# Parametric Equation Calculator

Evaluate parametric equations x(t) and y(t) at any t value. Convert between parametric and Cartesian forms for circles, ellipses, lines, and parabolas.

## What this calculates

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.

## Inputs

- **Parametric Curve Type** — options: Circle: x = r cos(t), y = r sin(t), Ellipse: x = a cos(t), y = b sin(t), Line: x = x₀ + at, y = y₀ + bt, Parabola: x = t, y = at² + bt + c — Select a standard parametric form.
- **Parameter 1** — Circle: radius r. Ellipse: semi-axis a. Line: starting x₀. Parabola: coefficient a.
- **Parameter 2** — Ellipse: semi-axis b. Line: starting y₀. Parabola: coefficient b. Circle: ignored.
- **Parameter 3** — Line: x-direction a. Parabola: constant c. Circle/Ellipse: ignored.
- **Parameter 4** — Line: y-direction b. Others: ignored.
- **Value of t** — The parameter value at which to evaluate. For angles, use radians.

## Outputs

- **x(t)** — The x-coordinate at the given t.
- **y(t)** — The y-coordinate at the given t.
- **Point (x, y)** — formatted as text — The point on the curve at the given t.
- **Parametric Equations** — formatted as text — The parametric equations used.
- **Cartesian Equation** — formatted as text — The equivalent equation with the parameter eliminated.

## Details

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.

## Frequently Asked Questions

**Q: What are parametric equations?**

A: Parametric equations express the x and y coordinates of a point as separate functions of a parameter (usually t). For example, x = cos(t) and y = sin(t) trace a unit circle as t varies from 0 to 2π. The parameter t does not appear in the final graph but controls which point you are at.

**Q: How do I eliminate the parameter?**

A: Solve one equation for t and substitute into the other. For a circle: from x = r cos(t), we get cos(t) = x/r. From y = r sin(t), sin(t) = y/r. Then cos²t + sin²t = 1 gives x²/r² + y²/r² = 1, or x² + y² = r². The strategy depends on the specific parametric form.

**Q: Should t be in degrees or radians?**

A: Use radians. Standard math functions in calculators and programming languages expect radians. For common angles: π/6 = 0.5236, π/4 = 0.7854, π/3 = 1.0472, π/2 = 1.5708, π = 3.1416, 2π = 6.2832.

**Q: Can parametric equations describe any curve?**

A: Yes. Any curve in the plane can be expressed parametrically. This is one of their main advantages over Cartesian equations: a circle fails the vertical line test and cannot be written as a single function y = f(x), but the parametric form x = cos(t), y = sin(t) works perfectly.

**Q: How is the parameter t related to time?**

A: In physics, t often represents time, and the parametric equations describe the position of a moving object. For a projectile: x = v₀ cos(θ)t and y = v₀ sin(θ)t - (1/2)gt² describe the horizontal and vertical positions as functions of time. But mathematically, t is just a variable and does not have to mean time.

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Source: https://vastcalc.com/calculators/math/parametric-equation
Category: Math
Last updated: 2026-04-08