{"id":305,"date":"2016-12-13T02:22:42","date_gmt":"2016-12-13T02:22:42","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=305"},"modified":"2016-12-13T02:22:42","modified_gmt":"2016-12-13T02:22:42","slug":"zuijindiantoushedejibenxingzhi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=305","title":{"rendered":"\u6700\u8fd1\u70b9\u6295\u5c04\u7684\u57fa\u672c\u6027\u8d28"},"content":{"rendered":"

\u5047\u8bbe$N$\u662f\u4e00\u4e2a\u5149\u6ed1\u6d41\u5f62\u7b49\u8ddd\u5730\u5d4c\u5165\u5230$\\mathbb{R}^K$, \u6211\u4eec\u77e5\u9053\u5b58\u5728$N$\u7684\u7ba1\u72b6\u90bb\u57df$N(\\delta)\\subset\\mathbb{R}^K$, \u4f7f\u5f97\u5b9a\u4e49\u5728$N(\\delta)$\u4e0a\u7684\u6620\u5c04$\\Pi$:
\n\\[
\n\\Pi:y\\mapsto x\\in N,\\quad \\mathrm{dist}(y,N)=|y-\\Pi(y)|.
\n\\]
\n\u5173\u4e8e$\\Pi$, \u6211\u4eec\u6709\u5982\u4e0b\u57fa\u672c\u6027\u8d28.
\n

\u547d\u9898 1<\/span>.<\/span> \u5047\u8bbe$\\Pi$\u5b9a\u4e49\u5982\u4e0a, \u5219
\n
  1. $D\\Pi|_y:\\mathbb{R}^K\\to T_xN$, $x=\\Pi(y)$;<\/li>
  2. \u5bf9$v_1,v_2\\in T_xN$, \u6211\u4eec\u6709$\\mathrm{Hess}\\Pi|_x(v_1,v_2)=-A(x)(v_1,v_2)$.<\/li><\/ol><\/div><\/p>\n
    \u8bc1\u660e<\/span>.<\/span> \u5047\u8bbe$x=\\Pi(y)$, \u4e14\u4ee4$v\\in T_xN$\u662f\u4efb\u4e00\u5207\u65b9\u5411\u800c$\\gamma(s)$\u662f$N$\u4e2d\u4e00\u5f27\u957f\u53c2\u6570\u66f2\u7ebf(\u6cd5\u622a\u7ebf), \u4f7f\u5f97$\\gamma(0)=x$, $\\gamma'(0)=v$. \u6211\u4eec\u59cb\u7ec8\u7528$\\perp$\u6765\u8bb0\u5782\u76f4\u65b9\u5411(\u5355\u4f4d). \u4e3a\u4e86\u8ba1\u7b97$D\\Pi(v+v^\\perp)$, \u6211\u4eec\u9009\u53d6\u4ee5$v+v^\\perp$\u4e3a\u5207\u5411\u7684\u4e00\u6761\u66f2\u7ebf\u5e76\u8ba1\u7b97\u5176\u5bfc\u6570\u5373\u53ef. \u4e00\u4e2a\u7b80\u5355\u7684\u9009\u53d6\u65b9\u6cd5\u662f
    \n\\[
    \n\\alpha(s)=\\gamma(s)+|y-x|[\\gamma'(\\lambda s)]^\\perp,
    \n\\]
    \n\u5373\u8fc7$y$\u4e14\u4e0e$\\gamma(s)$\u5e73\u884c\u7684\u66f2\u7ebf, \u8fd9\u91cc$\\lambda$\u662f\u4efb\u4f55\u5e38\u6570. \u6ce8\u610f\u5230
    \n\\[
    \n\\Pi(\\alpha(t))=\\Pi(\\gamma(t))=\\gamma(t),
    \n\\]
    \n\u4e24\u8fb9\u5bf9$s$\u5728$s=0$\u6c42\u5bfc\u5f97\u5230
    \n\\[
    \nv=\\left.\\frac{d\\gamma(s)}{ds}\\right|_{s=0}= D\\Pi\\cdot \\alpha'(0).
    \n\\]
    \n\u6ce8\u610f\u5230, \u7531\u66f2\u7ebf\u8bba\u57fa\u672c\u7406\u8bba
    \n\\begin{align*}
    \n\\gamma(s)&=\\gamma(0)+\\gamma'(0)s+\\gamma^\\prime\\prime(0)s^2\/2+o(s^2)\\\\
    \n&=x+sv+\\kappa(0)v^\\perp s^2\/2+o(s^2),
    \n\\end{align*}
    \n\u53ef\u89c1
    \n\\[
    \n\\gamma'(s)=v+\\kappa(0)v^{\\perp_0} s+o(s).
    \n\\]
    \n\u6545
    \n\\begin{align*}
    \n\\gamma'(\\lambda s)&=\\lambda\\gamma’|_{\\lambda s}=\\lambda v+\\kappa(0)v^{\\perp_0} \\lambda^2 s+o(s),\\\\
    \n[\\gamma'(\\lambda s)]^{\\perp_{\\lambda s}}&=\\lambda v^{\\perp_{\\lambda s}}+\\kappa(0)(v^{\\perp_0})^{\\perp_{\\lambda s}}\\lambda^2 s+o(s)
    \n\\end{align*}
    \n\u5373
    \n\\[
    \n\\alpha'(0)=v+\\lambda^2\\kappa(0)|y-x|v^\\perp.
    \n\\]
    \n\u7531$\\lambda$\u7684\u4efb\u610f\u6027\u77e5
    \n\\[
    \nD\\Pi(v+\\lambda v^\\perp)=v,\\quad\\forall \\lambda.
    \n\\]
    \n\u5373$D\\Pi|_y$\u662f\u5230\u5207\u7a7a\u95f4$T_{\\Pi(y)}N$\u7684\u6b63\u4ea4\u6295\u5f71.<\/p>\n

    \u6700\u540e, \u6ce8\u610f\u5230\u6309\u7167\u6620\u7167Hessian\u7684\u5b9a\u4e49(\u4eff\u7167\u51fd\u6570Hessian),
    \n\\begin{align*}
    \n\\mathrm{Hess}\\Pi(v_1,v_2)&=D_{v_1}D_{v_2}\\Pi-D_{D_{v_1}v_2}\\Pi
    \n=D_{v_1}(D\\Pi(v_2))-D\\Pi(D_{v_1}v_2)\\\\
    \n&=D_{v_1}v_2-[D_{v_1}v_2]^{\\top}
    \n=[D_{v_1}v_2]^\\bot.
    \n\\end{align*}
    \n\u800c\u6309\u7167\u7b2c\u4e8c\u57fa\u672c\u578b\u7684\u5b9a\u4e49, \u5bf9$T_xN$\u5904\u7684\u6cd5\u5411\u91cf$\\set{\\nu_i}$,
    \n\\[
    \nA^i(x)(v_1,v_2)=\\inner{D_{v_1}\\nu_i, v_2}
    \n=-\\inner{\\nu_i,D_{v_1}v_2}
    \n\\]
    \n\u6545
    \n\\[
    \nA(x)(v_1,v_2)=-\\inner{\\nu_i,D_{v_1}v_2}\\nu_i=-[D_{v_1}v_2]^\\perp.
    \n\\]
    \n\u8fd9\u8868\u660e\u7b2c\u4e8c\u6761\u6210\u7acb.
    \n<\/div>\n","protected":false},"excerpt":{"rendered":"

    \u5047\u8bbe$N$\u662f\u4e00\u4e2a\u5149\u6ed1\u6d41\u5f62\u7b49\u8ddd\u5730\u5d4c\u5165\u5230$\\mathbb{R}^K$, \u6211\u4eec\u77e5\u9053\u5b58\u5728$N$\u7684\u7ba1\u72b6\u90bb\u57df$N(\\del… \u7ee7\u7eed\u9605\u8bfb\u6700\u8fd1\u70b9\u6295\u5c04\u7684\u57fa\u672c\u6027\u8d28<\/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":[55,54,56],"class_list":["post-305","post","type-post","status-publish","format-standard","hentry","category-math","tag-hessian","tag-zuijindiantoushe","tag-dierjibenxing","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/305","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=305"}],"version-history":[{"count":26,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/305\/revisions"}],"predecessor-version":[{"id":334,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/305\/revisions\/334"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=305"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=305"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}