Parametric Curves

How to use?

Provide the x(t) and y(t) functions in terms of the parameter t. The curve is traced by animating t from its minimum value to its maximum. Everything from Math is at hand: sin, cos, PI, and the rest. Run the examples to see classic curves in action.

Parametric curves: x(t), y(t)

A parametric curve describes a path by writing each coordinate as a function of a parameter t: x = x(t), y = y(t). Classical examples include the circle (cos t, sin t), the ellipse (a cos t, b sin t), the cycloid (r(t − sin t), r(1 − cos t)), the lemniscate, the hypocycloid and the logarithmic spiral (eᵗ cos t, eᵗ sin t). The main advantage over y = f(x) is that parametric form can represent curves that fail the vertical line test, such as a full circle, and it naturally models the motion of a particle when t is interpreted as time. The velocity vector is (x'(t), y'(t)) and the arc length from t = a to t = b is L = ∫√(x'(t)² + y'(t)²) dt.

Applications: physics, graphics and engineering

Parametric curves model projectile motion in ballistics, are the foundation of computer graphics (Bézier and B-spline curves are parametric polynomials), drive CNC machining and 3D printing toolpaths, are used in animation to define trajectories over time, and appear in industrial design and robotics (end-effector paths).

FAQ

Why use parametric instead of y = f(x)? Many curves are not graphs of a single-valued function — a circle has two y values for most x. Parametric form encodes the full path and also the direction of traversal.

How do I find dy/dx from x(t), y(t)? By the chain rule, dy/dx = (dy/dt) / (dx/dt), valid whenever dx/dt ≠ 0.

What is the t range? Depends on the curve. For one full circle, t ∈ [0, 2π]. For spirals or open curves, t can grow unbounded.

What is the difference from polar coordinates? Polar is a special case where x = r(t) cos t, y = r(t) sin t and t is the angle.