Parametric Equation Calculator
Evaluate common parametric curves at a given parameter value t. This calculator computes the x and y coordinates and shows the equivalent Cartesian equation with the parameter eliminated.
Parametric equations define a curve by expressing x and y each as a function of a third variable, usually called t. Instead of writing y as a function of x directly, you get two equations: x = f(t) and y = g(t).
Circle (x = r cos t, y = r sin t):
As t goes from 0 to 2π, the point traces out a circle of radius r centered at the origin. Eliminating t using the identity cos²t + sin²t = 1 gives the Cartesian form x² + y² = r².
At t = π/2 (about 1.5708): x = 0, y = r. That is the top of the circle.
Ellipse (x = a cos t, y = b sin t):
Similar to the circle but with different semi-axes. Cartesian form: x²/a² + y²/b² = 1. When a = b, you get a circle.
Line (x = x₀ + at, y = y₀ + bt):
The point starts at (x₀, y₀) and moves in the direction (a, b). Eliminating t gives a standard linear equation. The direction vector (a, b) determines the slope: m = b/a.
Parabola (x = t, y = at² + bt + c):
The simplest parametrization: x equals t directly, so the Cartesian form is just y = ax² + bx + c. This makes it easy to evaluate the parabola at any x value.
Why use parametric form? Parametric equations can describe curves that fail the vertical line test (like circles), model motion (t = time), and simplify calculations in physics and computer graphics.