\u4eca\u5929\u53d1\u73b0\u4e00\u9053\u6709\u8da3\u7684\u4e60\u9898:\u00a0 \u8bbe{r(t); e1(t), e2(t), e3(t)}\u662f\u6cbf\u66f2\u7ebfr(t)\u5b9a\u4e49\u7684\u4e00\u4e2a\u5355\u4f4d\u6b63\u4ea4\u6807\u67b6\u573a, \u5047\u5b9a1\u2264i\u22643<\/p>\n
\\[
\ne’_{i}(t)=\\sum_{j=1}^{3}a_{ij}(t)e_{j}(t)
\n\\]<\/p>\n
\u8bc1\u660e:<\/p>\n
\\[
\na_{ij}(t)+a_{ji}(t)=0
\n\\]<\/p>\n
\u8bc1: \u5176\u5b9e\u5c31\u662f\u8bc1\u660e\u4e00\u4e2a\u91cd\u8981\u7684\u7ed3\u8bba:\u00a0 \u7a7a\u95f4\u66f2\u7ebf\u7684\u4efb\u610f\u4e00\u4e2a\u6b63\u4ea4\u6807\u67b6\u573a\u7684\u5bfc\u6570\u516c\u5f0f\u7684\u7cfb\u6570\u77e9\u9635\u90fd\u662f\u53cd\u5bf9\u79f0\u9635, \u663e\u7136\u662fFrenet\u6807\u67b6\u7684\u63a8\u5e7f. \u60f3\u4e86\u4e00\u4e0b\uff0c \u539f\u6765\u5c31\u662f\u5229\u7528\u4e00\u4e0b\u6b63\u4ea4\u5173\u7cfb\u5c31ok…\u00a0\u00a0 \u5982\u4e0b\u4f5c\u70b9\u79ef<\/p>\n
\\[
\ne_{i}(t) \\cdot e’_{i}(t) = e_{i}(t) \\cdot \\sum_{j=1}^{3}a_{ij}(t)e_{j}(t)=a_{ii}
\n\\]<\/p>\n
\u5de6\u8fb9\u663e\u7136\u7b49\u4e8e0, \u8fd9\u662f\u56e0\u4e3a\u5355\u4f4d\u5411\u91cf\u5728\u5355\u4f4d\u7403\u9762\u4e0a, \u5176\u5207\u5411\u91cf\u5728\u7403\u9762\u5207\u7a7a\u95f4\u4e0a, \u5fc5\u7136\u5782\u76f4\u4e8e\u8be5\u5411\u91cf, \u53e6\u5916\u5f53i\u2260k\u65f6:<\/p>\n
\\begin{align}
\ne_{k}(t) \\cdot e’_{i}(t) &= e_{k}(t) \\cdot \\sum_{j=1}^{3}a_{ij}(t)e_{j}(t)=a_{ik} \\\\
\ne_{i}(t) \\cdot e’_{k}(t) &= e_{i}(t) \\cdot \\sum_{j=1}^{3}a_{kj}(t)e_{j}(t)=a_{ki}
\n\\end{align}<\/p>\n
\u5c06\u4e0a\u9762\u4e24\u5f0f\u76f8\u52a0\u5f97\u5230\u6700\u7ec8\u7ed3\u8bba:<\/p>\n
\\[ a_{ik}+a_{ki} = e_{k} \\cdot e’_{i}(t) + e_{i} \\cdot e’_{k}(t) = (e_{i} \\cdot e_{k})’=0 \\]<\/p>\n
\u9644:\u00a0 \u8fd9\u4e2a\u6f02\u4eae\u7684\u53cd\u5bf9\u79f0\u9635\u4e3a: (\u4ee5\u4e0b\u7ea6\u5b9a\u90fd\u4f7f\u7528\u884c\u5411\u91cf)<\/p>\n
\\[
\nA=\\begin{bmatrix}
\n0 & a_{12} & a_{13}\\\\
\n-a_{12} & 0 & a_{23}\\\\
\n-a_{13} & -a_{23} & 0
\n\\end{bmatrix}
\n\\]<\/p>\n
\u5f53\u6807\u67b6\u573a\u521a\u597d\u4e3aFrenet\u6807\u67b6\u573a\u65f6\uff0c \u7ed3\u8bba\u5c31\u53d8\u6210\u4e86Frenet\u516c\u5f0f\uff0c \u53ef\u89c1Frenet\u516c\u5f0f\u53ea\u4e0d\u8fc7\u662f\u5176\u7279\u6b8a\u5f62\u5f0f, \u77e9\u9635\u7531\u66f2\u7ebf\u7684\u66f2\u7387\u548c\u6320\u7387\u7ec4\u6210:<\/p>\n
\\[
\nA=\\begin{bmatrix}
\n0 & \\kappa & 0\\\\
\n-\\kappa & 0 & \\tau\\\\
\n0 & -\\tau & 0
\n\\end{bmatrix}
\n\\]<\/p>\n
\u5f53\u8fd9\u6761\u66f2\u7ebf\u5728\u67d0\u4e2a\u66f2\u9762\u4e0a\u65f6,\u5e76\u4e14\u8ba9e3\u4e3a\u66f2\u9762\u7684\u6cd5\u5411\u91cf, e2 = e3 X e1, \u90a3\u4e48\u8fd9\u4e2a\u77e9\u9635\u5176\u5b9e\u5c31\u662f\u7531\u6cd5\u66f2\u7387\u3001\u6d4b\u5730\u66f2\u7387\u548c\u6d4b\u5730\u6320\u7387\u7ec4\u6210\u7684!<\/p>\n
\\[
\nA=\\begin{bmatrix}
\n0 & \\kappa_{g} & \\kappa_{n}\\\\
\n-\\kappa_{g} & 0 & \\tau_{g}\\\\
\n-\\kappa_{n} & -\\tau_{g} & 0
\n\\end{bmatrix}
\n\\]<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
\u4eca\u5929\u53d1\u73b0\u4e00\u9053\u6709\u8da3\u7684\u4e60\u9898:\u00a0 \u8bbe{r(t); e1(t), e2(t), e3(t)}\u662f\u6cbf\u66f2\u7ebfr(t)\u5b9a\u4e49\u7684\u4e00\u4e2a\u5355\u4f4d\u6b63\u4ea4\u6807\u67b6\u573a, \u5047\u5b9a1\u2264i\u22643 \\[ e’_{i}(t)=\\sum_{j=1}^{3}a_{ij}(t)e_{j}(t) \\] \u8bc1\u660e: \\[ a_{ij}(t)+a_{ji}(t)=0 \\] \u8bc1: \u5176\u5b9e\u5c31\u662f\u8bc1\u660e\u4e00\u4e2a\u91cd\u8981\u7684\u7ed3\u8bba:\u00a0 \u7a7a\u95f4\u66f2\u7ebf\u7684\u4efb\u610f\u4e00\u4e2a\u6b63\u4ea4\u6807\u67b6\u573a\u7684\u5bfc\u6570\u516c\u5f0f\u7684\u7cfb\u6570\u77e9\u9635\u90fd\u662f\u53cd\u5bf9\u79f0\u9635, \u663e\u7136\u662fFrenet\u6807\u67b6\u7684\u63a8\u5e7f. \u60f3\u4e86\u4e00\u4e0b\uff0c \u539f\u6765\u5c31\u662f\u5229\u7528\u4e00\u4e0b\u6b63\u4ea4\u5173\u7cfb\u5c31ok…\u00a0\u00a0 \u5982\u4e0b\u4f5c\u70b9\u79ef \\[ e_{i}(t) \\cdot e’_{i}(t) = e_{i}(t) \\cdot \\sum_{j=1}^{3}a_{ij}(t)e_{j}(t)=a_{ii} \\] \u5de6\u8fb9\u663e\u7136\u7b49\u4e8e0, \u8fd9\u662f\u56e0\u4e3a\u5355\u4f4d\u5411\u91cf\u5728\u5355\u4f4d\u7403\u9762\u4e0a, \u5176\u5207\u5411\u91cf\u5728\u7403\u9762\u5207\u7a7a\u95f4\u4e0a, \u5fc5\u7136\u5782\u76f4\u4e8e\u8be5\u5411\u91cf, \u53e6\u5916\u5f53i\u2260k\u65f6: \\begin{align} e_{k}(t) \\cdot e’_{i}(t) &= e_{k}(t) \\cdot \\sum_{j=1}^{3}a_{ij}(t)e_{j}(t)=a_{ik} \\\\ e_{i}(t) \\cdot e’_{k}(t) &= e_{i}(t) \\cdot \\sum_{j=1}^{3}a_{kj}(t)e_{j}(t)=a_{ki} \\end{align} \u5c06\u4e0a\u9762\u4e24\u5f0f\u76f8\u52a0\u5f97\u5230\u6700\u7ec8\u7ed3\u8bba: \\[ a_{ik}+a_{ki} = […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[20],"class_list":["post-269","post","type-post","status-publish","format-standard","hentry","category-math","tag-geo"],"_links":{"self":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/269","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/comments?post=269"}],"version-history":[{"count":0,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/269\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/media?parent=269"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/categories?post=269"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/tags?post=269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}