{"id":3554,"date":"2015-12-20T14:23:52","date_gmt":"2015-12-20T22:23:52","guid":{"rendered":"http:\/\/bendwavy.org\/wp\/?p=3554"},"modified":"2018-03-22T14:02:11","modified_gmt":"2018-03-22T22:02:11","slug":"scribbles-the-ensmoothening-part-ii","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=3554","title":{"rendered":"Scribbles: The Ensmoothening, Part II"},"content":{"rendered":"<p>One thing I noticed in <a href=\"?p=3529\">that last series of charts<\/a> is that more than one degree of discontinuity doesn&#8217;t help: the best-looking curves are mostly on the diagonal, where only the last nonzero derivative is discontinuous.  Here, therefore, are those curves all together.<br \/>\n<img decoding=\"async\" src=\"\/takana\/chart-jb.svg\"><br \/>\nIn column zero, the tangent angle is piecewise constant; in column one, it is a piecewise linear function of path length, resulting in six circular arcs; in column two it is piecewise quadratic, resulting in six clothoid arcs with continuous curvature; and so on.<\/p>\n<p>Of course the arcs are approximated by cubics; to improve the match, I put a knot wherever any derivative crosses zero, as well as at the discontinuities.  (<a href=\"\/takana\/chart-jb-dots.svg\">See the knots<\/a>.)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>One thing I noticed in that last series of charts is that more than one degree of discontinuity doesn&#8217;t help: the best-looking curves are mostly on the diagonal, where only the last nonzero derivative is discontinuous. Here, therefore, are those &hellip; <a href=\"https:\/\/bendwavy.org\/wp\/?p=3554\">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-3554","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\/3554","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=3554"}],"version-history":[{"count":7,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/3554\/revisions"}],"predecessor-version":[{"id":3954,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/3554\/revisions\/3954"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3554"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3554"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3554"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}