Parametric Equations
Parametric equations define curves using x(t) and y(t) as functions of a parameter t, enabling complex curves like circles and cycloids.
The Formula
Parametric equations describe a curve by expressing the x and y coordinates separately as functions of a third variable called a parameter, usually t. As t varies over some range, the point (x, y) traces out the curve.
This approach is more flexible than writing y as a function of x. It can describe curves that loop, cross themselves, or are vertical — things that are impossible with a single function y = f(x).
Common Parametric Curves
| Curve | x(t) | y(t) | Parameter Range |
|---|---|---|---|
| Circle (radius r) | r cos(t) | r sin(t) | 0 ≤ t ≤ 2π |
| Ellipse (a, b) | a cos(t) | b sin(t) | 0 ≤ t ≤ 2π |
| Line through (x₁,y₁) to (x₂,y₂) | x₁ + t(x₂−x₁) | y₁ + t(y₂−y₁) | 0 ≤ t ≤ 1 |
| Cycloid | r(t − sin(t)) | r(1 − cos(t)) | t ≥ 0 |
| Projectile (angle θ, speed v₀) | v₀ cos(θ) t | v₀ sin(θ) t − ½gt² | t ≥ 0 |
Useful Formulas for Parametric Curves
| Property | Formula |
|---|---|
| Slope (dy/dx) | (dy/dt) / (dx/dt) |
| Arc length | ∫√((dx/dt)² + (dy/dt)²) dt |
| Speed | √((dx/dt)² + (dy/dt)²) |
Example 1
A circle of radius 5 centered at the origin. Write parametric equations and find the point at t = π/4.
Parametric form: x = 5 cos(t), y = 5 sin(t)
At t = π/4: x = 5 cos(π/4) = 5 × (√2/2) = 5 × 0.7071
y = 5 sin(π/4) = 5 × (√2/2) = 5 × 0.7071
(x, y) ≈ (3.54, 3.54) — a point on the circle at 45°
Example 2
A projectile is launched at 30 m/s at 60° from the ground. Find the position at t = 2 seconds. (Use g = 9.81 m/s²)
x(t) = v₀ cos(θ) × t = 30 × cos(60°) × 2 = 30 × 0.5 × 2 = 30 m
y(t) = v₀ sin(θ) × t − ½gt² = 30 × sin(60°) × 2 − ½(9.81)(4)
y(2) = 30 × 0.866 × 2 − 19.62 = 51.96 − 19.62
Position at t = 2s: (30 m, 32.3 m) — the projectile is 30 m downrange and 32.3 m high
When to Use It
Parametric equations are used whenever standard y = f(x) form is too restrictive.
- Describing the path of a moving object (projectiles, planets, robots)
- Computer graphics and animation (Bezier curves, splines)
- Physics simulations with time as the parameter
- Curves that cannot be written as single functions (circles, figure-eights)
- CNC machining and 3D printing toolpaths