Reorganize a bit, write some more

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

@@ -51,7 +51,7 @@ First, $S$. We're generating images, so everything is in two dimensions: $S \in
 all points that are "in the system." To generate our final image, we just plot every point in the system
 like a coordinate chart.
 
-TODO: What is a stationary point? How does it relate to the chaos game?
+TODO: What is a stationary point? How does it relate to the chaos game? Why does the chaos game work?
 
 For example, if we say $S = \{(0,0), (1, 1), (2, 2)\}$, there are three points to plot:
 
@@ -188,3 +188,5 @@ import Gasket from '!!raw-loader!./Gasket'
 Note: The image our chaos game generates is different than the fractal flame paper, but I think the version displayed
 here is correct. As confirmation, the next post will re-create the same image using a different method.
 </small>
+
+TODO: Explanation of function weights $w_i$

blog/2024-11-15-playing-with-fire/2-transforms/BaselineRender.ts

@@ -0,0 +1,5 @@
+import {useColorMode, useThemeConfig} from "@docusaurus/theme-common";
+
+export default function BaselineRender({children}) {
+    const {colorMode, setColorMode} = useColorMode();
+}

blog/2024-11-15-playing-with-fire/2-transforms/baseline.ts

@@ -0,0 +1,22 @@
+// hidden-start
+import { Coefs } from './coefs'
+import { Variation } from './variations'
+// hidden-end
+export function applyTransform(
+    x: number,
+    y: number,
+    coefs: Coefs,
+    variations: [number, Variation][])
+{
+    const transformX = coefs.a * x + coefs.b * y + coefs.c;
+    const transformY = coefs.d * x + coefs.e * y + coefs.f;
+
+    var finalX = 0;
+    var finalY = 0;
+    for (const [blend, variation] of variations) {
+        const [variationX, variationY] = variation(transformX, transformY);
+        finalX += blend * variationX;
+        finalY += blend * variationY;
+    }
+    return [finalX, finalY];
+}

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

@@ -25,6 +25,10 @@ $$
 F_i(x,y) = (a_i \cdot x + b_i \cdot y + c_i, \hspace{0.2cm} d_i \cdot x + e_i \cdot y + f_i)
 $$
 
+import coefsSrc from '!!raw-loader!../src/coefs'
+
+<CodeBlock language={'typescript'}>{coefsSrc}</CodeBlock>
+
 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:
 
@@ -37,7 +41,7 @@ F_0(x,y) &= \left({x \over 2}, {y \over 2}\right) \\
 \end{align*}
 $$
 
-However, these transforms are pretty boring. We can build more exciting images by using some additional functions
+However, these transforms are pretty boring. We can build more exciting images by using additional functions
 within the transform. These "sub-functions" are called "variations":
 
 $$
@@ -52,18 +56,18 @@ Our reference image will focus on just four variations:
 
 ### Linear (variation 0)
 
-This variation returns the $x$ and $y$ coordinates as-is. As mentioned, the Sierpinski Gasket is
+This variation returns the $x$ and $y$ coordinates as-is:
-a fractal flame using only the linear variation:
 
 $$
 V_0(x,y) = (x,y)
 $$
 
-```typescript
+import linearSrc from '!!raw-loader!../src/linear'
-function linear(x: number, y: number) {
+
-    return [x, y];
+<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.
 
 ### Julia (variation 13)
 
@@ -86,18 +90,9 @@ V_{13}(x, y) &= \sqrt{r} \cdot (\text{cos} ( \theta / 2 + \Omega ), \text{sin} (
 \end{align*}
 $$
 
-```typescript
+import juliaSrc from '!!raw-loader!../src/julia'
-function julia(x: number, y: number) {
-    const r = Math.sqrt(Math.pow(x, 2) + Math.pow(y, 2));
-    const theta = Math.atan2(x, y);
-    const omega = Math.random() > 0.5 ? 0 : Math.PI;
 
-    return [
+<CodeBlock language={'typescript'}>{juliaSrc}</CodeBlock>
-        r * Math.cos(theta / 2 + omega),
-        r * Math.sin(theta / 2 + omega)
-    ]
-}
-```
 
 ### Popcorn (variation 17)
 
@@ -108,14 +103,9 @@ $$
 V_{17}(x,y) = (x + c \cdot \text{sin}(\text{tan }3y), y + f \cdot \text{sin}(\text{tan }3x))
 $$
 
-```typescript
+import popcornSrc from '!!raw-loader!../src/popcorn'
-function popcorn(coefs: {c: number, f: number}) {
+
-    return (x: number, y: number) => [
+<CodeBlock language={'typescript'}>{popcornSrc}</CodeBlock>
-        x + coefs.c * Math.sin(Math.tan(3 * y)),
-        y + coefs.f * Math.sin(Math.tan(3 * x))
-    ]
-}
-```
 
 ### PDJ (variation 24)
 
@@ -126,11 +116,27 @@ p_1 = \text{pdj.a} \hspace{0.2cm} p_2 = \text{pdj.b} \hspace{0.2cm} p_3 = \text{
 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))
 $$
 
-```typescript
+import pdjSrc from '!!raw-loader!../src/pdj'
-function pdj(a: number, b: number, c: number, d: number) {
+
-    return (x: number, y: number) => [
+<CodeBlock language={'typescript'}>{pdjSrc}</CodeBlock>
-        Math.sin(a * y) - Math.cos(b * x),
+
-        Math.sin(c * x) - Math.cos(d * y)
+### 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:
+

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

@@ -0,0 +1,13 @@
+// hidden-start
+import { Variation } from './variations'
+// hidden-end
+export const julia: Variation = (x, y) => {
+    const r = Math.sqrt(Math.pow(x, 2) + Math.pow(y, 2));
+    const theta = Math.atan2(x, y);
+    const omega = Math.random() > 0.5 ? 0 : Math.PI;
+
+    return [
+        r * Math.cos(theta / 2 + omega),
+        r * Math.sin(theta / 2 + omega)
+    ]
+}

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

@@ -0,0 +1,4 @@
+// hidden-start
+import {Variation} from "./variations"
+//hidden-end
+export const linear: Variation = (x, y) => [x, y];

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

@@ -3,8 +3,12 @@
  * translated into something that's easier to work with.
  */
 
-import {Coefs, Transform} from "./types";
+import { Coefs } from './coefs';
-import { julia, linear, pdj, popcorn } from "./variation";
+import { Transform } from './transform';
+import { linear } from './linear'
+import { julia } from './julia'
+import { popcorn } from './popcorn'
+import { pdj } from './pdj'
 
 export const identityCoefs: Coefs = {
     a: 1, b: 0, c: 0,
@@ -12,53 +16,51 @@ export const identityCoefs: Coefs = {
 }
 
 export const xform1Weight = 0.56453495;
-export const xform1: Transform = {
+export const xform1Coefs = {
-    coefs: {
+    a: -1.381068, b: -1.381068, c: 0,
-        a: -1.381068, b: -1.381068, c: 0,
+    d: 1.381068, e: -1.381068, f: 0,
-        d: 1.381068, e: -1.381068, f: 0,
-    },
-    coefsPost: identityCoefs,
-    variations: [[1, julia]],
-    color: 0
 }
+export const xform1CoefsPost = identityCoefs;
+export const xform1Variations = [
+    [1, julia]
+]
+export const xform1Color = 0;
 
-const xform2Weight = 0.013135;
+export const xform2Weight = 0.013135;
-export const xform2: Transform = {
+export const xform2Coefs = {
-    coefs: {
+    a: 0.031393, b: 0.031367, c: 0,
-        a: 0.031393, b: 0.031367, c: 0,
+    d: -0.031367, e: 0.031393, f: 0,
-        d: -0.031367, e: 0.031393, f: 0,
-    },
-    coefsPost: {
-        a: 1, b: 0, c: 0.241352,
-        d: 0, e: 1, f: 0.271521,
-    },
-    variations: [
-        [1, linear],
-        [1, popcorn]
-    ],
-    color: 0.844
 }
+export const xform2CoefsPost = {
+    a: 1, b: 0, c: 0.241352,
+    d: 0, e: 1, f: 0.271521,
+}
+export const xform2Variations = [
+    [1, linear],
+    [1, popcorn(xform2Coefs)]
+]
+export const xform2Color = 0.844;
 
 export const xform3Weight = 0.42233;
-export const xform3: Transform = {
+export const xform3Coefs = {
-    coefs: {
+    a: 1.51523, b: -3.048677, c: 0.724135,
-        a: 1.51523, b: -3.048677, c: 0.724135,
+    d: 0.740356, e: -1.455964, f: -0.362059,
-        d: 0.740356, e: -1.455964, f: -0.362059,
-    },
-    coefsPost: identityCoefs,
-    variations: [[1, pdj(1.09358, 2.13048, 2.54127, 2.37267)]],
-    color: 0.349
 }
+export const xform3CoefsPost = identityCoefs;
+export const xform3Variations = [
+    [1, pdj(1.09358, 2.13048, 2.54127, 2.37267)]
+];
+export const xform3Color = 0.349;
 
-export const xformFinal: Transform = {
+export const xformFinalCoefs = {
-    coefs: {
+    a: 2, b: 0, c: 0,
-        a: 2, b: 0, c: 0,
+    d: 0, e: 2, f: 0
-        d: 0, e: 2, f: 0
-    },
-    coefsPost: identityCoefs,
-    variations: [[1, julia]],
-    color: 0
 }
+export const xformFinalCoefsPost = identityCoefs;
+export const xformFinalVariations = [
+    [1, julia]
+]
+export const xformFinalColor = 0;
 
 export const palette =
     "7E3037762C45722B496E2A4E6A2950672853652754632656" +

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

@@ -0,0 +1,9 @@
+// hidden-start
+import { Variation } from './variations'
+//hidden-end
+export function pdj(a: number, b: number, c: number, d: number): Variation {
+    return (x, y) => [
+        Math.sin(a * y) - Math.cos(b * x),
+        Math.sin(c * x) - Math.cos(d * y)
+    ]
+}

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

@@ -0,0 +1,10 @@
+// hidden-start
+import {Coefs} from './coefs'
+import {Variation} from './variations'
+// hidden-end
+export function popcorn({c, f}: Coefs): Variation {
+    return (x, y) => [
+        x + c * Math.sin(Math.tan(3 * y)),
+        y + f * Math.sin(Math.tan(3 * x))
+    ];
+}

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

@@ -0,0 +1,9 @@
+import { Coefs } from './coefs'
+import { Variation } from './variations'
+
+export interface Transform {
+    coefs: Coefs,
+    variations: [number, Variation][],
+    coefsPost: Coefs,
+    color: number
+}

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

@@ -0,0 +1 @@
+export type Variation = (x: number, y: number) => [number, number];

blog/2024-11-15-playing-with-fire/types.ts

@@ -1,24 +0,0 @@
-/**
- * Affine transformation coefficients
- */
-export type Coefs = {
-    a: number;
-    b: number;
-    c: number;
-    d: number;
-    e: number;
-    f: number;
-};
-
-export type Variation = (
-  x: number,
-  y: number,
-  coefs: Coefs
-) => [number, number];
-
-export type Transform = {
-    coefs: Coefs,
-    coefsPost: Coefs,
-    variations: [number, Variation][],
-    color: number
-}

blog/2024-11-15-playing-with-fire/variation.ts

@@ -1,40 +0,0 @@
-import { Variation } from "./types";
-
-export const linear: Variation = (x, y) => [x, y];
-
-function r(x: number, y: number) {
-  return Math.sqrt(Math.pow(x, 2) + Math.pow(y, 2));
-}
-
-function theta(x: number, y: number) {
-  return Math.atan2(x, y);
-}
-
-function omega(): number {
-  return Math.random() > 0.5 ? Math.PI : 0;
-}
-
-export const julia: Variation = (x, y) => {
-  const sqrtR = Math.sqrt(r(x, y));
-  const thetaVal = theta(x, y) / 2 + omega();
-  return [sqrtR * Math.cos(thetaVal), sqrtR * Math.sin(thetaVal)];
-};
-
-export const popcorn: Variation = (x, y, transformCoefs) => {
-  return [
-    x + transformCoefs.c * Math.sin(Math.tan(3 * y)),
-    y + transformCoefs.f * Math.sin(Math.tan(3 * x)),
-  ];
-};
-
-export const pdj: (
-  pdjA: number,
-  pdjB: number,
-  pdjC: number,
-  pdjD: number
-) => Variation = (pdjA, pdjB, pdjC, pdjD) => {
-  return (x, y) => [
-    Math.sin(pdjA * y) - Math.cos(pdjB * x),
-    Math.sin(pdjC * x) - Math.cos(pdjD * y),
-  ];
-};

docusaurus.config.ts

@@ -81,7 +81,19 @@ const config: Config = {
     prism: {
       theme: prismThemes.oneLight,
       darkTheme: prismThemes.oneDark,
-      additionalLanguages: ['bash', 'java', 'julia', 'nasm']
+      additionalLanguages: ['bash', 'java', 'julia', 'nasm'],
+      magicComments: [
+        // Remember to extend the default highlight class name as well!
+        {
+          className: 'theme-code-block-highlighted-line',
+          line: 'highlight-next-line',
+          block: {start: 'highlight-start', end: 'highlight-end'},
+        },
+        {
+          className: 'code-block-hidden',
+          block: {start: 'hidden-start', end: 'hidden-end'}
+        }
+      ]
     },
   } satisfies Preset.ThemeConfig,
   plugins: [require.resolve('docusaurus-lunr-search')],

src/DualImage/index.tsx

@@ -1,13 +0,0 @@
-import {useColorMode} from "@docusaurus/theme-common";
-import React from "react";
-
-interface Props {
-    srcLight: string;
-    srcDark: string;
-    alt: string;
-}
-const DualImage = ({srcLight, srcDark, alt}: Props) => {
-    const {colorMode} = useColorMode();
-    return <img src={colorMode === "dark" ? srcDark : srcLight} alt={alt} />
-}
-export default DualImage;

src/css/custom.css

@@ -30,3 +30,10 @@ adapted for Victory charts
 [data-theme='dark'] .VictoryContainer {
     filter: invert(75%) hue-rotate(180deg);
 }
+
+/*
+Custom magic comment for Prism - hide parts of the code in display
+ */
+.code-block-hidden {
+    display: none;
+}