More writing for the math and browser APIs

blog/2024-11-15-playing-with-fire/1-introduction/GasketWeighted.tsx

@@ -31,7 +31,7 @@ export default function GasketWeighted() {
         <>
             <div className={styles.inputElement}>
                 <p><TeX>{title}</TeX>: {weight}</p>
-                <input type={'range'} min={0} max={1} step={.01} style={{width: '100%'}} value={weight}
+                <input type={'range'} min={0} max={1} step={.01} style={{width: '100%', background: 'transparent'}} value={weight}
                     onInput={e => setWeight(Number(e.currentTarget.value))}/>
             </div>
         </>

blog/2024-11-15-playing-with-fire/1-introduction/cameraGasket.ts

@@ -0,0 +1,10 @@
+export function camera(
+    size: number,
+    x: number,
+    y: number
+): [number, number] {
+    return [
+        Math.floor(x * size),
+        Math.floor(y * size)
+    ];
+}

blog/2024-11-15-playing-with-fire/1-introduction/chaosGame.js

@@ -30,5 +30,4 @@ function* chaosGame({width, height}) {
   yield img;
 }
 
-// Wiring so the code above displays properly
 render(<Gasket f={chaosGame}/>)

blog/2024-11-15-playing-with-fire/1-introduction/index.mdx

@@ -36,9 +36,7 @@ can understand without too much prior knowledge.
 ## Iterated function systems
 
 :::note
-
 This post covers section 2 of the Fractal Flame Algorithm paper
-
 :::
 
 As mentioned, fractal flames are a type of "[iterated function system](https://en.wikipedia.org/wiki/Iterated_function_system),"
@@ -50,15 +48,18 @@ $$
 S = \bigcup_{i=0}^{n-1} F_i(S)
 $$
 
-### Fixed set
+TODO: I'm not sure what the intuitive explanation here is. Is the idea that the solution is all points
+produced by applying each function to all points in the solution? And the purpose of the chaos game is
+that if we find one point in the solution set, we can effectively discover all the other points?
 
-First, $S$. $S$ is the set of points in two dimensions (in math terms, $S \in \mathbb{R}^2$) that represent
+### Solution set
-a "solution" of some kind. Our goal is to find all points in the set $S$, plot them, and display that image.
+
+First, $S$. $S$ is the set of points in two dimensions (in math terms, $S \in \mathbb{R}^2$)
+that represent a "solution" of some kind to our equation.
+Our goal is to find all points in $S$, plot them, and display that image.
 
 For example, if we say $S = \{(0,0), (1, 1), (2, 2)\}$, there are three points to plot:
 
-<!-- TODO: What is a stationary point? How does it relate to the chaos game? Why does the chaos game work? -->
-
 import {VictoryChart, VictoryTheme, VictoryScatter, VictoryLegend} from "victory";
 export const simpleData = [
     {x: 0, y: 0},
@@ -71,35 +72,52 @@ export const simpleData = [
 </VictoryChart>
 
 However, this is a pretty boring image. With fractal flames, rather than listing individual points,
-we use functions to describe which points are part of the solution.
+we use functions to describe which points are part of the solution. This means there are an infinite
-
-TODO: Explain characteristics of the solution - fixed set
-
-This means there are an infinite
 number of points, but if we find _enough_ points to plot, we'll end up with a nice picture.
-And if we choose different functions to start with, our solution set changes, and we'll end up
+And if we change the functions, our solution changes, and we'll get a new picture.
-with a new picture.
 
-However, it's not clear which points belong in the solution just by staring at the functions.
+### Transform functions
-We'll need a computer to figure it out.
 
-### Transformation functions
+Second, the $F_i(S)$ functions, also known as "transforms."
+At their most basic, each $F_i$ takes in a 2-dimensional point and gives back a new point
+(in math terms, $F_i \in \mathbb{R}^2 \rightarrow \mathbb{R}^2$).
+While you could theoretically use any function, we'll focus on a specific kind of function
+known as an "[affine transformation](https://en.wikipedia.org/wiki/Affine_transformation)."
 
-Second, $F_i(S)$. At their most basic, each $F_i$ is a function that takes in a 2-dimensional point and transforms
+The general form of an affine transformation is:
-it into a new 2-dimensional point: $F_i \in \mathbb{R}^2 \rightarrow \mathbb{R}^2$. It's worth discussing
-these functions, but not critical, so **this section is optional**.
-
-In mathematical terms, each $F_i$ is a special kind of function called an [affine transformation](https://en.wikipedia.org/wiki/Affine_transformation).
-We can think of them like mapping from one coordinate system to another. For example, we can define a coordinate system
-where everything is shifted over:
 
 $$
-F_{shift}(x, y) = (x + 1, y)
+F_i(a_i \cdot x + b_i \cdot y + c_i, d_i \cdot x + e_i \cdot y + f_i)
 $$
 
-That is, for an input point $(x, y)$, the output point will be $(x + 1, y)$:
+import transformSource from "!!raw-loader!../src/transform"
 
-export const shiftData = simpleData.map(({x, y}) => { return {x: x + 1, y} })
+<CodeBlock language="typescript">{transformSource}</CodeBlock>
+
+The parameters ($a_i$, $b_i$, etc.) are values we get to choose.
+For example, we can represent a "shift" function like this:
+
+$$
+\begin{align*}
+a &= 1 \\
+b &= 0 \\
+c &= 0.5 \\
+d &= 0 \\
+e &= 1 \\
+f &= 1.5 \\
+F_{shift}(x,y) &= (1 \cdot x + 0 \cdot y + 0.5, 0 \cdot x + 1 \cdot y + 0.5) \\
+F_{shift}(x, y) &= (x + 0.5, y + 0.5)
+\end{align*}
+$$
+
+Applying this function to our original points will give us a new set of points:
+
+import {applyCoefs} from "../src/transform"
+
+export const coefs = {a: 1, b: 0, c: 0.5, d: 0, e: 1, f: 1.5}
+export const toData = ([x, y]) => ({x, y})
+
+export const shiftData = simpleData.map(({x, y}) => toData(applyCoefs(x, y, coefs)))
 
 <VictoryChart theme={VictoryTheme.clean}>
     <VictoryScatter data={simpleData} size={5} style={{data: {fill: "blue"}}}/>
@@ -115,32 +133,37 @@ export const shiftData = simpleData.map(({x, y}) => { return {x: x + 1, y} })
     />
 </VictoryChart>
 
-This is a simple example designed to illustrate the principle. In general, $F_i$ functions have the form:
+Fractal flames use more complex functions, but they all start with this structure.
+
+<details>
+    <summary>If you're interested in more math...</summary>
+
+    TODO: Contractive functions, attractors, etc.?
+</details>
+
+### Fixed set
+
+With those definitions in place, we can try stating the original problem in
+a more natural way:
 
 $$
-F_i(x,y) = (a_i \cdot x + b_i \cdot y + c_i, d_i \cdot x + e_i \cdot y + f_i)
+S = \bigcup_{i=0}^{n-1} F_i(S)
 $$
 
-The parameters ($a_i$, $b_i$, etc.) are values we get to choose. In the example above, we can represent our shift
+> The solution, $S$, is the union of all sets produced by applying each function, $F_i$,
-function using these parameters:
+> to points in the solution.
 
-$$
+There's just one small problem: to solve the equation, we must already know what the solution is?
-a_i = 1 \hspace{0.5cm} b_i = 0 \hspace{0.5cm} c_i = 1 \\
-d_i = 0 \hspace{0.5cm} e_i = 1 \hspace{0.5cm} f_i = 0 \\
-$$
 
-$$
+TODO: Phrase it another way?
-\begin{align*}
+A point is in the solution if it can be reached by applying
-F_{shift}(x,y) &= (1 \cdot x + 0 \cdot y + 1, 0 \cdot x + 1 \cdot y + 0) \\
+one of the functions to another point in the solution?
-F_{shift}(x,y) &= (x + 1, y)
+Is that the definition of a fixed set?
-\end{align*}
-$$
-
-Fractal flames use more complex functions to produce a wide variety of images, but all follow this same format.
 
 ## Sierpinski's gasket
 
-Using these definitions, we can build the first image. The paper defines a function system for us:
+With the math out of the way, we're ready to build our first IFS.
+The Fractal Flame paper provides us three functions we can use for our system:
 
 $$
 F_0(x, y) = \left({x \over 2}, {y \over 2} \right) \\
@@ -169,7 +192,7 @@ $$
 
 Let's turn this into code, one piece at a time.
 
-First, the "bi-unit square" is the range $[-1, 1]$. We can pick a random point like this:
+First, the "bi-unit square" is the range $[-1, 1]$. We can :
 
 import biunitSource from '!!raw-loader!../src/randomBiUnit'
 
@@ -181,14 +204,36 @@ import randintSource from '!!raw-loader!../src/randomInteger'
 
 <CodeBlock language="typescript">{randintSource}</CodeBlock>
 
-Finally, implementing the `plot` function. Web browsers have a [Canvas API](https://developer.mozilla.org/en-US/docs/Web/API/Canvas_API)
+### Plotting
-we can use for 2D graphics. In our case, the plot function will take an $(x,y)$ coordinate and plot it by
+
-coloring the corresponding pixel in an [ImageData](https://developer.mozilla.org/en-US/docs/Web/API/ImageData):
+Finally, implementing the `plot` function. This blog series
+is designed to be interactive, so everything shows
+real-time directly in the browser. As an alternative,
+software like `flam3` an Apophysis can also save an image.
+
+To display the results, we'll use the [Canvas API](https://developer.mozilla.org/en-US/docs/Web/API/Canvas_API).
+This allows us to manipulate individual pixels an image,
+and display it on screen.
+
+First, we need to convert from Fractal Flame coordinates to pixel coordinates.
+To simplify things, we'll assume that we're plotting a square image,
+and we'll focus on the range $[0, 1]$ for both $x$ and $y$:
+
+import cameraSource from "!!raw-loader!./cameraGasket"
+
+<CodeBlock language="typescript">{cameraSource}</CodeBlock>
+
+Next, we'll use an [`ImageData` object](https://developer.mozilla.org/en-US/docs/Web/API/ImageData)
+to store the pixel data.
+Each pixel in the image on screen has a corresponding index in the `data` array.
+To plot our image, we set that pixel to be black:
 
 import plotSource from '!!raw-loader!./plot'
 
 <CodeBlock language="typescript">{plotSource}</CodeBlock>
 
+Putting it all together, we have our first image:
+
 import Playground from '@theme/Playground'
 import Scope from './scope'
 
@@ -199,8 +244,8 @@ import chaosGameSource from '!!raw-loader!./chaosGame'
 <hr/>
 
 <small>
-Note: The image here is slightly different than the fractal flame paper; I think the paper has an error,
+Note: The image here is slightly different than the one in the paper.
-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).
+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

blog/2024-11-15-playing-with-fire/1-introduction/plot.ts

@@ -1,10 +1,6 @@
-/**
+// hidden-start
- * ImageData is an array that contains
+import {camera} from "./cameraGasket"
- * four elements per pixel (one for each
+// hidden-end
- * red, green, blue, and alpha value).
- * This maps from pixel coordinates
- * to the array index
- */
 function imageIndex(
   width: number,
   x: number,
@@ -18,17 +14,9 @@ export function plot(
   y: number,
   img: ImageData
 ) {
-  // Translate (x,y) coordinates
+  let [pixelX, pixelY] = camera(img.width, x, y);
-  // to pixel coordinates.
-  // Also known as a "camera" function.
-  //
-  // The display range is:
-  //  x=[0, 1]
-  //  y=[0, 1]
-  let pixelX = Math.floor(x * img.width);
-  let pixelY = Math.floor(y * img.height);
 
-  const index = imageIndex(
+  const i = imageIndex(
     img.width,
     pixelX,
     pixelY
@@ -36,17 +24,18 @@ export function plot(
 
   // Skip pixels outside the display range
   if (
-    index < 0 ||
+    i < 0 ||
-    index > img.data.length
+    i > img.data.length
   ) {
     return;
   }
 
   // Set the pixel to black by writing 0
-  // to the first three elements,
+  // to the first three elements at the index
-  // and 255 to the last element
+  // (red, green, and blue, respectively),
-  img.data[index] = 0;
+  // and 255 to the last element (alpha)
-  img.data[index + 1] = 0;
+  img.data[i] = 0;
-  img.data[index + 2] = 0;
+  img.data[i + 1] = 0;
-  img.data[index + 3] = 0xff;
+  img.data[i + 2] = 0;
+  img.data[i + 3] = 0xff;
 }

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

@@ -12,22 +12,15 @@ shapes and patterns that fractal flames are known for.
 <!-- truncate -->
 
 :::note
+This post uses a set of [reference parameters](../params.flame) to demonstrate the fractal flame algorithm.
+If you're interested in tweaking the parameters, or generating your own art, [Apophysis](https://sourceforge.net/projects/apophysis/)
+can load that file and you can try tweaking things yourself!
 
-This post uses a set of [reference parameters](../params.flame) to demonstrate a working
+This post covers section 3 of the Fractal Flame Algorithm paper
-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/)
-can load that file and gives full control over the image.
-
 :::
 
 ## Transforms and variations
 
-:::note
-
-This post covers section 3 of the Fractal Flame Algorithm paper
-
-:::
-
 import CodeBlock from '@theme/CodeBlock'
 
 We previously introduced transforms as the "functions" of an "iterated function system," and showed how

blog/2024-11-15-playing-with-fire/src/applyTransform.ts

@@ -1,5 +1,4 @@
-import {Transform} from "./transform";
+import {Transform, Coefs, applyCoefs} from "./transform";
-import {applyCoefs, Coefs} from "./coefs";
 import {blend, VariationBlend} from "./blend";
 
 export const applyTransform = (coefs: Coefs, variations: VariationBlend): Transform =>

blog/2024-11-15-playing-with-fire/src/coefs.ts

@@ -1,11 +0,0 @@
-export interface Coefs {
-    a: number, b: number, c: number,
-    d: number, e: number, f: number
-}
-
-export function applyCoefs(x: number, y: number, coefs: Coefs): [number, number] {
-    return [
-        (x * coefs.a + y * coefs.b + coefs.c),
-        (x * coefs.d + y * coefs.e + coefs.f)
-    ]
-}

blog/2024-11-15-playing-with-fire/src/transform.ts

@@ -1 +1,13 @@
-export type Transform = (x: number, y: number) => [number, number];
+export type Transform = (x: number, y: number) => [number, number];
+
+export interface Coefs {
+    a: number, b: number, c: number,
+    d: number, e: number, f: number
+}
+
+export function applyCoefs(x: number, y: number, coefs: Coefs): [number, number] {
+    return [
+        (x * coefs.a + y * coefs.b + coefs.c),
+        (x * coefs.d + y * coefs.e + coefs.f)
+    ]
+}