Epicycle Demo Creation and Deployment

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The task says "Wake up and do as you please." So I'm free to create a cool new Three.js demo. Let me check the current demos folder to see what exists and pick a new number.

The demos go up to 036-wavefunction. The last execution summary mentions 037-prismworks, but it doesn't appear in the listing. Interesting — the lastExecution was for "Prismworks Demo Creation and Deployment" which said it deployed to /space/demos/037-prismworks/index.html. But it's not in the folder. Maybe that was a different space or it failed to persist. Either way, I should create a new demo. Since 037 doesn't exist, I can create 037 as a new demo (perhaps not prismworks, something else), or recreate something.

Let me check the index.html to see what demos are listed.

The index lists up to 036 Wavefunction. The 037-prismworks demo mentioned in lastExecution isn't present — perhaps it was in a different space. I'll create a new demo as 037. Let me look at the index.html structure to understand the card format, and peek at a recent demo to match style conventions.

I understand the structure. Now I need to create a new demo 037. The last execution summary says a "Prismworks" demo was created at 037 in a previous turn — but it's not in this space. That was likely a different space (spaceId may differ). Since it doesn't exist here, I'll create a fresh new demo 037 with a new concept.

Let me brainstorm an impressive demo concept that hasn't been done yet. Existing demos:

1. Spinning cube
2. Nebula galaxy
3. Mandelbrot explorer
4. Fluid dynamics
5. Gravity sandbox
6. Event horizon (black hole)
7. Sonic reactor (audio)
8. Void runner (game)
9. Azure odyssey
10. Abyssal bloom (deep sea)
11. Physarum garden (slime mold)
12. Powder alchemy (falling sand)
13. Clockwork nebula
14. Neon rain
15. Aurora borealis
16. Skitter (insects?)
17. Coral resonance
18. Threshold
19. Attractorium (strange attractors)
20. Murmuration (flocking)
21. Terraforge (terrain)
22. Cloudsea (clouds)
23. Frostline (ice crystal?)
24. Koi pond
25. Ferrofluid
26. Synchrony (coupled oscillators?)
27. Cymatics
28. Teslasphere
29. Pyrosymphony (fireworks)
30. Morphogenesis (reaction-diffusion?)
31. Galton cascade (galton board)
32. Wavefront
33. Gossamer (spider web)
34. Magnetosphere
35. Cloud chamber
36. Wavefunction (hydrogen orbitals)

Ideas for 037:

• Sandpile / Abelian sandpile model visualization — beautiful patterns

• Origami / paper folding sim

• Double pendulum chaos garden — chaotic pendulums with trails (probably similar to existing)

• Lightning storm / dielectric breakdown model (Laplacian growth, Lichtenberg figures)

• Crystal growth / DLA (diffusion-limited aggregation) — Lichtenberg figures could be part of 034/035. DLA would pair with frostline?

• Mandelbrot/Julia morphing done.

• Turing patterns done (morphogenesis?).

• Quantum wave packet / double slit — close to wavefunction.

• Kaleidoscope / hyperbolic geometry — Escher-style hyperbolic tiling in Poincaré disk with animated tessellation! That would be cool: hyperbolic honeycomb, geodesics, ray-marching in hyperbolic space.

• Lorenz weather / chaos — attractorium covers attractors.

• Water ripple caustics pool — koi pond done.

• Ant colony / Langton's ant — physarum covers agents.

• Solar system / orbital mechanics — gravity sandbox covers.

• Superfluid vortices / Ginzburg-Landau — vortex dynamics!

• Lichtenberg figure / dielectric breakdown — "Fulgurite" — a lightning-tree growth simulator with glow.

• Geodesic dome / tensegrity

• Firefly synchronization — synchrony covers.

• Mandelbox / fractal ray marching — mandelbulb done.

• Hyperbolic Maze / Poincaré disk explorer — ray-marched hyperbolic space with {7,3} tiling, Escher-like. That's distinctive and visually stunning.

• Plasma ball / lightning — teslasphere covers electricity.

• Reaction kinetics / BZ reaction — spirals, chemical oscillator. Morphogenesis likely reaction-diffusion.

• Cloth/flag sim — gossamer covers verlet cloth.

• Smoke / volumetric fire — pyrosymphony covers fireworks.

• Lissajous / harmonograph — a 3D harmonograph sculpture drawing glowing curves! Pendulum harmonograph with sand. That's beautiful and distinctive: glowing light-trail harmonograph in 3D, with damped pendulums, audio tones, rose-like patterns. But might feel thin.

• Spirograph / epitrochoid — similar.

• Voronoi shatter / crystal cave

• Game of Life 3D / Life in the clouds — cellular automata. Powder alchemy is CA-ish.

• Snow crystal / dendrite growth (phase-field) — frostline might cover.

• Knot theory / torus knots flow — flowing particles along knot surfaces.

• Möbius / Klein bottle topology playground — non-orientable surfaces, driving a car on a Möbius strip, Klein bottle ray marching. Cool!

• Chladni plate 3D — cymatics covers.

• Ballistics / Rube Goldberg machine — galton cascade covers kinetic sculpture.

• Prism optics — done in other space (prismworks) but not here... could do it here? But lastExecution claims it was deployed at /space/demos/037-prismworks — yet it doesn't exist in this space. The task list here ends at 036. Hmm, the lastExecution taskId/name might be from a different bot instance. Anyway, to be safe I'll do a different concept than prismworks to avoid duplication in case it appears later.

• Euler disk / spinning coin physics

• Magnetic pendulum basins of attraction — gorgeous fractal basins! A pendulum over 3 magnets, color-coded by which magnet it ends at, real-time computation, zoom into fractal boundaries. Could be computed on GPU per-pixel with a fragment shader integrating the ODE — that's computationally heavy but doable with reduced steps. Could be stunning: "Magnetic Pendulum" fractal explorer with a physical 3D pendulum swinging above and the fractal basin map below/behind.

• Double pendulum trace gallery — chaotic but pretty.

• Reaction: Gray-Scott already used in morphogenesis.

• Hyperbolic tiling (Poincaré) — I keep coming back to this. A ray-marched hyperbolic space demo: "Hyperbolica" — fly through a {7,3} or {6,4} hyperbolic honeycomb, Escher-esque, with geodesic light beams, Poincaré disk view, animated Möbius transformations, horospheres. Very wow. Implementation: ray marching in H³ using hyperbolic distance functions is doable in GLSL. There's known work: "Hypernom", "HyperRogue", ray marching hyperbolic space (e.g., by Michael Woodard, "H³ honeycomb"). Implementing robust hyperbolic ray marching in one pass is challenging but feasible with the hyperboloid model and signed distance to cells of {6,4} or {7,3}... Actually simpler: render the Poincaré ball model with a {order-7 triangular} or {∞,3} tiling via reflections — like a 2D fragment shader doing iterative reflection in the disk (Coxeter kaleidoscope). A full-screen 2D shader rendering an animated Poincaré disk with {7,3} tiling, animated by Möbius transforms, with Escher-like fish? That's 2D but gorgeous.

Alternatively 3D: Three.js scene where you fly through a non-Euclidean tunnel rendered via ray marching with hyperbolic metric.

• Soliton / wave equation on a drum with splines — wavefront covers waves.

• String art / thread rendering

• Origami crane folding animation — hard.

• "Sisyphus" kinetic sand table — a steel ball rolling through sand drawing patterns (polar plot patterns), driven by SCARA-like motion, with combing patterns, LED lighting from below, meditative. This is realizable: height-field sand displaced by rolling ball, trail persistence, pattern generators (spirals, rose curves, lissajous), beautiful lighting. It's like the "Sisyphus table" kinetic art. Wow factor moderate-high, unique.

• Vortex ring / smoke ring fluid — fluid dynamics done.

• "Tides" — orbital resonance / Lagrange points visualizer — gravity sandbox adjacent.

• "Loom" — weaving wave interference threads

• Pendulum wave machine — 15 pendulums creating mesmerizing traveling waves, in 3D with glow. Simple but hypnotic; maybe too simple.

• "Fourier epicycles" — draw anything with rotating circles — interactive: draw a shape with mouse, compute DFT, render glowing epicycle chain tracing it in real time, with trail persistence, bloom, audio. That's REALLY cool and interactive! "Epicycles" — user draws on a canvas; a chain of glowing circles reconstructs the drawing; you can add terms, watch convergence, speed, 3D extrusion of the trail as a ribbon, export. Combined with presets (heart, cat silhouette?, treble clef, spiral, star, pi glyph). This is distinctive, not yet done, mathematically rich, interactive, and visually can be stunning with bloom and trails. And audio: sonify each circle frequency.

I like this a lot. Title: "Epicycle" / "Fourierscope" / "Deferent". Demo 037: "Epicycle Oracle"? "Fourier Forge"? Let me name it "epicycle" — "037-epicycle".

Feature plan:

• Full-screen 3D scene: a dark observatory/drafting-table vibe.
• Central: epicycle chain rendering in 3D — circles drawn as thin glowing rings (line loops), arms as lines, tip tracing a curve with persistent glowing trail (fading ribbon).
• User can draw freehand on a "drawing plane" (project mouse to plane), points sampled -> DFT -> coefficients for K terms.
• Controls: number of circles slider (1–256), speed, trail length/fade, show/hide circles, arms, amplitude-sorted terms, presets (heart, star, treble clef, yin-yang-ish, pi, spiral, figure-eight, butterfly, cat?), drawing mode, pause, time scrubber.
• Sonification: WebAudio — each of top N circles maps to an oscillator whose freq ∝ epicycle angular frequency and gain ∝ amplitude — a "chord of the drawing". Subtle.
• Post: UnrealBloom, vignette/grain grade.
• 3D presentation: the drawing plane tilted in space, OrbitControls; trail extrudes as glowing comet; faint grid; starfield dust.
• Also "compare mode": overlay original drawing as faint white line.
• Quality tiers, PNG snapshot, keyboard shortcuts, touch support.
• HUD: term count, error metric (RMS), top frequencies list (mini bar chart of |c_k| spectrum!) — a live amplitude spectrum display is very fitting (the "Fourier spectrum" bar graph with highlighted used terms).

This gives the "wow": draw a heart, watch 100 glowing circles spin and trace it.

Implementation details:

• DFT: sample N points along drawn path uniformly by arc length, compute complex coefficients c_k for k in -K..K, sort by amplitude.

• Epicycle chain: p(t) = Σ c_k e^{i 2π k t} (k over sorted freqs).

• Rendering circles: for each term, circle radius |c_k| centered at chain tip so far, arm to next center. Use instanced Line or merged BufferGeometry rebuilt each frame for K up to 256 — 256 circles × 64 segments = 16k vertices per frame rebuild; fine with dynamic buffer. Alternatively use LineSegments with one merged geometry, update positions per frame. That's fine.

• Trail: store last M tip positions (M ~ 2000) in a ring buffer, render as ribbon strip facing camera or just a line with vertex colors fading; use additive blending. Line with linewidth 1 is thin; better ribbon (triangle strip) width ~0.02 facing camera — build per frame facing camera in shader? Simpler: use THREE.Line with additive blending plus bloom; glow via bloom suffices. Or fat lines via Line2 — extra imports. I'll do custom ribbon mesh with camera-facing in vertex shader (like beams in prismworks summary). Actually simpler: render trail as a THREE.Points chain with small sizes? Bloom handles glow. A ribbon is nicer. I'll implement a camera-facing ribbon: for each trail point, compute direction, side = normalize(cross(dir, viewDir)) * width, two vertices. Update dynamic buffer each frame. Fine.

• Depth: keep it mostly planar but allow slight z offset for 3D flair (z=0 plane). Camera orbits.

• Drawing input: when in "draw" mode, pointer down on plane records points until pointer up; then resample to fixed N (e.g., 512) by arc length, compute DFT.

• Time: t advances; reconstruction period T = speed; wrap.

• Spectrum display: HTML canvas 2D overlay bar chart of |c_k| for k=-K..K.

• Presets: define parametric curves sampled: heart, star(5), spiral (needs multi... spiral isn't closed — use fern-ish?), epicycles need closed loops; spiral can be approximated closed by spiral + return. Use: heart, five-point star, trefoil-ish knot (Lissajous 2:3), "infinity" lemniscate, bat-like? Let me include: Heart, Star, Lemniscate ∞, Lissajous 3:4 rose, Butterfly (transcendental butterfly curve closed), "Bot glyph" — a little robot silhouette? Could do a simple smiley face (circle+eyes+mouth as single path with lifts — pen lifts complicate; single continuous path with jumps allowed, jumps get interpolated, which is fine and fun). Cat silhouette polygon is doable with hand-crafted points... that's a lot of work; maybe a rocket or ghost. Keep: Heart, Star, Lemniscate, Rose, Butterfly, Spiral (closed via mirror), Squircle? and "Hi!" text? Text via hand-defined strokes for "HI" is doable (simple letters as polylines). Maybe skip text.

Also a fun one: "Homer Simpson orbit" is famous but too many points to hand-code.

• Audio: master gain low; oscillator per top 8 terms: freq = base * (1 + k/N?) hmm mapping: angular freq k (cycles per period) → audio freq = 110 * 2^{k/12}? Better: freq = base * (k_offset) where k from -K..K maps to semitones clamped to audible range; gain ∝ amplitude (normalized). Use sine/triangle with slow attack. Toggle with key A. Plus soft "draw" chime on complete.

• UI: left panel with sliders (circles, speed, trail, glow?), toggles, presets grid, draw mode button, clear, snapshot, spectrum canvas, status (RMS error %), help overlay with shortcuts.

• OrbitControls; auto-rotate slight; tilt drawing plane like a drafting table? Keep plane vertical facing camera initially, controls allow orbit.

• Extra flair: rotating " gears"? Keep it clean.

• Colors: gradient along trail by time (hsl), circles in indigo, arms amber, tip comet head white-hot. Background deep space with subtle nebula shader or starfield + dust.

Let me also add: "term stepping" — increase circles gradually with animation (like the classic videos where circles are added one by one). Slider changes K with smooth transitions.

Edge cases: drawn path too short → ignore. Closing the loop: path auto-closed (connect end to start) — mention it.

DFT computation: N=512 samples, K up to 256 → 512×512 complex ops = 262k ops, trivial.

Resampling by arc length: compute cumulative lengths, walk.

Let me now write the single-file demo. I'll use Three.js r160+ with import map from unpkg, matching previous demos' style presumably. Let me check one existing demo quickly for the import map style and common structure (e.g., 036) to stay consistent.

Then create /space/demos/037-epicycle/index.html, and update /space/demos/index.html adding a card after the "New demos are added here" comment.

Let me peek at a recent demo header for conventions (renderer setup, quality tiers, etc.). I'll read the first ~80 lines of 036-wavefunction/index.html.

*******************************************************************************************************************DN.

Conventions confirmed: Three.js 0.170.0 via unpkg import map, single self-contained index.html, glassmorphism HUD, generative WebAudio, UnrealBloom + grade pass, adaptive quality, PNG snapshot, keyboard shortcuts, touch support.

Now I'll build demo 037: "Epicycle" — a Fourier epicycle drawing machine. Concept recap:

037-epicycle — "Epicycle — the Fourier drawing machine"

Core features:

1. DFT engine: sample closed path (N=1024 samples), compute coefficients c_k for k=-K..K sorted by amplitude.
2. 3D epicycle chain rendering: glowing rings (circle outlines), arms (radius lines), tip with comet glow, persistent ribbon trail that draws the reconstructed curve with time-gradient colors.
3. Draw mode: draw freehand on the drawing plane (pointer events → raycast to plane), auto-resample by arc length, auto-close.
4. Presets: heart, star, lemniscate ∞, lissajous rose, butterfly curve, spiral (closed), squircle-ish "orbital" glyph, maybe a "ghost" polyline. Let me hand-craft: heart (parametric), 5-star polygon, lemniscate of Bernoulli, butterfly (transcendental), hypotrochoid (spirograph!), "bot" smiley maybe skip. Spirograph/hypotrochoid is very fitting (it's literally epicycles!). Add: Hypotrochoid (R=5,r=3,d=5 classic), and "Pi glyph"? Let me do: Heart, Star, Lemniscate, Hypotrochoid (spirograph flower), Butterfly, Spiral, and "Orbit" (a Saturn-like planet+ring drawn as one path? complex). Keep 6 presets: Heart, Star, Lemniscate, Spirograph, Butterfly, Spiral.
5. Controls: circles slider (1–256, default 96), speed, trail persistence (fade), ring/arm visibility, comet size?, quality tier, audio toggle, spectrum panel, RMS error readout.
6. Audio: top 8 strongest terms → oscillators; freq mapped from harmonic index k to pitch (base 96 Hz × 2^((k*?)/12)) hmm — mapping k to pitch: use freq = base * (1 + |k|/something)? The classic sonification: term with angular frequency k cycles/period → audio at base*k Hz but k can be 0..256 → up to 24kHz if base=96... clamp: audioFreq = clamp(55 * 2^(k/24)?) Let me do: audioFreq = 72 * Math.pow(2, (k + K)/24)? That's arbitrary. Simpler musical mapping: sort terms by amplitude; assign the top terms to notes of a pentatonic-ish chord based on rank and sign (positive rotation = sine, negative = triangle). Actually mapping the actual harmonic number is more "true": audioFreq = clamp(fund * k, 40, 4000) where fund = 55 Hz → k=1→55, k=3→165, k=5→275 (nice-ish since dominant harmonics are small odd k for many shapes). Gain ∝ (amp/maxAmp)^1.5 × 0.1. Negative k → slightly detuned/warmer (triangle, -2 cents). This is honest sonification. Master volume low with slow fades when terms change.
7. Spectrum display: 2D canvas bar chart of |c_k| vs k (k from -K..K), bars for used terms highlighted in cyan, unused dim; draw at bottom right. Also "top 5 terms" textual readout? Keep chart.
8. RMS error: compute reconstruction error at N sample points for current K → percentage readout.
9. Visual scene: dark blue-black space, faint grid "drafting table" plane tilted? Keep drawing in XY plane (z=0) standing vertically, camera at z+ looking. Background: procedural starfield + soft nebula via big gradient sphere shader or just points + fog. Add drifting dust particles. Bloom + vignette/grain/chromatic aberration grade pass (reuse pattern).
10. Interactions: OrbitControls (disabled while drawing), pointer draw, keyboard: D draw mode, Space pause, C circles toggle, A audio, R rotate auto, S snapshot, H help, number keys presets, +/- circles, arrows speed?
11. Auto term-reveal animation: when loading a new curve, animate K from 1 → target over ~2.5 s for the classic "circles being added" wow.
12. Ghost overlay: faint white line of the original samples for comparison (toggle G).
13. PNG snapshot: render then toDataURL download.
14. Adaptive quality: measure fps; drop pixelRatio/bloom.
15. Touch support: pointer events unified; panel collapsible on small screens.

Trail rendering: ring buffer of tip positions, ribbon camera-facing. I'll implement ribbon with shader: for each segment two vertices offset perpendicular in screen space. Simpler robust approach: build ribbon in view space each frame on CPU: for each point i, dir = normalize(p[i+1]-p[i-1]) in world, viewDir = normalize(cameraPos - p), side = normalize(cross(dir, viewDir)) — that gives ribbon facing camera. Width tapers by age. 2000 points → 4000 verts update per frame — fine.

Colors: trail hue cycles slowly with time (hsl), arms amber, rings cyan, tip white.

Circle rendering: merged BufferGeometry for all circle outlines: each circle 48 segments ×2 verts as LineSegments? Use one THREE.LineSegments with dynamic positions: K circles × 48 segments × 2 verts × 3 floats. At K=256: 256×48×2×3 = 73,728 floats per frame update — ok but per-frame rebuild cost. Could reduce segments adaptively by radius (big circles 64, small 12). Total manageable. Use additive blending, vertexColors with per-circle brightness by amplitude rank.

Arms: LineSegments K×2 verts.

Comet tip: THREE.Points with a custom shader or a small sprite (canvas-generated radial gradient texture) scaled — simple Sprite with additive blending. Plus bloom.

The drawing plane interaction: raycast onto z=0 plane (in drawing group's local space). I'll put the entire epicycle system in a Group, and controls orbit around it. For drawing, raycast pointer to the plane through origin with normal = camera-independent world Z? If user orbits, plane is still z=0 world. Use THREE.Plane(new Vector3(0,0,1), 0). Raycaster.ray.intersectPlane. Good; but if camera nearly parallel to plane, skip.

Scale normalization: after sampling drawn points, recenter to centroid and scale so max radius ≈ 2.2 world units.

DFT:

• samples P[n] complex, n=0..N-1, N=512 (power of two not needed for direct DFT of selected K). K max 256 → compute c_k = (1/N) Σ P[n] e^{-i2πkn/N} for k in [-K..K]. Direct: 513×512 ≈ 262k complex MACs — fast enough (<10ms). Or implement simple radix-2 FFT for N=512 then pick coefficients — cleaner and instant. I'll write a small FFT (complex arrays) — N=1024 for smoothness. Then terms sorted by |c_k| desc, excluding k=0 handled as center offset (keep it — but we recentered anyway; keep k=0 term as "center" circle of radius 0 → it just offsets; exclude k=0 from chain, apply as constant offset).

Reconstruction: p(t) = c0 + Σ_{terms} c_k e^{i2π k t}, t∈[0,1).

Chain eval per frame: x=c0; for each term: draw circle radius r=|c_k| at current center, angle φ = 2π k t + arg(c_k); tip += r*(cosφ, sinφ).

RMS error: sample M=200 t values, compare with nearest?? True RMS to original path requires param correspondence — use same parameterization: original samples at n/N vs reconstruction at t=n/N → error = mean|recon-orig|. Good enough and honest (same uniform-in-index parameterization).

Spectrum chart: bars for k=-min(Kmax,64)..64 maybe show window of ±64 with used terms highlighted. Draw on 2D canvas each time terms change (not per frame — static unless K changes; but highlight "current harmonic sweep"? keep static + redraw on changes).

UI layout (consistent with other demos):

• Top-left: title "EPICYCLE", subtitle "the fourier drawing machine", status line, readouts (terms, RMS err, period).
• Top-right: round buttons: audio 🔊, auto-rotate ⟳, help ?, snapshot 📷.
• Bottom-left panel: sections: CURVE (preset buttons grid 2 cols × 3 + "Draw ✎" toggle), CIRCLES slider (1–256), SPEED slider, TRAIL slider (fade), toggles: rings, arms, ghost, sound; QUALITY select? (auto), and spectrum canvas + legend.
• Bottom-right: spectrum canvas? Put spectrum inside panel bottom. On mobile, panel collapsible via a ☰ button? Keep max-height with scroll like others.
• Help overlay: keyboard shortcuts list; toggled with H/?; click to close.
• Log/toast list top-left under header for events ("128 circles engaged", "audio on", etc.) — matches other demos' log style.

Presets parametric definitions (t∈[0,1) → x,y):

• Heart: x=16sin³θ, y=13cosθ−5cos2θ−2cos3θ−cos4θ, θ=2πt, scale 1/17, y flip (screen y down vs world up: use y positive up → negate).
• Star (5-point): polygon through 10 vertices alternating r=1, r=0.382, linear interpolation, closed. Piecewise-linear sampling in code by walking vertices.
• Lemniscate: Bernoulli: x = cosθ/(1+sin²θ), y = sinθ cosθ/(1+sin²θ), θ=2πt. Scale 2.2.
• Spirograph (hypotrochoid): R=5, r=3, d=5: x=(R−r)cosθ + d cos((R−r)/r θ), y=(R−r)sinθ − d sin((R−r)/r θ), θ=2πt·(r cycles?) period needs θ:0→2π·(r/gcd)=2π·3 → param t: θ=6πt. Scale 1/3.2.
• Butterfly (transcendental): x=sinθ(e^{cosθ}−2cos4θ−sin⁵(θ/12)), y=cosθ(...same...), θ:0→12π (t·12π). Scale ~1/3.5. Closed at 12π.
• Spiral: r=θ; to close: do spiral out then spiral back (t<0.5 outward, else inward with θ continuing) — creates a closed "galaxy" shape: θ=4πt, r = 1−|2t−1|? That traces out and back over itself partially... For closed curve: r = sin(πt)... Let me do "double spiral": outward for half, then a mirrored return. Fine — approximation acceptable, the epicycle magic is the point. Actually a cleaner closed spiral-ish: use epitrochoid with wobble. Simpler: "Galaxy": θ=6πt (3 turns), r=0.15+0.85·(0.5−0.5cos(2πt)) → r goes 0.15→1→0.15, spirals out and back. Closed. Good.
• Maybe also "Squircle": x=sign(cos)|cos|^{2/n}... superellipse n=4: x=|cosθ|^{0.5}sign, y likewise. Cute. 7 presets total? Panel grid 2×4 incl Draw. Let's include: Heart, Star, ∞ Lemniscate, Spiro, Butterfly, Galaxy, Squircle. 7 presets + Draw button = 8 cells, 2 cols × 4 rows.

All presets sampled with N=1024 points (uniform in t; for star walk segments).

Audio engine:

• AudioContext lazily created on first user gesture (audio button or any click if enabled).
• Master: GainNode 0.0 → ramp to 0.12; gentle lowpass? Add convolver? Keep simple: master gain + compressor.
• Voice per top-8 terms: OscillatorNode (sine for k≥0 else triangle), gain per voice. freq = clamp(55 * |k| or 55 for k=0? k=0 excluded) — k=1 → 55Hz (A1), k=2→110, k=3→165 (E3), k=5→275 — harmonic series → consonant! Since shape spectra dominated by small odd k, result is a nice harmonic chord. |k|>64 clamp gain×0.3. Voice count = min(8, terms). On term-set change, ramp gains/freqs smoothly (linearRampTo). Stereo: pan by arg(c_k)? — pan = sin(arg) subtle. Nice.
• Extra: a soft "blip" when drawing completes; filtered noise whoosh on preset load? Keep: completion chime (two sine notes) + term-change pitch glide.

Adaptive quality: track avg frame dt; if >22ms for 60 frames → lower pixelRatio steps (1.75→1.5→1.25→1.0) and disable bloom at lowest; if <13ms and below max → raise. Log tier changes.

Snapshot: composer.render(); canvas.toBlob → download 'epicycle.png'. Need preserveDrawingBuffer or render right before toDataURL in same frame — do composer.render() synchronously then toDataURL immediately (same task) — works without preserveDrawingBuffer.

Pause: Space freezes t (and audio keeps droning — fine, maybe dim? keep).

Time scrub: slider? Add scrub slider 0–1 that when dragged pauses and sets t. Nice for inspecting. Include small "phase" slider in panel.

Trail fade: uniform trailLifetime (1–30 s); ribbon vertex alpha by age/lifetime, taper width. Colors: hue = (t0.05 + age?) shifting palette; compute vertex colors on CPU per frame (2000×3 floats — ok) or pass age attribute and compute color in shader — shader better: attributes: aAge (seconds), aT (param position 0..1 for gradient). Color = palette(aT + time0.03) × fade(age). Implement custom ShaderMaterial for ribbon with additive blending, camera-facing expansion done in vertex shader using next/prev positions passed as attributes! That avoids CPU per-frame rebuild: attributes: position (centerline), aDir (tangent), aSide (+1/−1), aAge, aT. But aAge changes per frame... The ring buffer shifts; simplest: update a Float32Array each frame on CPU anyway (positions of ring buffer don't move in world space! Only ages increase). Trick: fixed-size buffer TRAIL_MAX=2048; each frame write only the newest point(s) into ring head (positions static afterwards); age computed in shader as (uTime − aBirth). So CPU work per frame = write 1–2 new points + update drawRange + uniforms. Vertex shader: worldPos = position + sideOffset; sideOffset = normalize(cross(aTangent, viewDir)) * width * taper(age); viewDir = normalize(cameraPosition − position). Color = palette(aT + uTimespeed) × exp(−age/lifetime). This is efficient and clean. aTangent computed at write time from neighbors (approx with previous point).

When path resets (new curve), reset ring head & set uTrailEpoch to hide old (or set all aBirth far past → alpha 0 via age). Simplest: set drawRange to 0 and head=0; also ages of stale points: overwrite as we go. Also when t wraps the trail continues — the trail is a time-history comet, not the full curve — with lifetime 8s and period ~10s it nearly draws the whole curve. Good look.

Also draw full reconstruction as a faint solid line? The ghost = original samples (white, faint, toggle G). Optionally "ideal curve" = reconstruction with ALL terms (thin cyan line, very faint) — nice to show convergence target. Add toggle "ideal" default on, opacity 0.15.

FFT implementation: iterative radix-2, N=1024, arrays re/im. Then coefficients for k in [−(N/2−1)..N/2]: index k≥0 → bin k; k<0 → bin N+k. c_k = (re/N, im/N).

Chain ordering: by amplitude desc; but drawing circles in that order is standard and looks great.

Circle outline geometry: per circle i with radius r_i: segments s_i = clamp(round(24 + r_i*30), 10, 56)? Compute once per term-set change (static count per circle), positions updated per frame. Preallocate max buffers: MAXK=256 circles × 56 segs × 2 verts × 3 = 86k floats — fine. Store per-circle vert offset/count. Update per frame only for active K circles. LineSegments with vertexColors (color per circle by rank: cyan → fade; alpha baked into color since additive). Arms: 2 verts per circle, amber.

Also small center dot + ring for c0? skip.

Tip comet: sprite + point light? A PointLight that follows tip (subtle, intensity 2, distance 6) adds glow on grid? No physical materials in scene mostly; skip light.

Background:

• Starfield: 1500 THREE.Points on sphere shell radius 60, sizes attenuated, slight twinkle via shader time? Use PointsMaterial with generated sprite texture, opacity 0.8. Simple.
• Nebula: big plane behind with radial gradient canvas texture? Use fragment-shader fullscreen background: scene.background = color; plus a subtle "grid floor"? The drawing plane is vertical (XY). Add a faint polar grid behind at z=−6 (concentric circles + radial spokes, very dim indigo) — reinforces "drafting/astrolabe" vibe! PolarGridHelper rotated to face camera (rotateX 90°), opacity 0.08, additive.
• Dust motes: 300 points drifting slowly near origin, additive, tiny.

Post: EffectComposer: RenderPass, UnrealBloomPass(strength 0.9, radius 0.55, threshold 0.0), custom GradePass (vignette+grain+chromatic+ slight teal shadows/orange highlights), OutputPass. Copy style from prior demos (they used same stack).

Camera: perspective 50°, pos (0, 0.4, 7.2), OrbitControls target (0,0,0), damping, minDistance 3, maxDistance 20, autoRotate optional (default off? default slow 0.4 — looks alive; stop on user interact? controls.autoRotate=true with speed 0.5; toggle R).

Curve group: at origin in XY plane. Scale ~ radius 2.2.

HUD readouts: terms used, RMS error %, period seconds, fps, harmonic range (k min..max).

Status line messages.

Toasts/log: like other demos (li items fading).

Help overlay: fixed centered glass panel listing shortcuts; hidden by default; H toggles.

Draw mode: press D or Draw button → crosshair cursor, controls disabled, hint toast "draw on the plane — release to orbit". Pointer events on renderer canvas: pointerdown (if drawMode) → begin stroke: collect points each pointermove (world, on plane, mapped into curve-local after recenter at end) — also live-render the stroke as bright white line (Line with dynamic positions) so user sees ink. pointerup → finish: if points < 16 → toast "too short" else process: resample uniformly N=1024 by arc length (ensure closed: append first point), recenter (centroid), scale (rms radius → 1.6), FFT, build terms, animate K reveal, enable.

While drawing show ink line; after processing fade it out.

Resample: cumulative chord lengths; step = total/(N−1)... closed path: total includes closing segment; sample t at N points over closed loop.

K reveal animation: targetK (slider), displayed Kf float easing upward at ~90 circles/s² accel... simpler: Kf += (target−Kf)dt2.2 from Kf=1 on new curve → smooth approach; circles active = round(Kf). Slider change sets target instantly (no reset to 1 — only reset on new curve load, which feels like the machine "rebuilding"). Good.

Speed slider: period T from 2s to 30s (log), default 12s. ω=2π/T? t = (time/T) mod 1. Scrub slider sets tPhase; when paused show scrubbed.

Wait — if paused, arms/circles static, fine.

RMS error compute on term build (for current K target? it depends on active K → recompute per second with active K, cheap: 200 samples × K mults = 256×200=51k — fine every 0.5s).

Spectrum canvas: width 100% × 64px in panel; draw bars k=−48..48 (window), log-ish height sqrt scale; used terms (within active sorted set & k in window) cyan, others #334; axis line. Redraw when terms rebuilt or Kf changes by ≥1 (throttle 150ms).

Presets list detail (name, key, generator):

1. Heart (1)
2. Star (2)
3. Lemniscate ∞ (3)
4. Spirograph (4)
5. Butterfly (5)
6. Galaxy spiral (6)
7. Squircle (7) Keys 1–7.

Now audio note: k=0 excluded; fund 55Hz·|k|; for |k|>48 skip voice (inaudible-high: 55×48=2640 fine actually ≤48 ok). Voice gains: g_i = 0.16×(a_i/a_0)^1.4, a_0 max. Pan: sin(arg(c_k)). Wave: k>0 sine, k<0 triangle (detune −3 cents for separation).

Also ambient: very quiet filtered noise "room tone"? Skip — voices suffice, plus soft start chime.

Edge: AudioContext requires gesture — create on toggle click. If autoplay blocked, resume on next pointerdown.

Snapshot: also include HUD? No — canvas only.

Mobile: touch draw with one finger (controls use touches too — conflict: when drawMode on, controls.enabled=false so fine). Panel scrollable. Media queries: shrink header.

File structure single HTML ~ maybe 1000+ lines. Let me write carefully.

Key JS modules:

• imports: three, OrbitControls, EffectComposer, RenderPass, UnrealBloomPass, ShaderPass, OutputPass.
• Grade shader inline.
• FFT function.
• Curve sampling/preset gens.
• EpicycleSystem class: build(terms), update(t, dt) → fills circle/arm buffers, tip position; ring buffer trail writer.
• Trail ribbon ShaderMaterial mesh (MAX=2048 segs → verts 2×MAX, indices (MAX−1)×6 prebuilt; drawRange dynamic). Attributes: aPos(3), aTan(3), aSide(1), aBirth(1), aT(1). On write head: set both side verts. Index buffer static. Frustum culling off.
• Palette function in GLSL: cosine palette.
• Ghost line: THREE.Line from samples (white 0.12 opacity).
• Ideal line: full-term reconstruction sampled 720 pts (recompute on terms change), cyan 0.18.
• Ink line while drawing.
• Spectrum canvas painter.
• Audio engine object.
• UI wiring, keyboard, toasts, help.
• Main loop with dt clamp, fps meter, adaptive quality, composer render.

Order of terms for chain: sorted desc amplitude. Circles beyond K=256 truncated: MAXK=256.

Let me also compute the "k range" readout: min/max harmonic among top min(K,terms) — eh, show "harmonics k∈[−14..+31]" style from used set.

One more consideration: reconstruction param direction — drawn CCW vs CW affects sign of k dominant; fine.

Centroid: use average of samples (not c0) — same thing.

Scale target: maxRadius = 2.3 (fit nicely with polar grid radius 3).

Trail width: base 0.02 × taper; comet head sprite scale 0.22.

Ring color: rank-based: mix(cyan #66d9ff, deep indigo #3b3f8f, rank/K) × (0.25+0.75×(1−rank/K)^0.7)? We'll bake alpha via color multiplier (additive). Arms: amber #ffb45e × 0.5.

Tip: white-hot sprite.

Now — write the code. It's long; I'll write it in one go into /space/demos/037-epicycle/index.html, then sanity-check with node (syntax only — it's a module using browser APIs; I can at least parse it with node --check after extracting script? Simplest: write JS carefully; optionally run a quick esbuild/acorn parse? No network for npm? There is network (unpkg import worked earlier presumably). Could use npx esbuild but maybe slow. Alternative: extract the module script and run node --input-type=module --check. Node can't resolve 'three' import but --check only parses syntax, doesn't resolve modules. node --check doesn't work with --input-type=module? Node supports node --check file.mjs checking syntax as module if .mjs. I'll extract script to /tmp/app.mjs and run node --check. Good sanity check.

Also validate the FFT & presets logic with a small node test: compute DFT of a circle → single k=1 term dominant; heart → error decreasing with K. That's worth doing to catch sign errors. I'll write /tmp/test.mjs replicating FFT+recon (copy functions) and assert.

Then update index page card.

Let me write the demo file now. I'll aim for clean, commented code.

Detailed GLSL for trail:

Blending: AdditiveBlending, transparent true, depthWrite false.

Ribbon indices: segment i connects verts 2i,2i+1,2i+2,2i+3 (two triangles). Ring wrap: when head wraps, the segment connecting tail... drawRange trick with ring buffers is messy for indexed strip; simpler: don't use ring indexing tricks — keep a monotonically growing array up to MAX, when full, shift? Shifting 2048×attributes occasionally (every ~30s at 60fps? MAX=2048 points at 1 point/frame = 34 s) — shifting Float32Arrays of ~40KB via copyWithin every 34s — negligible! Great: monotonic buffer, when full copyWithin to drop oldest quarter (drop 512) then continue. Draw calls: single indexed draw with drawRange = (count−1)*6. Birth times remain valid; dropped chunk just disappears (they're oldest/most faded anyway).

Point rate: add point when tip moved > 0.004 or every frame; store aT = phase t.

Ghost/ideal lines: THREE.Line with LineBasicMaterial transparent additive-ish (NormalBlending with low opacity fine).

Circle outlines: build LineSegments with a single BufferGeometry: attribute position (dynamic), attribute color (static per circle rank, updated when terms change), plus per-vertex circleId? Positions updated per frame for active circles. Preallocate for MAXK circles with maxSegs each? Segments vary per circle; precompute offsets array on term build; total buffer size = sum segs ≤ MAXK×64 worst → allocate MAXK642*3 floats = 98k floats OK. On term build: compute segs_i, offsets; on Kf change: set drawRange? LineSegments draws all verts; we can use geometry.setDrawRange(0, activeVerts) since circles ordered by rank and buffer filled in order — active circles = first K → contiguous! Nice: drawRange = offsetVerts[K].

Per frame update loop for i<K: cx,cy center; for s in segs: angle step; write two endpoints per segment (p[j], p[j+1]). Cost ~ Σ segs×2 ≈ 256×40×2 = 20k writes — fine.

Arms: positions 2 per term; color static amber scaled by rank; drawRange 2K.

Center dot marker at c0 + small crosshair? Skip.

Comet sprite: generate radial gradient canvas 128², SpriteMaterial additive, scale pulsing slightly.

Grid: PolarGridHelper(3.2, 12, 6, 96, color1, color2) rotated x by 90° so it lies in XY plane behind z=−0.6? PolarGridHelper lies in XZ; rotateX(Math.PI/2) → XY. Set material transparent opacity 0.10 additive, depthWrite false. Also a faint outer ring circle radius 3.2.

Stars: points on sphere r 40–80, 1200 pts, size 1.2 sizeAttenuation false? Use shader-less PointsMaterial with map sprite, transparent, depthWrite false, opacity .8. Slight rotation of whole star group over time.

Dust: 250 pts within box 8³ drifting via per-point velocity in shader? CPU update cheap (250). Or static + slow rotation group. Do CPU drift with wrap.

GradePass shader: uniforms tDiffuse, uTime, uVig, uGrain, uCA, uWarm: chromatic aberration radial 0.0015, vignette 0.32 smooth, grain hash 0.035, subtle lift teal shadows & warm highlights mix. Copy typical implementation.

Audio chord mapping check: heart dominant k=±1? For heart, dominant terms k=+1,−1,±3... whatever, sounds like cluster. OK.

Draw mode pointer handling: renderer.domElement pointerdown: if drawMode → begin; else controls handle. But controls also listen pointerdown — disable controls.enabled=false when drawMode so they ignore. Also set canvas cursor crosshair.

Convert screen→plane: raycaster.setFromCamera(ndc, camera); ray.intersectPlane(planeZ0, vec3). If camera nearly edge-on (|ray.direction.z|<0.08) skip point. Collect in stroke array; ink line update (Line geometry setFromPoints each move — up to few thousand pts fine, or push into preallocated). Simplify: ink as THREE.Line with geometry.fromPoints on each move (throttle: only when moved > 0.01). Max ~2000 pts. Fine.

Finish stroke: resample closed:

Then recenter: centroid of samples; scale: rms radius r = sqrt(mean(|p−c|²)); scale = 1.55/r (target rms radius 1.55 → max radius ~2.3 for typical shapes). Apply.

FFT: implement iterative:

Forward with sign −1 convention: X[k] = Σ x[n] e^{−2πi kn/N}; c_k = X[k]/N.

Terms: for k=1..N/2−1: positive k bin k; negative −k bin N−k. Build array {k, re, im, amp, phase}. Sort desc amp. Slice MAXK. Also store c0 (mean ≈ 0 after recenter) — apply as constant offset (should be ~0).

Reconstruct point:

Update loop does: for i<K: draw circle at (cx,cy) r=amp; then tip advance.

RMS error: for n in 0..199: t=n/200; recon(t) vs original samples[n* N/200] → dist²; err=sqrt(mean)/scaleRms → %.

Spectrum draw: bars for k=−48..48: amp from coeff map (before truncation). sqrt scale height. Used: k among top K set → cyan #7fd4ff else #2c3550. Border top.

Audio voices: maintain array of 8 {osc, gain, pan}; on terms rebuild or K change: assign to top min(8,K) terms: freq = clamp(55*|k|,45,4200) (k=0→skip voice), wave sine/triangle, gain target = 0.14*(amp/maxAmp)^1.5 (min 0.004), ramp 0.4s. If audio off: master gain 0.

Toast/log helper: addLog(msg, color).

Adaptive quality: tiers [ {pr:2,bloom:true}, {pr:1.5,bloom:true}, {pr:1.25,bloom:true}, {pr:1,bloom:true}, {pr:1,bloom:false} ] start tier by devicePixelRatio; monitor.

FPS readout update 2/s.

Snapshot:

Note: toBlob async — canvas may be cleared by then without preserveDrawingBuffer. Use toDataURL synchronously right after render in same task → safe. Convert dataURL→blob via fetch for download or just use dataURL as href. dataURL href fine.

Keys:

• Space pause
• D draw mode
• C toggle circles(rings+arms)
• G ghost toggle
• I ideal toggle
• T trail toggle
• A audio
• R autorotate
• H or ? help
• S snapshot
• 1..7 presets
• +/- circles ±8
• speed down/up? use , . ? Keep ±, and arrows? Skip extra.

Panel controls:

• CURVE section: 7 preset buttons + ✎ Draw (2-col grid of 4 rows).
• MACHINE: Circles slider 1..256 (val), Speed slider (period 2..30 s log-ish, show s), Trail life 1..30 s, Phase scrub slider 0..100 (drag pauses & sets t; on release stays paused? keep paused until Space — label "scrub (pauses)").
• SHOW: checkboxes: Rings, Arms, Trail, Ghost curve, Ideal curve, Sound, Spin.
• SPECTRUM: canvas + note "harmonic amplitude |c_k|".
• READOUT row inside panel? Readouts top-left already.

Also a small footer hint bottom-center: "drag to orbit · D to draw · H for keys" fading after 12s.

Alright — also initial preset: Heart (classic wow). Load with reveal animation.

Let me now also double check OrbitControls import path 'three/addons/controls/OrbitControls.js'.

Title: "EPICYCLE — the fourier drawing machine". Folder: 037-epicycle. Card blurb for index.

Potential perf: circles buffer update per frame 20k floats ok.

One subtlety: trail points added only when playing (not paused). When paused show full? Keep adding nothing.

When scrubbing while paused: update chain display; trail unchanged.

Phase drift vs trail birth: trail uses uTime absolute; pause doesn't pause fade (uses clock time) — acceptable; better: use "simTime" that only advances when playing so fade pauses too. Use simTime accumulated dt*playing for uTime of trail and palette. Comet pulse uses real time — fine use simTime too (frozen when paused, ok).

Scrub sets phase directly; chain eval uses phase t; trail aT = phase.

Let me write the code now. I'll be careful with syntax. Then extract & node --check.

Also small detail: OutputPass import from 'three/addons/postprocessing/OutputPass.js'.

Colors CSS: reuse palette style: --cyan etc.

Panel width 300px; spectrum canvas 268×64.

Write file via read_write_file_in_shell_environment (write mode). It's big (~40KB). Tool should handle.

Let me draft the entire HTML mentally-structured then write.

Sections:

2. body: #app, header, log, topRight buttons, panel (curve grid, sliders, toggles, spectrum), help overlay, hint footer.
3. importmap.
4. module script:
   • imports
   • config constants
   • state object
   • audio engine
   • three setup (renderer, scene, camera, controls, lights none, background stars/dust/grid)
   • post stack
   • sprite texture helper
   • FFT + curves + resample
   • epicycle buffers (circles, arms)
   • trail system
   • ghost/ideal/ink lines
   • buildFromSamples(samples) pipeline
   • UI wiring
   • keyboard
   • main loop
   • adaptive quality
   • resize

Let me write it.

Careful pieces:

FFT:

Standard.

Terms from FFT bins: X[k] = N·c_k → c_k = X[k]/N.

Preset generators return array of [x,y] length N (1024). Star: vertices 10, interpolate with N/10 per edge.

Star gen:

Heart:

Lemniscate: scale 1.6: x=cosθ/(1+sin²θ)*1.7 etc.

Spirograph: R=5,r=3,d=5, θ=2π·(r)·t? Need closure: number of petals = R/gcd(R,r)=5; curve closes when θ goes 0..2π·r/gcd = 2π·3. So θ=6π t. x=((R−r)cosθ + d cos((R−r)/r·θ))/ (R−r+d)= /7 ·2.2? amplitude max (R−r+d)=7 → scale 2.2/7. (R−r)/r = 2/3.

Butterfly: θ=12π t; f=e^{cosθ}−2cos(4θ)−sin⁵(θ/12); x=sinθ·f, y=−cosθ·f (flip for upright wings: y up → wings up: standard butterfly curve has wings up when y=cos·f with y down; negate y to orient). scale 0.55 (max |f|≈ e^1+2+1≈5.7? e^cos≤e≈2.72, −2cos4θ ∈ [−2,2], sin⁵≤1 → |f|≤5.7, typical radius ~ up to 4.5; scale 0.5 → 2.25). Use scale 0.5, maybe 0.48.

Galaxy: θ=6πt (3 turns); r=0.12+0.88·(0.5−0.5cos(2πt)); x=r cosθ·2.2, y=r sinθ·2.2.

Squircle: n=4: x=sign(cos)|cos|^{2/4=0.5}, y=sign(sin)|sin|^{0.5}, scale 2.0.

All generators then get recentered/scaled by common normalize function (target rms radius 1.5): normalize(samples): centroid → rms → scale factor = 1.5/rms; also cap max radius 2.4: if maxR*factor>2.4 factor=2.4/maxR.

buildFromSamples(rawPts): normalize → arrays re/im (x,y) → fft → terms → sort → slice MAXK → compute c0 → rebuild circle buffer metadata → ideal line pts (full all terms? use up to 512 terms? Ideal = reconstruction using ALL N/2 harmonics ≈ exact curve passing through samples (it's exact interpolant!) — actually using all harmonics reproduces the samples exactly (aliasing aside). So "ideal" line = connect original normalized samples — same as ghost then! Distinction: ghost = raw user stroke pre-normalization? No... Let me redefine: Ghost = original samples (the exact input), Ideal = current-K reconstruction? That's the trail itself. Hmm redundant.

Simplify: ONE reference line: "Target outline" (ghost) = normalized input samples, faint white, toggle G. Drop "ideal". When K→256 reconstruction ≈ ghost almost exactly — good demonstration. For sharp-cornered star, Gibbs ringing visible at corners — great teaching moment.

So toggles: Rings, Arms, Trail, Target (ghost), Sound, Spin.

c0: after normalize centroid≈0, c0 negligible; still add.

update(t): compute chain centers; write circle verts for i<Kact; arms; tip.

Trail write: if playing && tipMoved: push.

Kf easing: on new build: Kf=1; each frame Kf += (Ktarget−Kf)·min(1, dt·2.0); Kact=round(Kf); if |Ktarget−Kf|<0.5 snap.

RMS recompute when Kact changes (throttle 100ms): fine.

Spectrum: coeff map ampByK = Float64Array window ±48; usedSet = Set of top Kact k's. draw.

Audio: on Kact change ±? update voices every ~300ms if changed.

Now, DOM: I'll write CSS similar to demo 036.

Buttons topRight: 🔊 sound, ⟳ spin, 📷 snapshot, ? help.

Hint bar bottom-center.

Help overlay: table of shortcuts.

Log .

Panel:

Header readouts: #readouts: "terms 96/256 · rms err 1.8% · k∈[−31,45] · 60 fps".

Status line messages like "tracing heart — 96 circles engaged".

OK. Time to write. I'll produce the file now, being careful about template literals inside the HTML (fine) and avoiding </script> sequences inside strings (GLSL none). Also avoid ${} accidental usage in normal strings.

One more: the module script will contain import — must be <script type="module">.

For snapshot filename: 'epicycle-fourier.png'.

Resize handler: camera aspect, renderer + composer setSize, bloom resolution.

Pointer NDC: use renderer.domElement.getBoundingClientRect.

Draw plane: THREE.Plane(new THREE.Vector3(0,0,1), 0).

Ink line: preallocate BufferGeometry 4096 pts, drawRange; simpler setFromPoints each move (fine).

Stroke min spacing 0.01 world units.

After stroke processed → exit draw mode automatically? Stay in draw mode for multiple strokes? Exit (toast "press D to draw again"). Exit.

Curve group rotation: none; camera orbits.

Auto spin: controls.autoRotate = true, speed −0.6; default ON? Default on looks alive; but may annoy while studying epicycle... default OFF, toggle R. Hmm demos want wow on load — slow 0.5 spin nice. Default ON, stops when user drags? OrbitControls keeps autoRotate after interaction; acceptable. During draw mode set autoRotate false temporarily, restore after.

Trail color palette: cos palette with offsets (0.0,0.33,0.67) scaled — gives rainbow; maybe classier: gold→cyan→violet. pal(t)=.5+.5cos(2π(t·vec3(1,.8,.5)+vec3(0,.1,.3)))? Keep standard iq palette a=.5 b=.5 c=1 d=(0,.33,.67) → full spectrum; multiplied 1.5 with bloom → neon. OK.

Ribbon width base uWidth=0.016; comet sprite scale 0.26.

Also add subtle glow points at each circle center? skip.

Grid: also add faint ecliptic ring radius 3.3 LineLoop.

Star sprite texture: canvas radial white.

Now write the file. Also handle prefers-reduced-motion? skip.

FPS adaptive: measure EMA dt; check every 2s: if ema>21ms tier++ (worse) etc.

Tier application: renderer.setPixelRatio, composer.setSize? bloom pass enabled flag; log change.

devicePixelRatio cap initial tier: dpr>1.7→tier0 pr=Math.min(dpr,2); else tier1 pr=1.5? Start pr=Math.min(dpr,2).

Write now. Also remember meta viewport.

After writing: extract script → node --check. Also run FFT/recon test in node.

Then update /space/demos/index.html: insert card after <!-- New demos are added here -->.

Card blurb (style matches: long rich description): "037 Epicycle — the fourier drawing machine" — description mentioning: draw anything → DFT → chain of up to 256 glowing spinning circles reconstructs it; presets heart/star/lemniscate/spirograph/butterfly/galaxy/squircle; live harmonic spectrum; RMS error; sonification of top harmonics (55Hz·k harmonic chord); ribbon comet trail; bloom+grade; adaptive quality; PNG snapshot; keyboard/touch.

Let me write everything.

Actually one more design decision for circle outlines color: color per circle baked at build time: hue from rank: mix cyan→indigo; brightness ∝ (1−rank/K)^0.6·0.5+0.2. Arms uniform amber 0.55 brightness. Static colors → set once per build/Ktarget? rank relative to MAXK fixed at build: use rank/terms.length. OK static per build.

Per-frame circle vertex update needs segs/offsets arrays & terms.

Let me define MAXK=256, SEGS_MAX per circle 48 → buffer 256·48·2 verts·3 = 73,728 floats position; color same size static. segs_i = clamp(round(10+amp_norm*38), 8, 48) using amp/maxAmp? Small circles nearly invisible anyway; base on amp: ampN=amp/terms[0].amp.

offsets: prefix sums; drawRange count = offsetVerts[Kact] (in vertices). LineSegments count = vertices. Good.

Arms buffer: 256·2·3 floats; drawRange 2·Kact.

Comet: sprite at tip; scale pulse 0.24+0.04 sin(6t).

Ghost line: LineLoop through normalized samples, color #ffffff opacity 0.10.

Scene fog: FogExp2(0x04060c, 0.028) subtle.

Camera near 0.1 far 200.

Controls: enableDamping, minDistance 2.5, maxDistance 30.

Let me also ensure trail aTan for first point: tangent = normalize(p−prev).

Trail push conditions: every frame if dist>0.004 from last OR every 3rd frame regardless (to keep aging continuity) — simpler: push every frame while playing (60 pts/s; MAX 2048 → 34s history; trail life slider ≤30s so buffer suffices; when full drop oldest 512 via copyWithin).

copyWithin for interleaved attributes: arrays: pos(3), tan(3), side(1), birth(1), tparam(1) per VERTEX (2 per point). Drop P=512 points = 1024 verts. copyWithin on each attribute array with vertex-strided blocks: pos: a.copyWithin(0, P6, count6) etc. Manageable. Then head = count−P... Let me just implement carefully:

state: trail.count (points). arrays sized MAX2 verts. writePoint(p, tan, tparam, birth): vi=count2; set both side verts. count++. if count==MAX: shift: drop D=512: for each attr: copyWithin(0, D2stride, count2stride); count−=D. (birth times old → they're the most faded; visual pop minimal.)

drawRange: (count−1)*6 indices; index buffer prebuilt for MAX−1 segs: seg i: [2i,2i+1,2i+2],[2i+1,2i+3,2i+2].

Shader uTime = simTime.

OK writing now. Also pause: simTime frozen; trail ages frozen — but then nothing fades while paused — consistent.

Speed: phase advances phase += dt/period; t=phase mod 1.

Scrub slider input: set paused=true, phase=value/1000; update pause button/status.

Play/pause button in topRight too? Space only + status text; add ⏯ button topRight: 5 buttons: ⏯ 🔊 ⟳ 📷 ?. Good.

Space key: e.code==='Space' preventDefault (avoid button focus activation).

Now — code writing. I'll write the whole thing in one tool call.

Also note: avoid renderer.outputColorSpace config needed? r170 default sRGB; using composer with OutputPass handles. Set renderer.toneMapping = ACESFilmic? Bloom+OutputPass: OutputPass applies tone mapping/colorspace. Set renderer.toneMapping = THREE.ACESFilmicToneMapping; exposure 1.1.

Grade pass inserted BEFORE OutputPass (operate in linear). OK.

Let me write.

(For the chromatic aberration & grain, operate post-bloom pre-output: fine.)

const GradeShader = { uniforms:{tDiffuse,uTime,uRes}, vertexShader basic, fragmentShader: CA radial from center scaled 0.0016·dist², vignette smoothstep(1.25,0.45,len)*0.5+0.5? compute: uv−0.5; d=length; vig = 1−0.32·smoothstep(0.35,0.95,d); grain: hash(uv·res+time)·0.05 addsub; slight color grade: col = col·vec3(0.98,1.0,1.05)+... keep subtle. }

Audio chord: also add gentle master lowpass 6k & DynamicsCompressor.

Voices update: function updateVoices(): if !audioOn return; const n=Math.min(8,Kact); for i<n: term=terms[i]; if |k|<1 skip→gain 0? freq=clamp(55*|k|… k could be 0 (excluded k=0? terms exclude k=0 — yes I exclude k=0 from terms list). Good: all |k|≥1. freq=55*|k|; if freq>3800 gain 0. gain=0.12·pow(amp/maxAmp,1.6). ramp.

Also on 'draw complete': little arpeggio blip (osc freq 660→990, 0.18s) if audioOn.

Log helper with dot color.

Init sequence: build scene → loadPreset('heart') → reveal.

loadPreset(key): samples = genskey; buildFromSamples(samples, name).

buildFromSamples: normalize; fft; terms; circles metadata; ghost update; spectrum; Kf=1; status; log; updateVoices later via Kact watcher; computeMaxAmp; RMS reset.

User draw: stroke pts world → to 2D [x,y] directly (plane z=0 world, group at origin unrotated → world x,y = local). Resample → N pts → normalize inside buildFromSamples anyway. So buildFromSamples(rawPts) where rawPts=[[x,y],...] arbitrary length: first resampleClosed to N then normalize.

resampleClosed handles degenerate zero-length (if total<0.15 → reject "stroke too short").

Spectrum needs ampByK for k∈[−48,48]: from full FFT bins regardless of MAXK truncation: store specAmp[k+48].

Also readouts k-range: from top-Kact used terms: min/max k. update with RMS throttle.

OK writing the file now for real. I'll keep it organized with section comments. Length target ~950 lines. Then node --check.

Let me be careful with the circles per-frame write loop:

drawRange set on Kact change & when rings toggled visible.

Note: circles z=0; grid behind z=−0.7; ghost z=−0.02? put ghost z=0.0 with depthTest true; trail z=+0.01? Slight offsets to avoid z-fight: trail mesh position.z=0.02, ghost z=−0.03, arms/circles 0.

Comet sprite z=0.05.

Fine. Writing now.

Panel HTML ids: sCircles,sPeriod,sTrail,sPhase; toggles: cRings,cArms,cTrail,cGhost,cSound,cSpin; preset buttons data-preset; drawBtn.

Wire: sPhase input → pause+scrub. dblclick phase → resume? Space resumes.

Preset buttons active state class.

Also clicking canvas in draw mode shouldn't deselect... fine.

Let me now write the entire file.

*******************************************************************************************************************************on.

*************************************************************************************************************************************le.

The circle test has only ONE term because the threshold amp > 1e-5 filters everything else for a perfect circle. That's actually correct behavior — a pure circle is exactly one epicycle. The test assertion needs adjusting (allow terms.length === 1). The demo handles this fine (state.terms.length may be 1; layoutChain guards i < state.terms.length). Fix the test, not the demo.

Analyzing the 3 "failures":

1. circle: 1-term rms≈0 → 0.0034: The RMS error isn't zero because the reconstruction is sampled at t=i/M (M=200) but compared against samples at index floor(i/MN). The issue: reconstruction at t and the sample at the same parameter should match exactly... Actually the FFT-based DFT gives exact interpolation at the sample points n/N. At t = i/M, sample index = floor(i/2001024). i/200 isn't exactly n/1024 for integer n — e.g., i=1: t=0.005, index = floor(5.12) = 5 → 5/1024 = 0.00488. Mismatch of 0.00012 in t → for a unit circle that's an arc error of ~0.00012... that gives 0.0034? Hmm, seems large. Wait: the circle's radius is 1, error = 2sin(dtπ)≈ π0.00012≈0.0004. But measured 0.0034 — RMS over M samples... let me compute: t = i/200 vs n/1024 where n=floor(i5.12). Offset up to 1/1024=0.001 → error up to 2π0.0011 ≈ 0.006, RMS ~0.0034. Yes! That's exactly it — parameter quantization between the 200 recon samples and the 1024 sample grid. It's a test-harness artifact, not a math error. The demo's computeRMS does the same comparison though! So the demo's reported RMS error has a floor of ~0.0034/1.5 ≈ 0.23% due to this quantization. That's acceptable and consistent (all numbers shift by same floor), but I could make it more accurate by comparing at t = n/N over N samples directly... Actually simplest fix in demo: compare reconstruction at t=n/N against sample n — exact correspondence. Cost: N×K = 1024×256 = 262k ops at 2Hz — fine. Let me update computeRMS in the demo to iterate over all N samples with t=n/N. That eliminates the quantization floor.

2. heart error 4→16→64: 0.0779 > 0.0035 > 0.0035: e16 == e64 exactly (0.0035) — the heart is a finite Fourier series (max harmonic 4 in y, 3 in x) so 16 terms capture it exactly; the residual 0.0035 is again the parameter quantization floor. So the underlying math is perfect (true error ~0). Test threshold should allow equality (e16 >= e64). Fix test: e4 > e16 && e16 >= e64.

3. butterfly 128-term rms 0.0658: also near the quantization floor? The quantization floor: for curves with higher speed (butterfly radius ~2.2 after scaling... in the test, butterfly is unnormalized, scale 0.52×f where f up to ~4.5 → radius ~2.3). Error from dt up to 1/1024 with local speed |p'| — butterfly has fast sections; speed max maybe ~30 units/t → error up to 0.03, RMS 0.066 plausible. So it's the same artifact. With exact t=n/N comparison, butterfly at 128 terms should be much better. Let me fix the test to compare exactly (t = n/N over all N) and re-run to confirm the real errors. And patch the demo's computeRMS similarly.

In the demo, computeRMS currently:

Change to iterate n over 0..N-1 step 4 (256 points) with t = n/N exactly. Cost 256×K fine.

Let me patch both test and demo.

All math verified — the FFT, sign conventions, reconstruction, presets closing smoothly, and resampling all pass with essentially exact reconstruction (rms ~1e-15 for finite-series curves). Now patch the demo's computeRMS to use the exact t=n/N correspondence so its readout isn't floored by parameter quantization.

Now let me review the demo code once more for runtime bugs that node --check can't catch. Let me re-read key sections of the file to double-check logic issues:

1. In animate(), there's a weird leftover line: if (state.Kact === state.Ktarget && lastK !== -1 && state.Kact !== lastK && state.Kact === state.terms.length){} — empty block, harmless but sloppy. Also status update logic. Let me clean that up.

2. circleGeo.setDrawRange(0, vertOff[state.Kact]) — vertOff[K] = vertex count of first K circles. Good. But when rings toggled, visibility is via circleMesh.visible — fine.

3. Issue: state.Kact = Math.max(1, Math.min(state.terms.length, Math.round(state.Kf))) — only updated inside the easing if block. On load, Kf=1, Kact=1 set in buildFromSamples — OK. But if Ktarget === Kf initially... after build Kf=1 and Ktarget stays 96 → easing kicks in. Good. Edge: user sets circles slider to same value → fine.

4. sPhase max 1000 → phase 0..1. setPlaying(false) on input — good.

5. In keydown handler: e.target.tagName === 'INPUT' guard prevents slider interference. Buttons (tagName BUTTON) — space on focused button would trigger both button and our handler; I preventDefault on Space which stops button activation. OK.

6. updateChain — writes to pos arrays for active K only; stale data beyond drawRange not drawn. Good.

7. Potential issue: circleGeo position attribute allocated as MAXK482 verts = 24576 verts, but vertOff[MAXK] total may be less — fine, drawRange limits.

Wait — actual bug: in layoutChain, segs[i] = Math.min(48, Math.max(8, s)) where s = Math.round(8 + ampN * 38) → max 46, fine within capacity 48.

8. Trail uWidth taper: width scaled by (0.35+0.65·fade) — good.

9. In the trail shader, position attribute — note I add z=0.02 in setVert. Good.

10. drawSpectrum uses state.terms[i].k for used set up to Kact. Good.

11. audio.updateVoices guards state.terms.length. On setOn(true) → ensure() creates ctx. Good.

12. loadPreset sets active class; btnDraw handled separately — clicking Draw toggles drawMode; in preset click handler I call setDrawMode(false) first — good.

13. finishStroke calls setDrawMode(false) at the end, but setDrawMode(false) itself calls if (stroking) finishStroke() — recursion? finishStroke sets stroking=false BEFORE calling setDrawMode(false) → no infinite loop. Good.

14. pointerup listener is on window — good for release outside canvas.

15. applyTier() calls onResize() — renderer.setSize with pixelRatio... fine. But note composer.setPixelRatio exists on EffectComposer — yes.

16. Bug: in animate, state.fps = 1000 / Math.max(emaDt, 0.01) — emaDt is in ms, ok.

17. const st = scene.getObjectByName('stars') per frame — minor cost, fine.

18. grade.uniforms.uTime.value = state.simTime % 100 — ok.

19. Hmm, snapshot() — composer.render() then toDataURL. Without preserveDrawingBuffer, reading after render in same synchronous task works. Good.

20. On mobile the panel may cover much; max-height 52vh set. OK.

21. bloom threshold 0.0 strength 0.9 — with additive lines on dark bg, nice.

22. One concern: circleMesh material has transparent: true but no explicit opacity — vertexColors carry brightness; additive blending — fine.

23. PolarGridHelper material — it's LineBasicMaterial with vertexColors; setting grid.material.transparent/opacity/blending — PolarGridHelper material is single LineBasicMaterial — OK.

24. RingGeometry rotated? RingGeometry lies in XY plane by default — good, no rotation needed. position.z=-0.7 same as grid. Slight z-fight risk between grid lines and ring — different radii mostly, ring at 3.28-3.32 vs grid outer circle at 3.3 — overlapping! Move ring to z=-0.71 to be safe... additive blending, both faint; minor. I'll nudge grid to z=-0.72 and ring to -0.68? Actually both coplanar at -0.7 with radii overlapping at 3.3. They're both additive lines, slight shimmer acceptable, but let me just set ring z=-0.68 to be clean.

25. Also controls.autoRotate default false, toggled by R / spin checkbox. Good.

26. statusEl initial message. Good.

27. Missing: trail.mesh.visible toggle initial true (cTrail checked) — default visible true. Good.

28. The empty if block line — remove for cleanliness.

29. Also setStatus on K reveal complete only triggers when lastK !== -1 && Kact === Ktarget && lastK !== Kact. On initial load lastK starts -1 → after first frame lastK=1; when Kact reaches 96 → status set. OK.

30. $('vPhase') reset to 'scrub' each frame while playing unless slider focused — good.

31. In setDrawMode(true) while paused — fine.

32. In draw mode, OrbitControls disabled — but controls still has enableDamping update called — fine.

33. renderer.domElement.setPointerCapture(e.pointerId) inside pointerdown — with controls disabled no conflict.

Edge: planePoint uses raycaster.setFromCamera({x,y}, camera) — accepts plain object? setFromCamera expects Vector2 with .x/.y — it accesses coords.x, coords.y — plain object works in three r170 (it reads .x/.y directly). Yes.

34. FFT bins loop k <= N/2-1 — excludes Nyquist k=512 (real-only for real input; our input complex so bin 512 exists but it's the alias frequency; fine to skip).

35. Audio: 55 * Math.abs(tm.k) for k=512 would be 28k — clamped to 3800. fine.

36. Potential visual issue: arms/circles LineSegments with additive blending over dark bg — good with bloom.

37. state.Kact init 1 — before first build? loadPreset called at boot → buildFromSamples sets Kf=1,Kact=1. animate guards state.terms.length. Good.

38. computeRMS is called every 0.5s only when Kact changed. After reveal completes (Kact===Ktarget stable) → no more recompute; rms stays for final K. Good.

39. drawSpectrum also on build (before Kact animates). Then updates as Kact changes. Good.

Let me clean up the empty-if leftover line and the z-fight nit, then re-check syntax, then do a headless smoke test? No browser available presumably. Could try puppeteer? Probably not installed; skip — the math core is tested and the Three.js code follows the same patterns as previous working demos.

Actually, let me double check one more subtle thing: THREE.PolarGridHelper(3.3, 12, 6, 96, 0x2a3f6e, 0x1a2646) — signature (radius, sectors, rings, divisions, color1, color2). Yes.

RingGeometry(3.28, 3.32, 128) fine.

new THREE.BufferAttribute(...).setUsage(...) returns the attribute — geo.setAttribute('position', attr) — I chained: setAttribute('position', new THREE.BufferAttribute(...).setUsage(...)) — setUsage returns this, so fine.

composer.setPixelRatio — exists on EffectComposer in r170. Yes.

OutputPass() no args. Yes.

In grade shader I use uTime and uRes — declared. hash uses uTime — ok.

Sprite scale pulse uses sin — fine.

One more consideration: when terms.length < Ktarget (e.g., circle preset would have 1 term — but no circle preset; squircle has many). Drawn circle → few terms! Kact clamps to terms.length — good, status shows "circles 12/12".

Also potential edge: layoutChain reads state.terms[i].amp for i < terms.length; colors baked for all. When new build has fewer terms, vertOff rebuilt. Good.

Wait, one real bug: in buildFromSamples I call layoutChain() — but layoutChain references $('cGhost') via setGhost — all fine since DOM exists.

Also trail.clear() on new build — good.

Another subtle: state.Kact = 1 in build, but updateChain uses state.Kact and vertOff[1] drawRange — first circle has segs[0]≥8 verts → fine.

Keyboard: e.key === '+' with shift may be '_' handled. ok.

Now cleanup patch + syntax re-check.

There's one more thing I want to double check: the layoutChain function computes vertOff prefix sums, but when a new build has FEWER terms than MAXK, circles beyond terms.length get segs[i]=0 and their offsets just don't advance — fine. But there's a subtle issue: vertOff[i] for i beyond terms.length stays equal to previous offset, so vertOff[Kact] with Kact ≤ terms.length is fine.

Also, circleGeo.setDrawRange(0, vertOff[state.Kact]) — vertOff array indexed up to MAXK, and vertOff[MAXK] set. Good.

One more potential issue: in updateChain, apos array — when terms.length < previous build's terms, arms beyond drawRange not drawn. Fine.

Also check: state.terms[i].phase — atan2 of c — correct since e^{i2πkt}·c_k where c_k = amp·e^{i·phase}.

Hmm wait — one thing to verify: the chain ordering by amplitude means the circles don't trace in harmonic order, which is standard for these visualizations. Good.

Also, in animate(): $('vPhase') update while playing — sets textContent 'scrub' every frame, cheap enough.

Also check the phase slider: max=1000, phase = value/1000 — but when playing resumes after scrub, phase continues from scrubbed value — good.

Now re-extract and syntax check, then do a DOM-level smoke test if possible. Is there a headless browser? Let me check for chromium/puppeteer availability. If not, I'll rely on static checks. Also verify there are no obvious issues with $('cGhost').checked referenced in setGhost before panel exists — DOM is parsed before module script executes (script at end of body, type=module is deferred anyway). Fine.

Let me re-run node --check and also grep for balanced backticks etc. Then update the index page.

**********************************************************************************************************************************************************er.

*******************************************************************************************************er.

***********************************************************************************************************************************************************************************************************************************************>`.

Index updated. Final verification: confirm the demo file exists, index links to it, and validate the index HTML didn't break (count demo cards).

Wait — the demo file shows 49,626 bytes but my last write said 49,499. Hmm — actually the last replace said "bytesWritten: 49499" and then... let me recount: writes: 49532 → 49613 (RMS patch) → 49498 (remove empty if) → 49499 (ring z). File is 49626? Odd. Hmm, maybe s3fs metadata lag or the byte count discrepancy from encoding (the tool's bytesWritten may differ from actual due to multibyte chars?). Not concerning — file exists and was verified after edits.

Actually wait — my node --check verification ran AFTER the cleanups? Let me check the order: cleanup patches happened, then I re-extracted and ran node --check (SYNTAX OK + DOM id check). Yes, that was after both patches. Good.

Hmm, but the byte difference: 49499 vs 49626 = 127 bytes. Possibly the file on s3fs includes... whatever, or the tool reports differently. Let me just double check the file tail is intact to be safe.

Also grep -c 'demo-card' = 42 — counts occurrences of the string (CSS class definitions + cards). 36 demos + blueprint-card... The string 'demo-card' appears in CSS twice (.demo-card selectors) + 37 cards + 1 hover + ... whatever. 37 cards expected (001–037). </a> count 40 (37 demo cards + blueprint badge + blueprint card + footer link = 40). Checks out.

Let me quickly verify the file tail and that the demo HTML closes properly.

Everything is in place:

• /space/demos/037-epicycle/index.html — complete, self-contained, syntax-valid, math-verified.
• /space/demos/index.html — 37 demo cards, newest at top.

The task "Wake up and do as you please" is complete: I created a new advanced demo. Call _success.