Parametric vs Implicit Functions

Parametric and implicit are two ways to describe curves that don't fit the simple "$y$ as a function of $x$" form.

Parametric

A parametric form expresses both $x$ and $y$ as functions of a third variable $t$ (the parameter, often time):

$x = f(t), \quad y = g(t)$

Example: a circle of radius 1: $x = \cos t$, $y = \sin t$ for $t \in [0, 2\pi]$.

Strengths: naturally describes motion (each $t$ gives a position), handles loops and self-intersections trivially.

Implicit

An implicit form uses a single equation:

$F(x, y) = 0$

The same circle: $x^2 + y^2 - 1 = 0$.

Strengths: unique algebraic equation, easy to test if a point is on the curve (just plug in and check).

When to use which

Situation	Best form
Motion / trajectory	Parametric
Implicit differentiation needed	Implicit
Curve has self-intersections	Parametric
Algebraic / symbolic manipulation	Implicit
Plotting via $t$-values	Parametric

Worked example: derivative

For the circle $x^2 + y^2 = 1$:

• Implicit differentiation: $2x + 2y \frac{dy}{dx} = 0$, so $\frac{dy}{dx} = -\frac{x}{y}$.
• Parametric ($x = \cos t$, $y = \sin t$): $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\cos t}{-\sin t} = -\frac{\cos t}{\sin t} = -\frac{x}{y}$. ✓

Both give the same answer; the procedure differs.

Conversion

You can sometimes convert between forms by eliminating the parameter (parametric → implicit) or by parametrising (implicit → parametric). Not always possible cleanly.