Parametric Equation Calculator

Parametric equations describe a curve by expressing both x and y as functions of a third variable t (the parameter). For example, a circle of radius r is given by x(t) = r*cos(t) and y(t) = r*sin(t) for t from 0 to 2*pi. As t varies, the point (x(t), y(t)) traces the curve.

A cycloid is the curve traced by a point on the rim of a rolling circle. If the circle has radius r and rolls along the x-axis, the parametric equations are x(t) = r*(t - sin(t)) and y(t) = r*(1 - cos(t)). The cycloid has special properties: a ball sliding down a cycloid-shaped ramp reaches the bottom in minimum time (brachistochrone problem).

The lemniscate of Bernoulli is a figure-eight shaped curve with polar equation r^2 = a^2*cos(2*theta). In parametric form it is x(t) = a*cos(t)/(1+sin^2(t)) and y(t) = a*sin(t)*cos(t)/(1+sin^2(t)). It is the locus of points whose product of distances to two foci equals a^2/4.

A standard equation y = f(x) defines y as a function of x, which requires each x to map to exactly one y. Parametric equations have no such restriction: the curve can loop back on itself, cross itself, or fail the vertical line test entirely. Parametric form also encodes direction of travel as t increases, which y = f(x) does not.

For simple cases: from x = a*cos(t) and y = b*sin(t), use x^2/a^2 + y^2/b^2 = cos^2(t)+sin^2(t) = 1, giving the ellipse equation. For x = t, y = t^2, substitute to get y = x^2. In general, eliminating t may produce an implicit equation F(x,y) = 0 that describes the same curve.