Parametric Equations Calculator (x(t), y(t))

Curves built from a moving point

A parametric curve describes x and y separately as functions of a third variable, the parameter t. As t advances, the point (x(t), y(t)) traces out the curve. This is far more flexible than y = f(x), which cannot draw anything that loops back, crosses itself, or runs vertically. Think of t as time and the curve as the path of a moving point. This calculator evaluates the position, the velocity components, the speed, and the slope at any t for several classic curves, and plots the whole path with your point marked.

The curves it handles

A circle is x = r cos t, y = r sin t. An ellipse stretches that into x = a cos t, y = b sin t. A cycloid, the path traced by a point on the rim of a rolling wheel, is x = r(t − sin t), y = r(1 − cos t); it is the curve a pebble wedged in a tire tread follows down the road. Projectile motion is x = v₀ cos θ · t, y = v₀ sin θ · t − ½g t², the parabola of anything thrown or launched.

What you get back

The slope of the curve is (dy/dt) / (dx/dt), which lets you find a tangent even where the curve is vertical and y = f(x) would break down. The speed is √((dx/dt)² + (dy/dt)²), the rate the tracing point moves along the path. For a circle of radius r that speed works out to exactly r, constant the whole way around, which is why uniform circular motion is the textbook example.

Where they matter

Animation and computer graphics run on parametric Bézier curves and splines. Physics simulations use t for time. CNC machines and 3D printers follow parametric toolpaths. Orbits, robot-arm motion, and roller-coaster track are all far easier to describe by a parameter than by a single equation in x and y. For trig curves below, enter t in radians; for projectile motion, t is in seconds.