Drawing with trigonometry, polar coordinates, and curves

This module builds toward the classic pen-plotter aesthetic: mathematically pure line work — spirals, Lissajous knots, wave-clock mandalas — that reads as hand-drawn precision. In a live-visuals or print-edition practice, these forms are the workhorse vocabulary: they render fast, and pair naturally with slow ambient sets or gallery editions where crisp geometry matters more than pixel effects.

The arc starts supported: draw a single sine wave by mapping sin(angle) to canvas y-coordinates, then internalize the unit-circle move — cos for x, sin for y — until the polar-to-Cartesian conversion (x = cx + r·cos(a), y = cy + r·sin(a)) is automatic; the two drills here target exactly that recurrent conversion, framed inside real drawing tasks rather than isolated math. From there each new form is one twist on the same loop: grow the radius per step for an Archimedean spiral, cross two sinusoids at integer ratios for Lissajous figures, rotate a noise-length chord for the wave clock, and swap sin() for tan() in one family so the radius shoots off-scale near the asymptote and wraps back — a deliberately wilder sibling among the bounded forms. Just-in-time pointers for the composition phase: “Adding Perlin noise to a spiral’s radius” for organic wobble, “A custom pow(sin(x),n) shaping function” for deterministic character, and the beginShape/bezierVertex atoms for turning point streams into smooth paths.

Every required atom is load-bearing for the capstone: the trig and polar atoms generate the forms, the shaping/noise atoms give each family its variance — including the tan wave, whose unbounded spike-and-wrap behaviour drives the capstone’s wildest form family — and the vertex/bezier atoms build the paths. Supporting atoms — 3D sphere coordinates, organic geometry, iterative variance, and vector-vs-raster export for when a piece heads to a plotter or print — extend the same ideas into depth and craft without gating the piece.

Runnable examples

Generated from the context/ instrument corpus by concept (redistributable idioms only). Do not edit — regenerate with gen-module-examples.mjs.

modulation-warp

osc(4).modulate(src(o0), 0.6).out(o0)

_{hydra-0022 · CC0-1.0}

s0.initP5(); src(s0).modulate(noize(), 0.3).out()

_{p5live-0038 · CC0-1.0}

feedback-trail

osc(4).modulate(src(o0), 0.6).out(o0)

_{hydra-0022 · CC0-1.0}

function draw(){ fill(0, 20); rect(0, 0, width, height); circle(mouseX, mouseY, 40) }

_{p5live-0003 · CC0-1.0}

radial-symmetry

osc(10).kaleid(5).out()

_{hydra-0010 · CC0-1.0}

// sandbox
osc(10, 0.05, 1.3).kaleid(8).out()
// sandbox

_{p5live-0037 · CC0-1.0}

oscillation

let y = height/2 + sin(frameCount * 0.05) * 100

_{p5live-0004 · CC0-1.0}

float rings = abs(sin(length(uv)*20.0 - u_time*2.0));

_{glsl-0039 · public-domain}

polar-warp

float r = length(uv); float a = atan(uv.y, uv.x);

_{glsl-0008 · public-domain}

[rtx [fr, ft + 0.2*time], rty [fr, ft], 0.5] >> rgb

_{punctual-0032 · CC0-1.0}

vector-drawing

beginShape(); for(let p of pts) curveVertex(p.x, p.y); endShape()

_{p5live-0016 · CC0-1.0}