I pulled a .zip file from https://www.deviantart.com/clairejones/art/Apophysis-Starter-Flames-V-67029474 and the file I was interested was the .flame file which contains parameters for several different Fractal Flames.

One of them looks like this:

<flame name="CJ_Starter_Flame_1" version="Apophysis 2.06c beta" size="640 512" center="0.536884822382779 -0.626382357937026" scale="764.167485097157" angle="1.82997772071605" rotate="-104.85" oversample="2" filter="1" quality="5" background="0 0 0" brightness="2" gamma="5" >
   <xform weight="0.5" color="0" linear="0.98505982500501" spherical="0.0149401749949902" coefs="0.047466 -0.574133 0.524841 0.137706 0.620555 -0.116118" />
   <xform weight="0.5" color="1" spherical="0.17593513103202" bubble="0.82406486896798" coefs="0.268823 -0.261644 0.261644 0.268823 0.204257 0.083402" />
   <palette count="256" format="RGB">
      253625384A3348593D5C664C6E685B7A7260827363938169
...
   </palette>
</flame>

Having read the original Draves & Reckase paper some of this is obvious. The coefs= probably contains the parameters of the affine transform. The thing that really confuses me is how a single xform clause mentions two variants, e.g. linear and spherical. It appears that the values on each attribute sum to 1 so they are probably weights/probabilities. But why are there two xform clauses? Are they alternating weighted transform sets?

There is some information about the XML file format used for .flame files. There are a number of forks of this program written in different languages, the original site is by definition the place to go for answers; unless you are using a fork - the original seems to have been abandoned years ago.

Coefs is 6 floats which defines the coefficients for the affine part of the function, in row order (a d b e c f). Xform has an interpolation_type which is a string describing the method for interpolation of xform positions: options are "log", "linear", "old", etc. (a list of a few dozen options is here). There can be multiple xforms, each applies a transformation to the previous state.

The spherical transform is complex, there's a 19 page .docx tutorial with 9 pages of examples at: "Understanding Spherical Transforms: Apophysis Tut" by UniversalKinase (direct link to .ZIP file). Another tutorial is "Apo-Tuto Sphericals" by Fiery-Fire.

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