speice.io/blog/2024-11-15-playing-with-fire/2-transforms/index.mdx

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---
slug: 2024/11/playing-with-fire-transforms
title: "Playing with fire: Transforms and variations"
date: 2024-11-15 13:00:00
authors: [bspeice]
tags: []
---
Now that we have a basic chaos game in place, it's time to spice things up. Transforms and variations create the
interesting patterns that fractal flames are known for.
<!-- truncate -->
This blog post uses a set of reference parameters ([available here](../params.flame)) to demonstrate a practical
implementation of the fractal flame algorithm. If you're interested in tweaking the parameters, or generating
your own art, [Apophysis](https://sourceforge.net/projects/apophysis/) is a good introductory tool.
## Transforms and variations
import CodeBlock from '@theme/CodeBlock'
We previously introduced "transforms" as the "functions" of an "iterated function system." Their general format is:
$$
F_i(x,y) = (a_i \cdot x + b_i \cdot y + c_i, \hspace{0.2cm} d_i \cdot x + e_i \cdot y + f_i)
$$
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import coefsSrc from '!!raw-loader!../src/coefs'
<CodeBlock language={'typescript'}>{coefsSrc}</CodeBlock>
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We also introduced the Sierpinski Gasket functions ($F_0$, $F_1$, and $F_2$), demonstrating how they are related to
the general format. For example:
$$
\begin{align*}
F_0(x,y) &= \left({x \over 2}, {y \over 2}\right) \\
&= (a_0 \cdot x + b_0 \cdot y + c_o, d_0 \cdot x + e_0 \cdot y + f_0) \\
& a_0 = 0.5 \hspace{0.2cm} b_0 = 0 \hspace{0.2cm} c_0 = 0 \\
& d_0 = 0 \hspace{0.2cm} e_0 = 0.5 \hspace{0.2cm} f_0 = 0
\end{align*}
$$
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However, these transforms are pretty boring. We can build more exciting images by using additional functions
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within the transform. These "sub-functions" are called "variations":
$$
F_i(x, y) = V_j(a_i \cdot x + b_i \cdot y + c_i, \hspace{0.2cm} d_i \cdot x + e_i \cdot y + f_i)
$$
The fractal flame paper lists 49 variation functions ($V_j$ above), but the sky's the limit here.
For example, the official `flam3` implementation supports
[98 variations](https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/variations.c).
Our reference image will focus on just four variations:
### Linear (variation 0)
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This variation returns the $x$ and $y$ coordinates as-is:
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$$
V_0(x,y) = (x,y)
$$
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import linearSrc from '!!raw-loader!../src/linear'
<CodeBlock language={'typescript'}>{linearSrc}</CodeBlock>
Before we move on, it's worth mentioning the relationship between this variation and the Sierpinski Gasket.
Specifically, we can think of the Gasket as a fractal flame that uses only the linear variation.
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### Julia (variation 13)
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This variation still uses just the $x$ and $y$ coordinates, but does crazy things with them:
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<small>TODO: Is this related to the Julia set?</small>
$$
\begin{align*}
r &= \sqrt{x^2 + y^2} \\
\theta &= \text{arctan}(x / y) \\
\Omega &= \left\{
\begin{array}{lr}
0 \hspace{0.4cm} \text{w.p. } 0.5 \\
\pi \hspace{0.4cm} \text{w.p. } 0.5 \\
\end{array}
\right\} \\
V_{13}(x, y) &= \sqrt{r} \cdot (\text{cos} ( \theta / 2 + \Omega ), \text{sin} ( \theta / 2 + \Omega ))
\end{align*}
$$
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import juliaSrc from '!!raw-loader!../src/julia'
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<CodeBlock language={'typescript'}>{juliaSrc}</CodeBlock>
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### Popcorn (variation 17)
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This is known as a "dependent variation" because it depends on knowing the transform coefficients
(specifically, $c$ and $f$):
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$$
V_{17}(x,y) = (x + c \cdot \text{sin}(\text{tan }3y), y + f \cdot \text{sin}(\text{tan }3x))
$$
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import popcornSrc from '!!raw-loader!../src/popcorn'
<CodeBlock language={'typescript'}>{popcornSrc}</CodeBlock>
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### PDJ (variation 24)
This is known as a "parametric" variation because it has additional parameters given to it:
$$
p_1 = \text{pdj.a} \hspace{0.2cm} p_2 = \text{pdj.b} \hspace{0.2cm} p_3 = \text{pdj.c} \hspace{0.2cm} p_4 = \text{pdj.d} \\
V_{24} = (\text{sin}(p_1 \cdot y) - \text{cos}(p_2 \cdot x), \text{sin}(p_3 \cdot x) - \text{cos}(p_4 \cdot y))
$$
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import pdjSrc from '!!raw-loader!../src/pdj'
<CodeBlock language={'typescript'}>{pdjSrc}</CodeBlock>
### Blending
Now, one variation is fun, but we can also combine variations in a single transform by "blending."
First, each variation is assigned a value that describes how much it affects the transform function ($v_j$).
Afterward, sum up the $x$ and $y$ values respectively:
$$
F_i(x,y) = \sum_{j} v_{ij} V_j(a_i \cdot x + b_i \cdot y + c_i, \hspace{0.2cm} d_i \cdot x + e_i \cdot y + f_i)
$$
The formula looks intimidating, but it's not hard to implement:
import baselineSrc from '!!raw-loader!./baseline'
<CodeBlock language={'typescript'}>{baselineSrc}</CodeBlock>
TODO: Mention that the Sierpinski Gasket is just a blend with linear weight 1, all others 0. Maybe replace comment above about Sierpinski Gasket and linear transform?
And with that in place, we have enough to render a first full fractal flame: