A friendly guide to parametric equations
Most graphs you've seen describe a curve as y = f(x). The x-coordinate is the input; the y-coordinate is the output. This is fine for many curves, but it has a hidden assumption: only one y for each x. As soon as you want a curve that loops, crosses itself, or has vertical tangents, the explicit form breaks down.
Parametric equations escape this trap. Instead of treating x as input and y as output, we let both coordinates depend on a third variable, traditionally called t for "time". Imagine a particle moving in the plane: at each moment t, it has an x-coordinate x(t) and a y-coordinate y(t). The path it traces is the curve.
The simplest example: a circle
You can write the unit circle as x² + y² = 1, but that's implicit. As a parametric:
x(t) = cos(t) y(t) = sin(t)
As t sweeps from 0 to 2π, the point (cos(t), sin(t)) traces the circle exactly once, counter-clockwise. Want it the other way? Use (cos(t), -sin(t)). Want a bigger circle? Multiply by a radius.
Why parametric is more powerful
Parametric form unlocks several things you can't easily do explicitly:
- Curves that aren't functions (loops, self-intersections).
- Curves with timing — useful when modelling motion physically.
- Curves in 3D (just add a
z(t)). - Lissajous figures, where x and y oscillate at different frequencies.
Famous parametric curves
Lissajous figures
x(t) = sin(a·t + φ) y(t) = sin(b·t)
Vary the integer ratio a:b and the phase φ. Whole-number ratios produce closed, knotted shapes; irrational ratios fill the rectangle densely.
Cycloid
x(t) = r(t − sin(t)) y(t) = r(1 − cos(t))
The path of a point on the rim of a wheel rolling along a flat surface. Cycloids have astonishing properties — they are the brachistochrone curve, the shape that minimises descent time under gravity.
Projectile motion
x(t) = v · cos(θ) · t y(t) = v · sin(θ) · t − ½ · g · t²
The classic high-school physics curve. Gravity pulls in y; horizontal motion is uniform.
Parametric in DesmosGraph
For now, DesmosGraph supports polar parametric curves directly via r = f(θ). For general parametric curves you can use the polar form by converting, or break the parametric into separate explicit functions for visual exploration. Native parametric support is on our roadmap — let us know via our contact form if you'd like it prioritised.
From parametric to implicit
It's often possible to eliminate the parameter to get an implicit equation. For the circle, cos²(t) + sin²(t) = 1 gives x² + y² = 1. For a Lissajous figure, eliminating t produces complicated polynomial relations. The implicit form is often less revealing — the parametric form tells you not only what the shape is but also how it's traced.
Try this
- Trace the cycloid with
r = 5(1 − cos(θ))as a polar approximation. (Not the same curve, but a good visual relative.) - Plot two sine functions with offset frequencies and see how their phase relationship looks.
Parametric equations are a doorway. Once you start thinking of curves as paths rather than graphs, a whole world of motion-based mathematics opens up — and physics, animation and engineering all become friendlier.