{"id":4066,"date":"2020-03-30T10:35:14","date_gmt":"2020-03-30T18:35:14","guid":{"rendered":"https:\/\/bendwavy.org\/wp\/?p=4066"},"modified":"2020-12-16T05:56:17","modified_gmt":"2020-12-16T13:56:17","slug":"scribbles-the-ensmoothening-part-iii","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=4066","title":{"rendered":"Scribbles: The Ensmoothening, Part III"},"content":{"rendered":"<p>Many of the curves in <a href=\"?p=3554\">this chart<\/a> have some unsightly wiggles.  That&#8217;s because, when a function of degree 2 or higher tries to approximate a piecewise constant, it tends to go back and forth across the target.  So here instead I fitted each such function not to the piecewise constant directly but to the fit of the next lower degree.<br \/>\n<img decoding=\"async\" src=\"\/takana\/001122-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/001123-jk.svg\"> <!--more--> <img decoding=\"async\" src=\"\/takana\/001210-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/001212-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/001221-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/001233-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/011223-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/011232-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/011234-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/012123-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/012210-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/012321-jk1.svg\"> <img decoding=\"async\" src=\"\/takana\/012323-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/012332-jk.svg\"> <img decoding=\"async\" src=\"\/takana\/101232-jk.svg\"><br \/>\nSo some accuracy in fit is lost &ndash; but this was useful practice for an application in which, perhaps oddly, that doesn&#8217;t matter.<\/p>\n<p>Column 0 is the target.  Column 1 is made of circle segments with G1 continuity, and some of the joins are easy to see.  Column 2 is quadratic spirals with G2 continuity, and the improvement is clear.  It&#8217;s much harder to find differences between G2, G3, G4; that may be partly due to the limitations of the cubic splines that imitate these transcendental curves.<\/p>\n<p>This was also incidentally the first serious test of my piecewise polynomial object&#8217;s new integration method!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Many of the curves in this chart have some unsightly wiggles. That&#8217;s because, when a function of degree 2 or higher tries to approximate a piecewise constant, it tends to go back and forth across the target. So here instead &hellip; <a href=\"https:\/\/bendwavy.org\/wp\/?p=4066\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[72],"tags":[],"class_list":["post-4066","post","type-post","status-publish","format-standard","hentry","category-curve-fitting"],"_links":{"self":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/4066","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4066"}],"version-history":[{"count":7,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/4066\/revisions"}],"predecessor-version":[{"id":4145,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/4066\/revisions\/4145"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4066"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4066"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4066"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}