Parametric and Spirograph Art for Pen Plotters

Some of the most hypnotic pen plots come from pure mathematics: looping spirograph patterns, delicate rose curves, and spirals that seem to breathe. This is parametric art, and it’s both deeply satisfying to plot and surprisingly approachable — no advanced math required.

What “parametric” means

Normally we think of a curve as y depending on x. Parametric curves are different: both x and y depend on a third variable, usually called t (think of it as time). As t advances, the point (x, y) moves and traces a path.

A simple example — a circle:

x = cx + r * cos(t)
y = cy + r * sin(t)

As t goes from 0 to 2π, the point sweeps out a full circle. Change the equations and you get spirals, waves, flowers, and the endless loops of a spirograph.

The spirograph family

The classic spirograph patterns come from combining circular motions — one circle rolling around another. In parametric form, you add together sine and cosine terms at different frequencies:

x = R1 * cos(a * t) + R2 * cos(b * t)
y = R1 * sin(a * t) + R2 * sin(b * t)

The constants R1, R2, a, b control the shape. Change them and the pattern transforms completely — this experimentation is the art. Whole-number frequency ratios produce closed, repeating loops; irrational ratios produce dense, never-quite-repeating webs.

How to make and plot it

1. Generate in p5.js or Processing

Loop t over a range, compute x and y from your equations, and draw a line from each point to the next:

function setup() {
  createCanvas(800, 800, SVG);
  noFill(); stroke(0); noLoop();
}
function draw() {
  beginShape();
  for (let t = 0; t < TWO_PI * 50; t += 0.01) {
    let x = width/2 + 200*cos(3*t) + 100*cos(7*t);
    let y = height/2 + 200*sin(3*t) + 100*sin(7*t);
    vertex(x, y);
  }
  endShape();
}

2. Tune the constants

Change the frequencies and radii one at a time. Small changes produce dramatically different patterns — keep the ones that delight you.

3. Export and optimize

Export as SVG (with p5.svg) and run it through vpype to clean up before plotting.

4. Plot it

Parametric curves are continuous, so they plot efficiently — long flowing strokes with minimal pen lifts. Use a reliable fineliner and let the smooth motion do its thing.

Where to take it further

Layer multiple parametric curves in different colors. Animate the constants and plot frames as an evolving series. Combine parametric motion with noise for organic variation. Generative Design is full of parametric and math-driven techniques with source code you can adapt directly. Parametric art is the rare corner of generative work where a few equations and some curiosity produce genuinely stunning, plot-ready results.

Frequently asked questions

What is parametric art?

Parametric art is generated from mathematical equations where x and y are each defined as a function of a parameter (often called t). As t advances, the point traces a curve — circles, spirals, roses, and the looping spirograph patterns. It's a natural fit for plotters because the output is pure line.

Do I need to be good at math?

No. You can produce beautiful parametric plots by remixing a few standard equations and changing their constants. Understanding deepens your control, but you can start by tweaking numbers and watching what happens.

Why is parametric art good for plotting?

It's defined entirely as continuous curves, which is exactly what pens draw best — long flowing strokes with few lifts. The math also gives precise, repeatable control over the design.