{"id":366,"date":"2011-01-06T11:05:14","date_gmt":"2011-01-06T10:05:14","guid":{"rendered":"http:\/\/kmr.dialectica.se\/wp\/?page_id=366"},"modified":"2011-01-06T11:05:14","modified_gmt":"2011-01-06T10:05:14","slug":"inverse-curves","status":"publish","type":"page","link":"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/geometry-2\/metric-geometry\/euclidean-geometry\/geometry\/plane-curves\/inverse-curves\/","title":{"rendered":"Inverse Curves"},"content":{"rendered":"<p>This page is a sub-page of our page on <a href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/geometry-2\/metric-geometry\/euclidean-geometry\/geometry\/plane-curves\/\" title=\"Plane Curves at the KMR web site\" target=\"_blank\" rel=\"noopener\">Plane Curves<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/ <\/p>\n<p><strong>Related KMR-pages<\/strong>:<br \/>\n\u2022 &#8230;<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Other relevant sources of information<\/strong>: <\/p>\n<p>\u2022 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Inverse_curve\" target=\"_blank\" rel=\"noopener\">Inverse curves<\/a> at Wikipedia<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p>The interactive simulations on this page can be navigated with the <a href=\"http:\/\/www.pacifict.com\/FreeStuff.html\">Free Viewer<\/a><br \/>\nof the <a href=\"http:\/\/www.pacifict.com\/Home.html\">Graphing Calculator<\/a>.  <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Inversion of a polar grid<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/Yve3h5IBjoQ\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/626?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>.<\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/715?download\">Interactive simulation of 1\/z of a polar grid<\/a>.<br \/>\nThis is very similar to inversion, which would be 1\/Conjugate(z).<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Devil-inversion-xy.anim<\/strong> (or, rather, Devil transformed by 1\/z):<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/gX19DwdBuz4\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/1847?download\">The interactive simulation that created this movie<\/a>.<br \/>\nDrag the devil to study the dynamics of the transformation. Note the difference from the video clip. The interactive simulation is a true inversion.<\/p>\n<p><strong>Compare<\/strong>: <a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/1847?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">Interactive inversion of the same Devil in the unit circle<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Ellipse\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Ellipse<\/a>-focus-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/EhAIvlGRck8\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1778?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/ <\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Parabola\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Parabola<\/a>-focus-inverse<\/strong>:<\/p>\n<p>\/\/\/\/\/\/\/ MISSING<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hyperbola\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Hyperbola<\/a>-focus-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/KG-l9ERFg7Q\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1783?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Ellipse\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Ellipse<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/pVKQE6pBsQY\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1777?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/ <\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hyperbola\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Hyperbola<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/g_dWKatVEzM\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>  <\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1782?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Ellipse\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Ellipse<\/a>-vertex-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/Lf6zetPhUCY\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1779?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Parabola\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Parabola<\/a>-vertex-inverse<\/strong>:<\/p>\n<p>\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/ MISSING<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hyperbola\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Hyperbola<\/a>-vertex-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/hfwPL-a9WzY\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1784?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Logarithmic_spiral\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">LogarithmicSpiral<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/_KXLxl8loFI\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/856?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Strophoid#The_right_strophoid\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">RightStrophoid<\/a>-vertex-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/fuve2F2E9mQ\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/858\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Epitrochoid\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">EpiTrochoid<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/lE-O_6lR1Pg\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/849?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Epicycloid\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">EpiCycloid<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/djIq9cf_xXg\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1780?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hypotrochoid\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">HypoTrochoid<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/KJeCMTwJnlE\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1786?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hypocycloid\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">HypoCycloid<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/rMaC74TAgvU\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/my.confolio.org\/scam\/4\/resource\/1785?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Quadrifolium\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Quadrifolium<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/fMoSJbihSvs\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/857?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Sinusoidal_spiral\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Sinusoidal Spiral<\/a><\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Lemniscate_of_Bernoulli\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">LemniscateOfBernoulli<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/OLlfjG3rHD8\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/859?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Archimedean_spiral\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">ArchimedeanSpiral<\/a>-center-inverse<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"480\" height=\"385\" src=\"https:\/\/www.youtube.com\/embed\/gW-fB_jdYCc\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe> <\/p>\n<p><a href=\"http:\/\/telmap.confolio.org\/scam\/2\/resource\/844?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This page is a sub-page of our page on Plane Curves. \/\/\/\/\/\/\/ Related KMR-pages: \u2022 &#8230; \/\/\/\/\/\/\/ Other relevant sources of information: \u2022 Inverse curves at Wikipedia \/\/\/\/\/\/\/ The interactive simulations on this page can be navigated with the Free Viewer of the Graphing Calculator. \/\/\/\/\/\/\/ Inversion of a polar grid: The interactive simulation that &hellip; <a href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/geometry-2\/metric-geometry\/euclidean-geometry\/geometry\/plane-curves\/inverse-curves\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Inverse Curves<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":304,"menu_order":5,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-366","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/366","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/comments?post=366"}],"version-history":[{"count":0,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/366\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/304"}],"wp:attachment":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/media?parent=366"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}