{"id":11803,"date":"2023-10-08T22:25:59","date_gmt":"2023-10-08T14:25:59","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=11803"},"modified":"2025-08-15T20:13:41","modified_gmt":"2025-08-15T12:13:41","slug":"%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6%e5%b8%b8%e7%94%a8%e6%9b%b2%e7%ba%bf","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=11803","title":{"rendered":"\u9ad8\u7b49\u6570\u5b66\u5e38\u7528\u66f2\u7ebf\uff08\u7f16\u8f91\u4e2d\uff09"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_84 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blog.iyatt.com\/?p=11803\/#1_%E5%BF%83%E5%BD%A2%E7%BA%BF\" >1 \u5fc3\u5f62\u7ebf<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.iyatt.com\/?p=11803\/#2_%E9%98%BF%E5%9F%BA%E7%B1%B3%E5%BE%B7%E8%9E%BA%E7%BA%BF\" >2 \u963f\u57fa\u7c73\u5fb7\u87ba\u7ebf<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=11803\/#4_%E4%BC%AF%E5%8A%AA%E5%88%A9%E5%8F%8C%E6%89%AD%E7%BA%BF\" >4 \u4f2f\u52aa\u5229\u53cc\u626d\u7ebf<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=11803\/#5_%E6%91%86%E7%BA%BF\" >5 \u6446\u7ebf<\/a><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"1_%E5%BF%83%E5%BD%A2%E7%BA%BF\"><\/span>1 \u5fc3\u5f62\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\\rho=a(1-\\cos\\theta)\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09 \\\\\n\\\\\nx^2+y^2+ax=a\\sqrt{x^2+y^2}\uff08a\\gt0\uff09\\\\\n\\\\\n\\begin{cases}\nx=a(1-\\cos\\theta)\\cos\\theta \\\\\ny=a(1-\\cos\\theta)\\sin\\theta\n\\end{cases}\n\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09\n\\end{array}<\/code><\/pre>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\nimport numpy as np\n\nalpha = 1\ntheta = np.linspace(0, 2 * np.pi, 100)\n\nrho = alpha * (1 - np.cos(theta))\n\nplt.polar(theta, rho, color=&#039;r&#039;)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696772697049.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 465px; --smush-placeholder-aspect-ratio: 465\/434;\" \/><\/p>\n<hr \/>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\\rho=a(1+\\cos\\theta)\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09 \\\\\n\\\\\nx^2+y^2-ax=a\\sqrt{x^2+y^2}\uff08a\\gt0\uff09\\\\\n\\\\\n\\begin{cases}\nx=a(1+\\cos\\theta)\\cos\\theta \\\\\ny=a(1+\\cos\\theta)\\sin\\theta\n\\end{cases}\n\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09\n\\end{array}<\/code><\/pre>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\nimport numpy as np\n\nalpha = 1\ntheta = np.linspace(0, 2 * np.pi, 100)\n\nrho = alpha * (1 + np.cos(theta))\n\nplt.polar(theta, rho, color=&#039;r&#039;)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696772748850.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 452px; --smush-placeholder-aspect-ratio: 452\/430;\" \/><\/p>\n<hr \/>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\\rho=a(1-\\sin\\theta)\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09 \\\\\n\\\\\nx^2+y^2+ay=a\\sqrt{x^2+y^2}\uff08a\\gt0\uff09\\\\\n\\\\\n\\begin{cases}\nx=a(1-\\sin\\theta)\\cos\\theta \\\\\ny=a(1-\\sin\\theta)\\sin\\theta\n\\end{cases}\n\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09\n\\end{array}<\/code><\/pre>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\nimport numpy as np\n\nalpha = 1\ntheta = np.linspace(0, 2 * np.pi, 100)\n\nrho = alpha * (1 - np.sin(theta))\n\nplt.polar(theta, rho, color=&#039;r&#039;)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696772806151.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 453px; --smush-placeholder-aspect-ratio: 453\/432;\" \/><\/p>\n<hr \/>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\\rho=a(1+\\sin\\theta)\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09 \\\\\n\\\\\nx^2+y^2-ay=a\\sqrt{x^2+y^2}\uff08a\\gt0\uff09\\\\\n\\\\\n\\begin{cases}\nx=a(1+\\sin\\theta)\\cos\\theta \\\\\ny=a(1+\\sin\\theta)\\sin\\theta\n\\end{cases}\n\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09\n\\end{array}<\/code><\/pre>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\nimport numpy as np\n\nalpha = 1\ntheta = np.linspace(0, 2 * np.pi, 100)\n\nrho = alpha * (1 + np.sin(theta))\n\nplt.polar(theta, rho, color=&#039;r&#039;)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696772890191.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 465px; --smush-placeholder-aspect-ratio: 465\/439;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"2_%E9%98%BF%E5%9F%BA%E7%B1%B3%E5%BE%B7%E8%9E%BA%E7%BA%BF\"><\/span>2 \u963f\u57fa\u7c73\u5fb7\u87ba\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nr=a\\theta\uff08r\\gt0\uff0c\\theta\\ge0\uff09(a\\gt0\uff0c\\theta\\in[0,+\\infty)) \\\\\n\\\\\n\\begin{cases}\nx=a\\theta\\cos\\theta \\\\\ny=a\\theta\\sin\\theta\n\\end{cases}\n(a\\gt0\uff0c\\theta\\in[0,+\\infty))\n\\end{array}<\/code><\/pre>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\nimport numpy as np\n\nalpha = 1\ntheta = np.linspace(0, 2 * np.pi, 100)\nrho = alpha * theta\n\nx = rho * np.cos(theta)\ny = rho * np.sin(theta)\n\nplt.axis(&#039;equal&#039;)\nplt.plot(x, y, color=&#039;r&#039;)\nplt.axhline(y=0, color=&#039;k&#039;)\nplt.axvline(x=0, color=&#039;k&#039;)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696775229143.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 560px; --smush-placeholder-aspect-ratio: 560\/431;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"4_%E4%BC%AF%E5%8A%AA%E5%88%A9%E5%8F%8C%E6%89%AD%E7%BA%BF\"><\/span>4 \u4f2f\u52aa\u5229\u53cc\u626d\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nr^2=a^2\\cos2\\theta\uff08\\theta\\in[-\\frac{\\pi}{4},\\frac{\\pi}{4}]\\cup[\\frac{3\\pi}{4},\\frac{5\\pi}{4}]\uff0ca\\gt0\uff09\\\\\n\\\\\n(x^2+y^2)^2=2a^2(x^2-y^2)\uff08a\\gt0\uff09\n\\end{array}<\/code><\/pre>\n<pre><code class=\"language-python\">import numpy as np\nimport matplotlib.pyplot as plt\n\nalpha = 1\ntheta = np.concatenate((\n    np.linspace(-np.pi \/ 4, np.pi \/ 4, 100),\n    np.linspace(3 * np.pi \/ 4, 5 * np.pi \/ 4, 100)\n))\n\nrho = alpha * np.sqrt(np.abs(np.cos(2 * theta)))\n\nplt.polar(theta, rho, color=&#039;r&#039;)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696774576132.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 461px; --smush-placeholder-aspect-ratio: 461\/441;\" \/><\/p>\n<hr \/>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nr^2=a^2\\sin2\\theta\uff08\\theta\\in[0,\\frac{\\pi}{2}]\\cup[\\pi,\\frac{3\\pi}{2}]\uff0ca\\gt0\uff09\n\\\\\n\\\\\n(x^2+y^2)^2=2a^2xy\uff08a\\gt0\uff09\n\\end{array}<\/code><\/pre>\n<pre><code class=\"language-python\">import numpy as np\nimport matplotlib.pyplot as plt\n\nalpha = 1\ntheta = np.concatenate((\n    np.linspace(0, np.pi \/ 2, 100),\n    np.linspace(np.pi, 3 * np.pi \/ 2, 100)\n))\n\nrho = alpha * np.sqrt(np.abs(np.sin(2 * theta)))\n\nplt.polar(theta, rho, color=&#039;r&#039;)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696774678204.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 455px; --smush-placeholder-aspect-ratio: 455\/443;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"5_%E6%91%86%E7%BA%BF\"><\/span>5 \u6446\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<pre><code class=\"language-katex\">\\begin{cases}\nx=a(\\theta-\\sin\\theta) \\\\\ny=a(1-\\cos\\theta)\n\\end{cases}\n\uff08a\\gt0\uff09<\/code><\/pre>\n<pre><code class=\"language-python\">import numpy as np\nimport matplotlib.pyplot as plt\n\nalpha = 1\ntheta = np.linspace(0, 4 * np.pi, 100)\n\nx = alpha * (theta - np.sin(theta))\ny = alpha * (1 - np.cos(theta))\n\nplt.axis(&#039;equal&#039;)\nplt.plot(x, y, color=&#039;r&#039;)\nplt.axhline(y=0, color=&#039;k&#039;)\nplt.axvline(x=0, color=&#039;k&#039;)\nplt.axhline(y=2 * alpha, color=&#039;b&#039;)\nplt.axvline(x=np.pi * alpha, color=&#039;b&#039;)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696774960206.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 556px; --smush-placeholder-aspect-ratio: 556\/414;\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>1 \u5fc3\u5f62\u7ebf \\begin{array}{l} \\rho=a(1-\\cos\\theta)\uff08a&gt;0\uff0c\\the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,612],"tags":[],"class_list":["post-11803","post","type-post","status-publish","format-standard","hentry","category-all","category-612"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11803","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11803"}],"version-history":[{"count":2,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11803\/revisions"}],"predecessor-version":[{"id":21417,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11803\/revisions\/21417"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11803"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11803"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}