Closed Curve — Definition, Formula & Examples

A closed curve is a curve in a plane (or in space) whose starting point and ending point are the same, forming an unbroken loop. Circles, ellipses, and rectangles are all examples of closed curves.

A closed curve is a continuous function $\gamma: [a, b] \to \mathbb{R}^n$ such that $\gamma(a) = \gamma(b)$ . If the curve does not cross itself at any other point, it is called a simple closed curve (or Jordan curve).

How It Works

To determine whether a curve is closed, trace along it from any starting point and check whether the path returns exactly to that point without any breaks. A circle traced once is a simple closed curve because it never crosses itself. A figure-eight, on the other hand, is a closed curve that is not simple because it has a self-intersection. In calculus, closed curves matter when you compute line integrals around a loop or apply Green's theorem, which relates a line integral around a simple closed curve to a double integral over the region it encloses.

Worked Example

Problem: Determine whether the parametric curve defined by

x(t) = 3\cos t

y(t) = 3\sin t

for

0 \le t \le 2\pi

is a closed curve.

Step 1: Evaluate the curve at the starting point

t = 0

\gamma(0) = (3\cos 0,\; 3\sin 0) = (3, 0)

Step 2: Evaluate the curve at the ending point

t = 2\pi

\gamma(2\pi) = (3\cos 2\pi,\; 3\sin 2\pi) = (3, 0)

Step 3: Since

\gamma(0) = \gamma(2\pi)

and the cosine and sine functions are continuous, the curve is closed. It traces a circle of radius 3 with no self-intersections, so it is also a simple closed curve.

Answer: Yes, the curve is a simple closed curve — specifically, a circle of radius 3 centered at the origin.

Why It Matters

Green's theorem, Stokes' theorem, and many results in vector calculus require integration over closed curves. In topology, the Jordan curve theorem guarantees that every simple closed curve in the plane divides it into exactly two regions — an inside and an outside — which is foundational for understanding area and boundaries.

Common Mistakes

Mistake: Assuming every closed curve is simple (non-self-intersecting).

Correction: A figure-eight is closed because it starts and ends at the same point, but it crosses itself, so it is not a simple closed curve. Always check for self-intersections separately.