{"id":1166,"date":"2023-05-05T23:46:29","date_gmt":"2023-05-05T15:46:29","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=1166"},"modified":"2023-05-06T00:50:59","modified_gmt":"2023-05-05T16:50:59","slug":"mobiusdaidecanshuhuayijiyixiejisuan","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=1166","title":{"rendered":"Mobius\u5e26\u7684\u53c2\u6570\u5316\u4ee5\u53ca\u4e00\u4e9b\u8ba1\u7b97"},"content":{"rendered":"

1. \u53c2\u6570\u65b9\u7a0b<\/a><\/span>\n\n\u8003\u5bdf\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u4e00\u6839\u957f\u5ea6\u4e3a\u4e00\u7684\u7ec6\u68cd\uff0c\u5728\u521d\u59cb\u65f6\u523b\u5b83\u4f4d\u4e8e$P(a,0,0)$\u4e14\u5782\u76f4\u4e8e$xy$\u5e73\u9762, \u5176\u4e2d$a >0$. \u73b0\u5728\u6cbf\u7740$xy$\u5e73\u9762\u4e0a\u534a\u5f84\u4e3a$a$\u7684\u5706\u5468\u5300\u901f\u8f6c\u52a8\u7684\u540c\u65f6\uff0c\u8fd8\u5728\u5b83\u4e8e\u539f\u70b9\u5f62\u6210\u7684\u5e73\u9762\u4e0a\u7ed5\u7740$P$\u5300\u901f\u8f6c\u52a8\uff0c\u4e14\u8981\u6c42$t=2\\pi$\u65f6\uff0c\u6070\u597d\u8f6c\u52a8\u534a\u5468\u3002\u5219\u5728\u65f6\u523b$t$, \u7ec6\u68cd\u4e0a\u4e00\u70b9\u7684\u4f4d\u7f6e\u4e3a$(a\\cos t, a\\sin t,0)+u\\sin(t\/2)(\\cos t, \\sin t,0)+(0,0,\\cos(t\/2))$, \u5373
\n\\[
\n \\begin{cases}
\n x=\\cos t(a+u\\sin(t\/2)),\\\\
\n y=\\sin t(a+u\\sin(t\/2)),\\\\
\n z=u\\cos(t\/2),
\n \\end{cases}
\n\\]
\n\u8fd9\u91cc\uff0c$t\\in[0,2\\pi]$, $u\\in[-1,1]$. \u4e00\u4e2a\u56fe\u7247\u53ef\u4ee5\u53c2\u8003
\n

\"Mobius-band.png\"
\u56fe 1. Mobius\u5e26<\/figcaption><\/figure>
\n
\n\u5bb9\u6613\u9a8c\u8bc1\u5982\u4e0b\u4e8b\u5b9e\uff1a\u53c2\u6570$(0,-1+v)$\u4e0e\u53c2\u6570$(2\\pi,1-v)$\u5bf9\u5e94\u7684\u70b9\u76f8\u540c\uff0c\u5373\u5c06$t=0$\u65f6\u523b\u548c$t=2\\pi$\u65f6\u523b\u53cd\u5411\u7c98\u8d77\u6765\u3002<\/p>\n

2. \u5b9a\u5411\u6027<\/a><\/span>\n\n\u8003\u573aMobius\u5e26\u4e0a\u4e00\u4e2a\u6cbf\u7740\u4e2d\u5fc3\u66f2\u7ebf$\\gamma(t)=r(0,t)=(a\\cos t, a\\sin t, 0)$\u79fb\u52a8\u7684\u8d28\u70b9\uff0c\u5982\u679c\u8be5\u8d28\u70b9\u59cb\u7ec8\u5e26\u7740\u4e00\u4e2a\u53f3\u624b\u5207\u6807\u67b6\uff08\u5728\u5207\u5e73\u9762\u4e0a\uff0c\u5c06\u6cbf\u7740$\\gamma$\u7684\u5207\u5411\u91cf\u9006\u65f6\u9488\u65cb\u8f6c90\u5ea6, \u6ce8\u610f\u6211\u4eec\u7528\u5230\u4e86\u53c2\u6570\u66f2\u9762\u7684\u6cd5\u5411\u91cf$e_3$, \u8868\u8fbe\u5f0f\u6bd4\u8f83\u590d\u6742)\uff0c \u5373
\n\\[
\n e_1=(-\\sin t,\\cos t,0),\\quad e_2=e_3\\wedge e_1=(\\cos t\\sin(t\/2),\\sin t\\sin(t\/2),\\cos(t\/2)).
\n\\]
\n\u53ef\u89c1\uff0c\u5728$t=0$\u548c$t=2\\pi$\u65f6\uff0c\u6709
\n\\[
\n e_1(0)=e_1(2\\pi)=(0,1,0),\\quad
\n e_2(0)=-e_2(2\\pi)=(0,0,1).
\n\\]
\n\u5219\u8868\u660e\u6cbf\u7740\u66f2\u7ebf\u8fd0\u52a8\u4e00\u5468\u540e\u6807\u67b6\u53d1\u751f\u4e86\u7ffb\u8f6c\u3002\u8fd9\u5c31\u662fMobius\u5e26\u4e0d\u53ef\u5b9a\u5411\u7684\u6570\u5b66\u89e3\u91ca\u3002
\n3. Gauss\u66f2\u7387\u4e8e\u5e73\u5747\u66f2\u7387<\/a><\/span>\n\n\u5bb9\u6613\u8ba1\u7b97\u5f97\u5230\uff0c
\n\\[
\n K=-\\frac{4 a^2}{\\left(4 a^2+8au\\sin(t\/2)-2 u^2 \\cos t+3 u^2\\right)^2},\\quad
\n H=\\frac{4 \\cos (t\/2 ) \\left(u^2 \\cos t-2 \\left(a^2+2a u \\sin(t\/2)+u^2\\right)\\right)}{\\left(4 a^2+ 8au \\sin(t\/2)-2 u^2 \\cos t+3 u^2\\right)^{3\/2}}.
\n\\]<\/p>\n","protected":false},"excerpt":{"rendered":"

1. \u53c2\u6570\u65b9\u7a0b \u8003\u5bdf\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u4e00\u6839\u957f\u5ea6\u4e3a\u4e00\u7684\u7ec6\u68cd\uff0c\u5728\u521d\u59cb\u65f6\u523b\u5b83\u4f4d\u4e8e$P(a,0,0)$\u4e14\u5782\u76f4\u4e8e$… \u7ee7\u7eed\u9605\u8bfbMobius\u5e26\u7684\u53c2\u6570\u5316\u4ee5\u53ca\u4e00\u4e9b\u8ba1\u7b97<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[203],"class_list":["post-1166","post","type-post","status-publish","format-standard","hentry","category-math","tag-mobiusdai-canshuhua-weifenjihe","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1166","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1166"}],"version-history":[{"count":9,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1166\/revisions"}],"predecessor-version":[{"id":1176,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1166\/revisions\/1176"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1166"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1166"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}