Initial commit: add all skills files

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
2026-04-10 16:52:49 +08:00
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+## WebGL2 Adaptation Requirements
+
+Code templates in this document use ShaderToy GLSL style. When generating standalone HTML pages, you must adapt to WebGL2:
+
+- Use `canvas.getContext("webgl2")`
+- **IMPORTANT: Version directive must strictly be on the first line**: When injecting shader code into HTML, ensure nothing precedes `#version 300 es` — no newlines, spaces, comments, or other characters. Common pitfall: accidentally adding `\n` when concatenating template strings, causing the version directive to appear on line 2-3
+- First line of shader: `#version 300 es`, add `precision highp float;` for fragment shaders
+- Vertex shader: `attribute` → `in`, `varying` → `out`
+- Fragment shader: `varying` → `in`, `gl_FragColor` → custom `out vec4 fragColor`, `texture2D()` → `texture()`
+- ShaderToy's `void mainImage(out vec4 fragColor, in vec2 fragCoord)` must be adapted to standard `void main()` entry
+
+**IMPORTANT: GLSL Type Strictness Warning**:
+- `vec2 = float` is illegal: types must match exactly, e.g., `float r = length(uv)` not `vec2 r = length(uv)`
+- Function return types must match: commonly used `fbm()` / `noise()` return `float`, cannot be assigned to `vec2`
+- If you need a vec2 type, use `vec2(fbm(...), fbm(...))` or `vec2(value)` constructor
+
+# Polar Coordinates & UV Manipulation
+
+## Use Cases
+- Radially symmetric effects: flowers, kaleidoscopes, gears, radial patterns
+- Spiral patterns: galaxies, vortices, spiral staircases
+- Ring/tunnel effects: tube flying, torus twisting, circular UI elements
+- Polar coordinate shapes: cardioid, rose curves, stars, and other shapes defined by r(θ)
+- Vortex animations: swirls, rotational warping, card game backgrounds (e.g., Balatro)
+- Fractal/repetitive structures: recursive symmetric patterns based on angular subdivision
+
+## Core Principles
+
+Polar coordinates convert (x, y) to (r, θ):
+- **r = length(p)** — distance to origin
+- **θ = atan(y, x)** — angle from positive x-axis, range [-π, π]
+
+Inverse transform: x = r·cos(θ), y = r·sin(θ)
+
+Manipulation effects:
+- Modifying θ → rotation, warping, kaleidoscope
+- Modifying r → scaling, radial ripples
+- θ += f(r) → spiral effect
+
+| Spiral Type | Equation | Code |
+|------------|----------|------|
+| Archimedean spiral | r = a + bθ | `theta += radius` |
+| Logarithmic spiral | r = ae^(bθ) | `theta += log(radius)` |
+| Rose curve | r = cos(nθ) | `r - A*sin(n*theta)` |
+
+## Implementation Steps
+
+### Step 1: UV Normalization and Centering
+```glsl
+// Range [-1, 1], most commonly used
+vec2 uv = (2.0 * fragCoord - iResolution.xy) / min(iResolution.x, iResolution.y);
+
+// Range [-aspect, aspect] x [-1, 1]
+vec2 uv = (2.0 * fragCoord - iResolution.xy) / iResolution.y;
+
+// Pixelated style (Balatro style)
+float pixel_size = length(iResolution.xy) / PIXEL_FILTER;
+vec2 uv = (floor(fragCoord * (1.0/pixel_size)) * pixel_size - 0.5*iResolution.xy) / length(iResolution.xy);
+```
+
+### Step 2: Cartesian → Polar Coordinates
+```glsl
+float r = length(uv);
+float theta = atan(uv.y, uv.x); // [-PI, PI]
+
+// Reusable function
+vec2 toPolar(vec2 p) { return vec2(length(p), atan(p.y, p.x)); }
+
+// Normalized angle to [0, 1]
+vec2 polar = vec2(atan(uv.y, uv.x) / 6.283 + 0.5, length(uv));
+```
+
+### Step 3: Polar Space Operations
+
+**3a. Radial Swirl**
+```glsl
+float spin_amount = 0.25;
+float new_theta = theta - spin_amount * 20.0 * r;
+```
+
+**3b. Angular Twist**
+```glsl
+float twist_angle = theta + 2.0 * iTime + sin(theta) * sin(iTime) * 3.14159;
+```
+
+**3c. Archimedean Spiral**
+```glsl
+vec2 spiral_uv = vec2(theta_normalized, r);
+spiral_uv.y -= spiral_uv.x; // Unwrap into spiral band
+```
+
+**3d. Logarithmic Spiral**
+```glsl
+float shear = 2.0 * log(r);
+float c = cos(shear), s = sin(shear);
+mat2 spiral_mat = mat2(c, -s, s, c);
+```
+
+**3e. Kaleidoscope**
+```glsl
+float rep = 12.0;          // Number of symmetry axes
+float sector = TAU / rep;
+float a = polar.y;
+float c_idx = floor((a + sector * 0.5) / sector);
+a = mod(a + sector * 0.5, sector) - sector * 0.5;
+a *= mod(c_idx, 2.0) * 2.0 - 1.0; // Mirror
+```
+
+**3f. Spiral Arm Compression**
+```glsl
+float NB_ARMS = 5.0;
+float COMPR = 0.1;
+float phase = NB_ARMS * (theta - shear);
+theta = theta - COMPR * cos(phase);
+float arm_density = 1.0 + NB_ARMS * COMPR * sin(phase);
+```
+
+### Step 4: Polar → Cartesian Reconstruction
+```glsl
+vec2 new_uv = vec2(r * cos(new_theta), r * sin(new_theta));
+
+vec2 toRect(vec2 p) { return vec2(p.x * cos(p.y), p.x * sin(p.y)); }
+
+// Balatro-style round-trip (offset to screen center)
+vec2 mid = (iResolution.xy / length(iResolution.xy)) / 2.0;
+vec2 warped_uv = vec2(r * cos(new_theta) + mid.x, r * sin(new_theta) + mid.y) - mid;
+```
+
+### Step 5: Polar Coordinate Shape SDF
+```glsl
+// Cardioid
+float a = atan(p.x, p.y) / 3.141593; // atan(x,y) makes the heart face upward
+float h = abs(a);
+float heart_r = (13.0*h - 22.0*h*h + 10.0*h*h*h) / (6.0 - 5.0*h);
+float dist = r - heart_r;
+
+// Rose curve
+float rose_dist = abs(r - A_coeff * sin(PETAL_FREQ * theta) - 0.5);
+
+// Rendering
+float shape = smoothstep(0.01, -0.01, dist);
+```
+
+### Step 6: Coloring and Anti-Aliasing
+```glsl
+// fwidth adaptive anti-aliasing
+float aa = smoothstep(-1.0, 1.0, value / fwidth(value));
+
+// Resolution-based anti-aliasing
+float aa_size = 2.0 / iResolution.y;
+float edge = smoothstep(0.5 - aa_size, 0.5 + aa_size, value);
+
+// Radial gradient coloring
+vec3 color = vec3(1.0, 0.4 * r, 0.3);
+color *= 1.0 - 0.4 * r;
+
+// Inter-spiral-band anti-aliasing
+float inter_spiral_aa = 1.0 - pow(abs(2.0 * fract(spiral_uv.y) - 1.0), 10.0);
+```
+
+## Complete Code Template
+
+```glsl
+// === Polar Coordinates & UV Manipulation Complete Template ===
+// Paste directly into ShaderToy to run
+
+#define PI 3.14159265359
+#define TAU 6.28318530718
+
+// ===== Adjustable Parameters =====
+#define MODE 0            // 0=swirl, 1=spiral, 2=kaleidoscope, 3=rose curve
+#define SPIRAL_TYPE 0     // 0=Archimedean, 1=logarithmic (MODE=1)
+#define NUM_ARMS 5.0      // Number of spiral arms (MODE=1)
+#define KALEID_SEGMENTS 6.0 // Kaleidoscope segments (MODE=2)
+#define PETAL_COUNT 5.0   // Number of petals (MODE=3)
+#define SWIRL_STRENGTH 3.0 // Swirl intensity (MODE=0)
+#define ANIM_SPEED 1.0    // Animation speed
+#define COLOR_SCHEME 0    // 0=warm, 1=cool, 2=rainbow
+
+vec2 toPolar(vec2 p) {
+    return vec2(length(p), atan(p.y, p.x));
+}
+
+vec2 toRect(vec2 p) {
+    return vec2(p.x * cos(p.y), p.x * sin(p.y));
+}
+
+vec2 kaleidoscope(vec2 polar, float segments) {
+    float sector = TAU / segments;
+    float a = polar.y;
+    float c = floor((a + sector * 0.5) / sector);
+    a = mod(a + sector * 0.5, sector) - sector * 0.5;
+    a *= mod(c, 2.0) * 2.0 - 1.0;
+    return vec2(polar.x, a);
+}
+
+vec3 getColor(float t, int scheme) {
+    if (scheme == 1) return 0.5 + 0.5 * cos(TAU * (t + vec3(0.0, 0.33, 0.67)));
+    if (scheme == 2) return 0.5 + 0.5 * cos(TAU * t + vec3(0.0, 2.1, 4.2));
+    return vec3(1.0, 0.4 + 0.4 * cos(t * TAU), 0.3 + 0.2 * sin(t * TAU));
+}
+
+void mainImage(out vec4 fragColor, in vec2 fragCoord) {
+    vec2 uv = (2.0 * fragCoord - iResolution.xy) / min(iResolution.x, iResolution.y);
+    vec2 polar = toPolar(uv);
+    float r = polar.x;
+    float theta = polar.y;
+    float t = iTime * ANIM_SPEED;
+    vec3 col = vec3(0.0);
+    float aa = 2.0 / iResolution.y;
+
+    #if MODE == 0
+    // --- Swirl mode ---
+    float swirl_theta = theta - SWIRL_STRENGTH * r + t;
+    vec2 warped = toRect(vec2(r, swirl_theta));
+    warped *= 10.0;
+    float pattern = sin(warped.x) * cos(warped.y);
+    pattern += 0.5 * sin(2.0 * warped.x + t) * cos(2.0 * warped.y - t);
+    float val = smoothstep(-0.1, 0.1, pattern);
+    col = mix(
+        getColor(r * 0.5, COLOR_SCHEME),
+        getColor(r * 0.5 + 0.5, COLOR_SCHEME),
+        val
+    );
+    col *= exp(-r * 0.5);
+
+    #elif MODE == 1
+    // --- Spiral mode ---
+    #if SPIRAL_TYPE == 0
+        float spiral = theta / TAU + 0.5;
+        float bands = spiral + r;
+        bands -= t * 0.1;
+        float arm = fract(bands * NUM_ARMS);
+    #else
+        float shear = 2.0 * log(max(r, 0.001));
+        float phase = NUM_ARMS * (theta - shear);
+        float arm = 0.5 + 0.5 * cos(phase);
+        arm *= 1.0 + NUM_ARMS * 0.1 * sin(phase);
+    #endif
+    float brightness = smoothstep(0.0, 0.4, arm) * smoothstep(1.0, 0.6, arm);
+    col = getColor(theta / TAU + t * 0.05, COLOR_SCHEME) * brightness;
+    col *= exp(-r * r * 0.5);
+    col += 0.15 * exp(-r * r * 8.0);
+
+    #elif MODE == 2
+    // --- Kaleidoscope mode ---
+    vec2 kp = kaleidoscope(polar, KALEID_SEGMENTS);
+    vec2 rect = toRect(kp);
+    rect *= 4.0;
+    rect += vec2(t * 0.3, 0.0);
+    vec2 cell_id = floor(rect + 0.5);
+    vec2 cell_uv = fract(rect + 0.5) - 0.5;
+    float cell_hash = fract(sin(dot(cell_id, vec2(127.1, 311.7))) * 43758.5453);
+    float d = length(cell_uv);
+    float truchet = abs(d - 0.35);
+    if (cell_hash > 0.5) {
+        truchet = min(truchet, abs(length(cell_uv - 0.5) - 0.5));
+    } else {
+        truchet = min(truchet, abs(length(cell_uv + 0.5) - 0.5));
+    }
+    col = getColor(cell_hash + r * 0.2, COLOR_SCHEME);
+    col *= smoothstep(0.05, 0.0, truchet - 0.03);
+    col *= smoothstep(3.0, 0.0, r);
+
+    #elif MODE == 3
+    // --- Rose curve mode ---
+    float rose_r = 0.6 * cos(PETAL_COUNT * theta + t);
+    float dist = abs(r - abs(rose_r));
+    float ribbon_width = 0.04;
+    float rose_shape = smoothstep(ribbon_width + aa, ribbon_width - aa, dist);
+    float depth = 0.5 + 0.5 * cos(PETAL_COUNT * theta + t);
+    col = getColor(theta / TAU, COLOR_SCHEME) * depth;
+    col *= rose_shape;
+    float center = smoothstep(0.08 + aa, 0.08 - aa, r);
+    col += getColor(0.5, COLOR_SCHEME) * center * 0.5;
+    #endif
+
+    col = pow(col, vec3(1.0 / 2.2));
+    fragColor = vec4(col, 1.0);
+}
+```
+
+## Common Variants
+
+### Variant 1: Dynamic Vortex Background (Balatro Style)
+Cartesian→Polar→Cartesian round-trip + iterative domain warping
+```glsl
+float new_angle = atan(uv.y, uv.x) + speed
+    - SPIN_EASE * 20.0 * (SPIN_AMOUNT * uv_len + (1.0 - SPIN_AMOUNT));
+vec2 mid = (screenSize.xy / length(screenSize.xy)) / 2.0;
+uv = vec2(uv_len * cos(new_angle) + mid.x,
+           uv_len * sin(new_angle) + mid.y) - mid;
+uv *= 30.0;
+for (int i = 0; i < 5; i++) {
+    uv2 += sin(max(uv.x, uv.y)) + uv;
+    uv  += 0.5 * vec2(cos(5.1123 + 0.353*uv2.y + speed*0.131),
+                       sin(uv2.x - 0.113*speed));
+    uv  -= cos(uv.x + uv.y) - sin(uv.x*0.711 - uv.y);
+}
+```
+
+### Variant 2: Polar Torus Twist (Ring Twister Style)
+Direct rendering in polar space, angular slicing to simulate 3D torus
+```glsl
+vec2 uvr = vec2(length(uv), atan(uv.y, uv.x) + PI);
+uvr.x -= OUT_RADIUS;
+float twist = uvr.y + 2.0*iTime + sin(uvr.y)*sin(iTime)*PI;
+for (int i = 0; i < NUM_FACES; i++) {
+    float x0 = IN_RADIUS * sin(twist + TAU * float(i) / float(NUM_FACES));
+    float x1 = IN_RADIUS * sin(twist + TAU * float(i+1) / float(NUM_FACES));
+    vec4 face = slice(x0, x1, uvr);
+    col = mix(col, face.rgb, face.a);
+}
+```
+
+### Variant 3: Galaxy / Logarithmic Spiral (Galaxy Style)
+`log(r)` equiangular spiral + FBM noise + spiral arm compression
+```glsl
+float rho = length(uv);
+float ang = atan(uv.y, uv.x);
+float shear = 2.0 * log(rho);
+mat2 R = mat2(cos(shear), -sin(shear), sin(shear), cos(shear));
+float phase = NB_ARMS * (ang - shear);
+ang = ang - COMPR * cos(phase) + SPEED * t;
+uv = rho * vec2(cos(ang), sin(ang));
+float gaz = fbm_noise(0.09 * R * uv);
+```
+
+### Variant 4: Archimedean Spiral Band (Wave Greek Frieze Style)
+Polar unwrap into spiral band, creating vortex animation within the band
+```glsl
+vec2 U = vec2(atan(U.y, U.x)/TAU + 0.5, length(U));
+U.y -= U.x;                                    // Archimedean unwrap
+U.x = arc_length(ceil(U.y) + U.x) - iTime;     // Arc length parameterization
+vec2 cell_uv = fract(U) - 0.5;
+float vortex = dot(cell_uv,
+    cos(vec2(-33.0, 0.0)
+        + 0.3 * (iTime + cell_id.x)
+        * max(0.0, 0.5 - length(cell_uv))));
+```
+
+### Variant 5: Complex / Polar Duality (Jeweled Vortex Style)
+Complex arithmetic replaces explicit trigonometric functions for conformal mapping
+```glsl
+float e = n * 2.0;
+float a = atan(u.y, u.x) - PI/2.0;
+float r = exp(log(length(u)) / e);      // r^(1/e)
+float sc = ceil(r - a/TAU);
+float s = pow(sc + a/TAU, 2.0);
+col += sin(cr + s/n * TAU / 2.0);
+col *= cos(cr + s/n * TAU);
+col *= pow(abs(sin((r - a/TAU) * PI)), abs(e) + 5.0);
+```
+
+## Performance & Composition
+
+### Performance Tips
+- **Pole safety**: `float r = max(length(uv), 1e-6);` to avoid division by zero
+- **Trigonometric optimization**: When both sin/cos are needed, use a rotation matrix `mat2 ROT(float a) { float c=cos(a),s=sin(a); return mat2(c,s,-s,c); }`
+- **Kaleidoscope is naturally optimized**: All expensive computation happens in a single sector, visual complexity ×N
+- **Loop control**: Rose curves and other multi-loop effects work well with 4-8 loops; don't go too high
+- **Pixel downsampling**: `floor(fragCoord / pixel_size) * pixel_size` quantizes coordinates to reduce computation
+
+### Composition Tips
+- **Polar + FBM**: Sample noise in transformed space → organic spiral textures
+- **Polar + Truchet**: Lay Truchet tiles after kaleidoscope folding → geometric tunnel effects
+- **Polar + SDF**: `r(θ)` defines contour + SDF boolean operations / glow
+- **Polar + Checkerboard**: `sign(sin(u*PI*4.0)*cos(uvr.y*16.0))` → circular checkerboard
+- **Polar + Post-Processing**: Gamma + vignette + contrast enhancement for improved visual quality
+
+## Further Reading
+
+For complete step-by-step tutorials, mathematical derivations, and advanced usage, see [reference](../reference/polar-uv-manipulation.md)