Parametric Equations

Definition

Parametric equations describe a curve by expressing both the $x$ and $y$ coordinates as functions of a third variable, called the parameter (usually $t$ ). This allows for the representation of curves that are not functions in the Cartesian sense (e.g., loops or vertical lines).

How to read: “The variable x is equal to the function f evaluated at t; and the variable y is equal to the function g evaluated at t.”
Meaning: The curve is traced over time (or angle)—decoupling horizontal and vertical motion from a single $y = f(x)$ constraint.

Why It Matters

The world does not move in $y = f(x)$ lines. Most natural and engineered motions—from the orbit of a satellite to the path of a fly—involve loops, vertical segments, and self-intersections that are mathematically impossible to describe with a single Cartesian function. Parametric equations are the solution to this “functionality crisis,” allowing us to decouple horizontal and vertical motion. Without them, robotics, ballistics, and animation would be restricted to primitive, non-looping paths.

Core Concepts

Definition: $x = f(t)$ $x = f (t)$ and $y = g(t)$ $y = g (t)$ for $t$ $t$ in some interval.
- How to read: “The variable x is equal to the function f of t and the variable y is equal to the function g of t.”
- Meaning: Both coordinates are functions of a parameter $t$ —the curve is a path traced as $t$ varies.
Parametric Differentiation: The slope of the curve is $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ $\frac{d y}{d x} = \frac{d y / d t}{d x / d t}$ , provided $dx/dt \neq 0$ $d x / d t \neq = 0$ .
- How to read: “The derivative of y with respect to x is equal to the derivative of y with respect to t divided by the derivative of x with respect to t, provided the derivative of x with respect to t is not zero.”
- Meaning / when to use: Chain rule for parametric curves—tangent slope from velocity components.
Second Derivative: $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{dx/dt}$ $\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d t} ( \frac{d y}{d x} )}{d x / d t}$ .
- How to read: “The second derivative of y with respect to x is equal to the derivative with respect to t of the first derivative of y with respect to x, all divided by the derivative of x with respect to t.”
- Meaning: Curvature/concavity of the parametric path—differentiate slope with respect to $t$ , then normalize.
Arc Length: The distance traveled along the curve from $t=a$ $t = a$ to $t=b$ $t = b$ is $L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$ $L = \int_{a}^{b} (\frac{d x}{d t})^{2} + (\frac{d y}{d t})^{2} d t$ .
- How to read: “The arc length L is equal to the integral from a to b of the square root of the quantity the derivative of x with respect to t squared plus the quantity the derivative of y with respect to t squared, integrated with respect to t.”
- Meaning: Integrate speed $||\mathbf{v}(t)||$ to get total path length.

Parametric Equations

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes