Editing Fractal flame (section)

===Creating the histogram===
[[File:Apophysis-100303-104.jpg|thumb|right|A fractal flame.]]
First, one iterates a set of functions, starting from a randomly chosen point ''P = (P.x,P.y,P.c)'', where the third coordinate indicates the current color of the point.

:Set of flame functions: <math>\begin{cases}
F_1(x,y), \quad p_1 \\
F_2(x,y), \quad p_2 \\
\dots \\
F_n(x,y), \quad p_n
\end{cases}</math>

In each iteration, choose one of the functions above where the probability that ''F<sub>j</sub>'' is chosen is ''p<sub>j</sub>''. Then one computes the next iteration of ''P'' by applying ''F<sub>j</sub>'' on ''(P.x,P.y)''.

Each individual function has the following form:

:<math>F_j(x,y) = \sum_{V_k \in Variations} w_k \cdot V_k(a_j x + b_j y +c_j,d_j x + e_j y +f_j)</math>

where the parameter ''w<sub>k</sub>'' is called the weight of the '''''variation''''' ''V<sub>k</sub>''. Draves suggests
<ref name="dravespdf">{{cite web|url=https://flam3.com/flame_draves.pdf|title=The Fractal Flame Algorithm}}&nbsp;{{small|(22.5&nbsp;MB)}}</ref> that all <math>w_k</math>:s are non-negative and sum to one, but implementations such as Apophysis do not impose that restriction.

The functions ''V<sub>k</sub>'' are a set of predefined functions. 
A few examples<ref name="dravespdf" /> are

* V<sub>0</sub>(''x'',''y'') = (''x'',''y'') (Linear)
* V<sub>1</sub>(''x'',''y'') = (sin ''x'',sin ''y'') (Sinusoidal)
* V<sub>2</sub>(''x'',''y'') = (''x'',''y'')/(''x''<sup>2</sup>+''y''<sup>2</sup>) (Spherical)

The color ''P.c'' of the point is blended with the color associated with the latest applied function ''F<sub>j</sub>'':

: P.c := (P.c + (F<sub>j</sub>)<sub>color</sub>)  / 2

After each iteration, one updates the histogram at the point corresponding to ''(P.x,P.y)''. This is done as follows:

<syntaxhighlight lang="cpp">
histogram[x][y][FREQUENCY] := histogram[x][y][FREQUENCY]+1
histogram[x][y][COLOR] := (histogram[x][y][COLOR] + P.c)/2
</syntaxhighlight>

The colors in the image will therefore reflect what functions were used to get to that part of the image.