{"id":2934,"date":"2021-07-12T21:29:47","date_gmt":"2021-07-12T15:59:47","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=2934"},"modified":"2021-10-30T07:40:34","modified_gmt":"2021-10-30T07:40:34","slug":"what-is-curvature-and-radius-of-curvature","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-curvature-and-radius-of-curvature\/","title":{"rendered":"What is curvature and radius of curvature?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overview<\/strong><\/h2>\n\n\n\n<p>The degree of curvedness is used to determine the shape of a plane curve. Curvature is nothing but the measure of rate of change in of curvedness.<\/p>\n\n\n\n<p>Here note that there is no bending in a straight line while in a circle there is constant bending.<\/p>\n\n\n\n<p>Angle of contingence of the arc AB of a curve c is the angle between the tangents at A and B to the curve c<\/p>\n\n\n\n<p>Given two arcs of the same length, the arc with greater angle of contingence is said to be more curved.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/koJPhv_iC99OAAiZiFoTeO7RyOhZQuespNzq-XUzvZdh_77R_Ep0a65Y-40Vl2ZgfTCYGZ78UVRO4bYTMRJ7Dq1-TFHDbh1Pvb3SBSpYzTxzynurBUkMyvcMyec9dWkF9mh1jAsy\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>How do we find<\/strong> it<strong>?<\/strong><\/h2>\n\n\n\n<p>Let the equation of the curve is given in Cartesian form y = f(x), then<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/SDg8vlSiQmNcMlWsKoiOAWC9pxjw0A7hrzqn7ZrQI9xOdETiE38nrUs2TxpBUaom9mqvtKlZ6qn_WjweSl9gWv_UONGqmoXOgsotqbNiV8H4U-SaKBZu9alpVAFUmBeQTxdWNaGk\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p><strong>Formula for polar form-<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/tNRFUQbZNOPD-sL9be0kHvG_U2zBCGB53df-pG3idl5ZoRL1o_PcsOfIaycr-_yrhaTh5vGY2npTiZFD5mtYZn5nErihYR6jYuWtja7d7WFkawEUx_HdeNUbneE3M3cbI3CWPdTv\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/HRcXH329vLut29zETXKBaullhMEjDYZQc_5npFtMYDmbCCYVj-SdxKXtMkZ6I3SBtcOTEbpfZlxKocGk2vWCMegZipIC0eP29MZxuGJI9d5JBcJryQEZ81rRS1HwkqQ2AEeWDRz4\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Curvature and Radius of Curvature in xy-Plane | Derivation of Formula | Differential Calculus\" width=\"1170\" height=\"658\" src=\"https:\/\/www.youtube.com\/embed\/VGcJv8tLPTU?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Radius of curvature<\/strong><\/h2>\n\n\n\n<p>Radius of curvature is the reciprocal of curvature at any point. It is denoted by (rho)<img loading=\"lazy\" decoding=\"async\" width=\"17\" height=\"25\" src=\"https:\/\/lh3.googleusercontent.com\/gs5Cpf0eeY5YRmKWi4fqK-hLyr-6NWf-ysMWpc8ECrn5rxoEMLhwHDw4V3F5m7ik73el_gkC6oo48MCkl5U9kpTKnval_ulj6z5SUx6oAXCY8-bztFmVZPnU0Ot_-D3bX8NMkz9i\">, then <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/3Um4H-aSK8mkVLIAlbc1qqRxbb11BUSxi90HPVOiSCOBh0CMTllxR9DAXAlpIUzi-zvnQjv6J8NzEPv70BCftN1MUe163Qc9qG7CsBHwN3ms7GJUL0d3qIBKxnEtNmfY3oc1g4lZ\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Or<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/VLc_4bO1zYETGgdhOLKfNnyHwjceAO5DUDhuxsukDFeOJN-MamFeZVqcWRYZvEjoiPBYPZ6bDLMhVBCcPTiSjaFEh1B8uSJ2PkNCSic2d8LJMBUK7UUl7VAm0mtBqqV7y90VdZNL\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p><strong>Parametric form:<\/strong><\/p>\n\n\n\n<p>Let the co-ordinates be defined in the form of functions depend on one independent variable t.<\/p>\n\n\n\n<p>Suppose <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/kiA5m9qTdBaKYwDGrwMkjboPQ4xnSBVo31iWkeVoPKC8bohoP4-0l_wi0A04h92S6l6NY67WA5JIxP7x-h-U0H5bAFYZuVGiCFShw1yeYVan0xb3vHWuSbJ2pTEvJDTQvhuK4cYS\" width=\"161\" height=\"24\"><\/p>\n\n\n\n<p>, Then we will calculate the values in term of variables depend on t.<\/p>\n\n\n\n<p>By chain rule we have<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/gZ01o1hRkkEqYg-5ww66HcrXEcDnn7mSHe4M9gUUlwrReKnaU2TjImpb8aYUyLY7KLuQfKLeLWYlE5fy_KLwXDTV5P_QOnK8CkBobGKUrFT3hBN9iFlbJ-sU9nIfEadNQ29dGGVv\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>And<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/WmiLnVUA41Q4On3n6GxiOj-e5vHJEsSoKBhTnv_zL3JwaRrmel-HDSmOm0RBVdpNT60YVOzdZ6EgFxw1KtQoB3XID1Zk3FgkZNge5pMndZD1BTnU5uwoGPEr3Xxl3FKeCznFSA7j\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>By<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/ocB-X85vr-sWCja1kiehlEd1miPVQQCF123-bBwfQ1MGWZ5DxryM1W2zu4Hi1tTW9uBupTB6hdWgvIs8bA7T9fp5lAx1HK1yk6L3mAw41T-lNFZlhEj1mRcvs86B14aCQwrtkTQ6\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p><strong>Newton\u2019s formula<\/strong><\/p>\n\n\n\n<p>a. &nbsp; &nbsp; &nbsp;If x-axis is tangent&nbsp; to a curve&nbsp; at the origin, then<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/hR5BJ7qYGQT5vqiHmMOnreTbS5ewxPWXeSxC2YSgp1HMirqn8SUZgt93Q019oO56yQRcL4q7M4GRPVWtddZ-HHtylButuZx4PVDA3R66yc5K0-TesUCta6xMEuOkHxD3sk4EwvMN\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>b. &nbsp; &nbsp; If y-axis is tangent&nbsp; to a curve&nbsp; at the origin, then<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/B63IduK_MLvn6cs-WYkpCDW417gk-Zn5NBNuZdB_fzlztWmeX25nlr1cp1Bv3SAobn_I3ZKmJmodVi6AJdLJ9mZXklgPPKKWv7d8X6RAD9T51sfEfKJGCfDvpHrxcK_QHQOwIjoG\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>c.&nbsp; &nbsp; &nbsp; When&nbsp; the tangent is&nbsp; at the origin and neither&nbsp; on x-axis nor on&nbsp; y-axis then<\/p>\n\n\n\n<p>Where .<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/jdr2t8G8g4FY8DIynn4KB0a_m2R5UNMDBfMgoya3Ndp0p24c_0pkICw0bBx23j_w00hbnFj8WQtH5UFrIs0rAjHD9QXIhsMXBNK2R7LNVq3B8U10H7FhTIUDsIVqhBfixNqhkzme\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/3VuXbGlvy_ydGEgujT2ESl-Ml6Sg0zqG5tYbWOjlIYYeCfluryR8PSmrOVJxIL1bs6j-FDmC0SZoGnibxWm8pgdCvEoTF7nEA1rKeYHNROsPcVNq1hSHooUiVwiFgk3RKRD8lYxm\" width=\"178\" height=\"20\"><\/p>\n\n\n\n<p><strong>Centre of curvature-<\/strong><\/p>\n\n\n\n<p>Centre of curvature at any point P (x, y) on the curve y = f(x) is given by-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/gO7l0ZbMYGQEQhoA0vFzNIZALbPjT_pBq1ZWOH3KR3_BDH2MYKsCSUf-FN3KuIroHj2hO80YMUIuZbERw1_Ka1fykw96pIiJi2hPE3MrQbxDKV1tKPMrUMfe93TyZK3nyuWV31GQ\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Circle of curvature at P-<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/Jz5dPg66T2VZb6PPnJ8JC_zXJ5wL2TYB3JAGExoFUFJpLqi4009f5wEsAdc9FFhAuLBWtkzB5eHc6QdHeBQIwyUrrTGuEJdnaWGDUR-sLC4-t6cO4e_9v079OkJVvnIhagGG357_\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p><strong>Example: Find the radius of curvature at (0, c) of the catenary<\/strong> <strong>y = c<\/strong> <strong>cosh(x\/c)?<\/strong><\/p>\n\n\n\n<p>Solution:<\/p>\n\n\n\n<p>Given catenary<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/T_krUWh7FPH2zooQpL5gbBav-VDWaDKLYVJZR1T7cPm4LM01Mo43yKYSMqPSVq0VMDwjwVkTxZoqVqcpfG2ioz9zXUKyl9AYo-tjAmYs7OaCQ5cqHsTgJUoWsgKIgJuLaPWNjboK\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Differentiate (i) with respect to x.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/EVH8apuw5jHkfrSuTJs9R0altNqhQLhD1xQe0e22pNq75op6IevHSv_1QxktrSX0V-gas2rGMUv7IdTnnPAYH_1WYNgJRbaRMSFFfxEXbF9tU_c5DkHwKQgjNU1hANLUzikPUUTu\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Also consider<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/m4nLeXQ4NJJWkMCa9_45ekQPas6XeajZ-cXsswy1f7wvzbigQaShCPuRL1dicjp3oL_ah7lggLYW4WwBWAgTA5kxxjwo4Xqsvjmjv216sbbw-D18r10AeZE7tarkRJeSToBehyxE\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Again differentiating with respect to x , we get<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/j5UhSNu0hELnoWZq0eqDIsc10tdgbSFHwx86hY3wYVBjT6Mu_ck9I4VFyoXq35thsICQllC7_noNF9tETwdEX45TBLyYZlt1nTiCLsCmcFjkTxkWNoRwqzCDFPAcOUD5HZt04WjL\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/ZmNrjT5mwn8m2wtIbbG-HM-GVcyMT_HUQEIDw67lwSLqDpLUwMFNpEHgql_fBnLydx93Ni2WQRwN79dBRfjcvkOLZjWVO0WvoUi60P1vXb0Iuvbn13jEa1yaGrIzGgo431FGk83I\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>We know that<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/AeewA8atXWi07zlrb--Jno06cDABLb8fcTh8pEHcdxtIOqhXyWmbu0m2e8DKE9k-cCwEyHy0JaL1LPH-sTRCE0fNXfzo8Kl6gNUuDbCGOLa0vuDMx-4p2l_8FYhMVzcfboLB5tFr\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Substituting values from (i),(ii) and (iii) we get<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/gLsL1HtJeBg8gF7CueHFxsFa8KSNbQ91JnY8eodImk4F9ja85yGuyernc5ALrUPn0uU0zsaKKyvFtryAGhUUcEbmU6tmDeibrK10iYghJXA06zbR_MDrPwo_1JfZyEWtqqZinJtl\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Hence<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/WneDVAbVCLPsvDaAA7Zw6Wr0hG-j3J0SKITZsj538JiWuTpaLNSTcDhYnWMGU_B6Hp4L4Jp1-Y4eE5u8OzvRoydieajPPiGpkklRzqn2MrZlqaQ1yrLDBe4fEec-2RupIO1ZS5rb\" alt=\"\" width=\"75\" height=\"25\"\/><\/figure>\n\n\n\n<p>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<br><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-are-the-asymptotes\" target=\"_blank\" rel=\"noreferrer noopener\">What are asymptotes?<\/a><\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The degree of curvedness is used to determine the shape of a plane curve. Curvature is the measure of rate of change in of curvedness.<\/p>\n","protected":false},"author":3,"featured_media":4084,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[47,50],"class_list":["post-2934","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths","tag-curvature","tag-radius-of-curvature"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is curvature and radius of curvature? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"The degree of curvedness is used to determine the shape of a plane curve. 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