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A simple closed curve<\/em> is a curve which does not intersect itself anywhere. Mathematically, a curve in the $xy$ plane is defined by the parametric equations $x = \\phi(t),\\ y = \\psi(t)$ where $\\phi$ and $\\psi$ are single-valued and continuous in an interval $t_1 \\le t \\le t_2$ . If $\\phi(t_1) = \\phi(t_2)$ and $\\psi(t_1) = \\psi(t_2)$ , the curve is said to be closed<\/em>. If $\\phi(u) = \\phi(v)$ and $\\psi(u) = \\psi(v)$ only when $u = v$ (except in the special case where $u = t_1$ and $v = t_2$ ), the curve is closed and does not intersect itself and so is a simple closed curve. We shall also assume, unless otherwise stated, that $\\phi$ and $\\psi$ are piecewise differentiable in $t_1 \\le t \\le t_2$ . <\/p>\n
If a plane region has the property that any closed curve in it can be continuously shrunk to a point without leaving the region, then the region is called simply-connected<\/em>, otherwise it is called multiply-connected<\/em>. <\/p>\n
$\"Fig6-xx\"$ <\/a><\/p>\n
As the parameter $t$ varies from $t_1$ to $t_2$ , the plane curve is described in a certain sense or direction. For curves in the $xy$ plane, we arbitrarily describe this direction as positive<\/em> or negative<\/em> according as a person traversing the curve in this direction with his head pointing in the positive $z$ direction has the region enclosed by the curve always toward his left or right respectively. If we look down upon a simple closed curve in the $xy$ plane, this amounts to saying that traversal of the curve in the counterclockwise direction is taken as positive while traversal in the clockwise direction is taken as negative. <\/p>\n<\/blockquote>\n
<\/iframe><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>A simple closed curve is a curve which does not intersect itself anywhere. Mathematically, a curve in the plan … <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5678\" class=\"more-link\"><span class=\"screen-reader-text\">“SIMPLE CLOSED CURVES. SIMPLY AND MULTIPLY-CONNECTED REGIONS” \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":6040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[113,108,112,111,62,109,61,60,110],"class_list":["post-5678","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","tag-clockwise","tag-closed","tag-counterclockwise","tag-multiply-connected","tag-negative","tag-piecewise-differentiable","tag-positive","tag-simple-closed-curve","tag-simply-connected"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5678","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5678"}],"version-history":[{"count":5,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5678\/revisions"}],"predecessor-version":[{"id":5728,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/5678\/revisions\/5728"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/media\/6040"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5678"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5678"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5678"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}