Having conquered equidistant spacing along a B\u00c3\u00a9zier curve, my thoughts now turn to the same problem for an ellipse. I have solved the problem for a circle of course, which is a special case of an ellipse. One would think that going from a circle to an ellipse would be mathematically easy: it’s easy to compute a point on an ellipse given centre and radii, and an ellipse is just a 2-way stretched circle, right?<\/p>\n
Well, as with the B\u00c3\u00a9zier curve, it’s not as simple as simply incrementing the angle parameter around the ellipse as one can with a circle, because obviously in that case the distance between successive points will vary.<\/p>\n
And then one comes up against the rather startling fact that there is no simple exact equation for the circumference of an ellipse<\/b>. No variant of 2πr here. As one reference puts it, “there are simple formulas but they are not exact, and there are exact formulas but they are not simple”. The exact formulas are infinite sums. The simple formulas can be glaringly inexact, depending on the ellipse.<\/p>\n
I’m thinking on it some more. If, as I suspect, there proves to be no easy closed form solution to how much to vary the angle (or other alternative parameters of the curve) to achieve uniform spacing of radial points, I can fall back on the same solution as for the B\u00c3\u00a9zier curve, i.e. sampling and interpolation.<\/p>\n","protected":false},"excerpt":{"rendered":"
Having conquered equidistant spacing along a B\u00c3\u00a9zier curve, my thoughts now turn to the same problem for an ellipse. I have solved the problem for a circle of course, which is a special case of an ellipse. One would think that going from a circle to an ellipse would be…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13,11,8],"tags":[],"class_list":["post-456","post","type-post","status-publish","format-standard","hentry","category-lisp","category-maths","category-programming"],"_links":{"self":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/456","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=456"}],"version-history":[{"count":0,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/456\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=456"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=456"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}