Functions
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Parametric Equations

Draw circles, ellipses, hearts, and curves that regular functions can't make

Most functions you've seen have the form y = f(x) — one y-value for each x. But what about circles, loops, and curves that double back on themselves? Those need a different approach: parametric equations.

Instead of y depending on x, both x and y depend on a third variable t (think of it as time). As t increases, the point (x(t), y(t)) moves and traces out a curve. The classic example is x = \cos(t),\; y = \sin(t) — as t goes from 0 to 2π, the point traces a perfect circle.

In this lesson, you'll start with that circle, stretch it into an ellipse, create wild Lissajous figures, and even draw a heart — all by changing the parametric formulas.

Graph

FAQ

What are parametric equations?
Parametric equations define a curve by expressing both x and y as functions of a third variable, usually called t (for time): x = f(t), y = g(t). As t varies over some interval, the point (x, y) traces out a curve. This allows you to draw shapes that aren't possible with a single y = f(x) equation, like circles, loops, and spirals.
How is the parameter t like time?
Think of t as time: at each moment t, a point sits at position (x(t), y(t)). As time moves forward, the point moves and draws a trail. At t = 0, the point starts somewhere; as t increases, it traces the curve. This is why parametric equations are natural for describing motion — projectiles, orbits, and animations all use them.
What is the difference between parametric and regular equations?
A regular equation y = f(x) gives one y for each x — the graph passes the vertical line test. Parametric equations can produce curves that fail the vertical line test (like circles, figure-eights, and spirals) because x and y are independent functions of t. Parametric is more general — any y = f(x) curve can be written parametrically as x = t, y = f(t).
What are some famous parametric curves?
Famous parametric curves include: Circle: (cos t, sin t). Ellipse: (a·cos t, b·sin t). Lissajous figures: (cos(at), sin(bt)) with different frequencies. Cycloid: (t - sin t, 1 - cos t) — the curve a point on a rolling wheel traces. Cardioid: heart-shaped curve. Spirograph patterns: nested circular motion.

Related Topics

The Unit CircleSine and CosinePythagorean Theorem