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Bradlee Speice
2024-12-13 20:03:53 -05:00
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blog/2024-11-15-playing-with-fire/1-introduction/index.mdx
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@ -170,11 +170,11 @@ Thus, by applying the functions in our system to "fixed points," we will find th
However, this is all a bit vague, so let's work through an example.
<details>
<summary>But if you want a bit more math...</summary>
<summary>If you want a bit more math first...</summary>
...then it's worth mentioning some information I've skipped over.
...then there are some details worth mentioning that I've glossed over so far.
First, the Hutchinson paper explains that the functions $F_i$ must be _contractive_ for this result to hold.
First, the Hutchinson paper requires that the functions $F_i$ be _contractive_ tor the solution set to exist.
That is, applying the function to a point must bring it closer to other points. However, as the Fractal Flame
algorithm demonstrates, we only need functions to be contractive _on average_. At worst, the system will
degenerate and produce a bad image.
@ -200,7 +200,7 @@ $$
### The chaos game
Next, how do we find the "fixed points" we mentioned earlier? The paper lays out an algorithm called the "[chaos game](https://en.wikipedia.org/wiki/Chaos_game)"
that will give us points in the solution set:
that will give us points in the solution set.
$$
\begin{align*}
@ -213,7 +213,11 @@ $$
\end{align*}
$$
Let's turn this into code, one piece at a time.
:::note
In effect, the chaos game algorithm implements the "finite compositions of $F_{i_1..i_p}$ mentioned earlier.
:::
Now, let's turn this into code, one piece at a time.
First, the "bi-unit square" is the range $[-1, 1]$. We can :
@ -272,10 +276,13 @@ I think the paper has an error, so I'm choosing to plot the image
in a way that's consistent with [`flam3` itself](https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/rect.c#L440-L441).
</small>
## Weights
### Weights
Finally, we'll introduce a "weight" parameter ($w_i$) assigned to each function, which controls
how often that function is used:
Finally, we'll introduce a "weight" ($w_i$) for each function that controls how often we choose
that function during the chaos game relative to each other function.
For Sierpinski's Gasket, we start with equal weighting,
but you can see how changing the function weights affects the image below:
import randomChoiceSource from '!!raw-loader!../src/randomChoice'
@ -288,4 +295,13 @@ import chaosGameWeightedSource from "!!raw-loader!./chaosGameWeighted";
import GasketWeighted from "./GasketWeighted";
import {SquareCanvas} from "../src/Canvas";
<SquareCanvas><GasketWeighted/></SquareCanvas>
<SquareCanvas><GasketWeighted/></SquareCanvas>
## Summary
Studying the foundations of fractal flames is challenging,
but we now have an understanding of both the mathematics
and implementation of iterated function systems.
In the next post, we'll study the first innovation that fractal flames
bring: variations.