Andrew,
- anyone out there ever played with non-linear IFS?
Flame fractals are IFS fractals with non-linear transforms. See http://flam3.com/ for Scott Draves' flames, http://www.apophysis.org/ for Mark Townsend's implementation, and of course there's the KPT FraxFlame plugin if you want to pay money.
Damien M. Jones \\ dmj@fractalus.com \\ Fractalus Galleries & Info: \\ http://www.fractalus.com/
Thanks for this. I vaguely remember visiting the first website some time back [before I understood how IFS works]. From what I can gather from the mathematical description, each function in the system is a simple linear map, optionally with a non-linear transformation applied afterwards. ("spherical", "sinusoidal", etc.) I suppose this would make it intuitively easier to understand what the various parameters are actually doing... Suppose I use the transformation f(x, y) = (Ax^2 + Bxy + Cx + Dy + E, Fy^2 + Gxy + Hx + Iy + J). If A=B=F=G=0 then we have an ordinary linear transformation determined by C, D, E, H, I, J. (Specifically, the vector (x, y) multiplied by the matrix ((C D) (H I)) and then adding the vector (E J)... unless I got my numbers the wrong way round!) Using the linear case, adjusting C and I controls the scaling factor, E and J control the translation, and by fiddling with C, D, H and I all at once you can do rotation. But what - geometrically - do the remaining parameters do? The main factor about linear functions is that the scaling factor is constant over the whole plane, the translation distance is constant over the whole plane, the rotation angle is constant over the whole plane, etc. I would imagine that with a quadratic map, at least one of these things could be made to vary over the plane... Thanks. Andrew. PS. The composition of two linear functions is a linear function. Is this true of heigher polynomials?