\u4f46\u662f\u901a\u5e38\u4e0d\u6703\u8aaa\u660e\u5b83\u548c\u9ea5\u5361\u6258\u5730\u5716\u6709\u4ec0\u9ebc\u95dc\u4fc2\u3002<\/p>\n

\u4e00\u822c\u4eba\u624b\u908a\u4e0d\u6703\u6709\u9ea5\u5361\u6258\u5730\u5716\uff0c\u4f46\u662f\u5982\u679c\u4f60\u6709\u4e00\u500b\u5730\u7403\u5100\u53ca\u4e00\u689d\u76ae\u5c3a\uff0c\u4f60\u5c31\u53ef\u4ee5\u6c7a\u5b9a\u5169\u9ede\u4e4b\u9593\u7684\u5927\u5713\u3002\u65b9\u6cd5\u662f\u9019\u6a23\u7684\uff1a\u628a\u76ae\u5c3a\u58d3\u5728\u9019\u5169\u9ede\u4e4b\u4e0a\uff0c\u8abf\u6574\u76ae\u5c3a\uff0c\u4f7f\u5f97\u76ae\u5c3a\u5e73\u58d3\u5728\u5730\u7403\u5100\u4e0a\uff0c\u800c\u7121\u7ff9\u8d77\u4f86\u7684\u5730\u65b9\uff0c\u5247\u76ae\u5c3a\u6240\u58d3\u7684\u5730\u65b9\u5c31\u662f\u7d93\u904e\u9019\u5169\u9ede\u7684\u5927\u5713\u3002\u4f60\u53ef\u4ee5\u8a66\u8a66\u53f0\u5317\u8207\u6d1b\u6749\u78ef\u9019\u5169\u5730\u9ede\uff0c\u5b83\u6703\u8b93\u4f60\u6709\u9a5a\u5947\u7684\u5927\u767c\u73fe\u3002<\/span><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u8b49\u660e\u7403\u4e0a\u5169\u9ede\u9593\u7684\u6700\u77ed\u6377\u5f91\u4e00\u5b9a\u662f\u5927\u5713\u7684\u5f27\u9577\uff0c\u4e26\u8a0e\u8ad6\u9ea5\u5361\u6258\u5730\u5716\u3002<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[222,219,111],"tags":[388,1613,1612],"class_list":["post-14544","post","type-post","status-publish","format-standard","hentry","category-math03-03","category-math03","category-mathematics00","tag-388","tag-1613","tag-mercators-projection","loop-entry","cat-222","cat-219","cat-111","no-thumbnail"],"views":6310,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/14544","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=14544"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/14544\/revisions"}],"predecessor-version":[{"id":89361,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/14544\/revisions\/89361"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=14544"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=14544"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=14544"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}