{"id":1514,"date":"2011-09-19T16:00:06","date_gmt":"2011-09-19T23:00:06","guid":{"rendered":"https:\/\/www.reenigne.org\/blog\/?p=1514"},"modified":"2011-08-28T21:40:04","modified_gmt":"2011-08-29T04:40:04","slug":"interactive-ifs","status":"publish","type":"post","link":"https:\/\/www.reenigne.org\/blog\/interactive-ifs\/","title":{"rendered":"Interactive IFS"},"content":{"rendered":"

I want to write a program that allows you to explore Iterated Function System (IFS) fractals interactively by moving points around with the mouse.<\/p>\n

There's a few different ways to do this, depending on what set of transformations you use.<\/p>\n

IFS fractals usually use affine transformations, which encompass translations, rotations, dilations and shears. This can be done with 3 points per transformation - if one considers the points as complex numbers we have the transformation f(x)=ax + b\\overline{x} + c. However, rather than controlling a, b and c directly I think it would work better to move the images of some interesting points like 0, 1 and i (i.e. move c, a+b+c and (a-b)i+c). Then the geometric interpretation of the control points would be easy to understand - they would just be the corners of a transformed rectangle.<\/p>\n

However, there are other possible transformations. We could reduce the number of control points to 2 and disallow non-orthogonal transformations, giving f(x)=ax + b and control points 0 and 1 mapping to a and a+b.<\/p>\n

We could move to quadratics f(x) = ax^2 + bx + c and move c, a+b+c and c+ib-a, and with 4 points we can do cubics f(x) = ax^3 + bx^2+cx+d (in which case we would probably use control points f(0), f(1), f(i) and f(1+i)).<\/p>\n

We can even go all the way to f(x) = ax^2 + b\\overline{x}^2 + c|x|^2 + dx + e\\overline{x} + g (6 control points) if we wanted to go really crazy - that might be a bit unwieldy though.<\/p>\n

I'd also like to be able to associate a colour with each transformation so that the colour of a point in the final image depends on the sequence of transformations that led to that point. Perhaps C = \\alpha C_{prev} + (1-\\alpha)C_{transform} for some \\alpha.<\/p>\n","protected":false},"excerpt":{"rendered":"

I want to write a program that allows you to explore Iterated Function System (IFS) fractals interactively by moving points around with the mouse. There's a few different ways to do this, depending on what set of transformations you use. IFS fractals usually use affine transformations, which encompass translations, rotations, dilations and shears. This can […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18],"tags":[],"class_list":["post-1514","post","type-post","status-publish","format-standard","hentry","category-fractals"],"_links":{"self":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/1514","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/comments?post=1514"}],"version-history":[{"count":11,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/1514\/revisions"}],"predecessor-version":[{"id":1523,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/1514\/revisions\/1523"}],"wp:attachment":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/media?parent=1514"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/categories?post=1514"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/tags?post=1514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}