Jordan Curve — Definition, Formula & Examples

A Jordan curve is a closed curve in the plane that does not cross or touch itself. It forms a single unbroken loop, like a circle or an ellipse, dividing the plane into an interior region and an exterior region.

A Jordan curve is the image of a continuous injective (one-to-one) function $\gamma: [0,1] \to \mathbb{R}^2$ satisfying $\gamma(0) = \gamma(1)$ , with $\gamma$ injective on $[0,1)$ . The Jordan Curve Theorem states that every such curve separates $\mathbb{R}^2$ into exactly two connected components — a bounded interior and an unbounded exterior — with the curve as their common boundary.

How It Works

A Jordan curve must satisfy two properties: it is closed (it returns to its starting point) and it is simple (it never intersects itself). The power of the Jordan Curve Theorem is that no matter how wildly the curve winds or how irregular its shape, it always splits the plane into exactly two regions. To determine whether a point lies inside or outside a Jordan curve, you can use the crossing number test: draw a ray from the point to infinity, and count how many times it crosses the curve. An odd count means the point is inside; an even count means it is outside.

Example

Problem: Determine whether the curve parameterized by

\gamma(t) = (\cos 2\pi t,\, \sin 2\pi t)

for

t \in [0,1]

is a Jordan curve, and decide if the point

(0,0)

is in the interior or exterior.

Check closure: Evaluate the endpoints of the parameterization.

\gamma(0) = (1, 0), \quad \gamma(1) = (\cos 2\pi, \sin 2\pi) = (1, 0)

Check simplicity: For distinct

t_1, t_2 \in [0,1)

, the function

\gamma

gives distinct points because

\cos

and

\sin

together are injective on one full period. The curve is the unit circle, which does not self-intersect. So it is a Jordan curve.

Apply the crossing number test: Cast a ray from

(0,0)

in the positive

x

-direction. This ray crosses the unit circle exactly once, at

(1,0)

. Since 1 is odd, the origin lies in the interior.

Answer: The curve is a Jordan curve (the unit circle), and

(0,0)

lies in its interior.

Why It Matters

The Jordan Curve Theorem is foundational in topology, complex analysis, and computational geometry. In complex analysis, contour integration relies on knowing that a simple closed contour separates the plane into inside and outside regions. In computer graphics and GIS, point-in-polygon algorithms are direct applications of the crossing number test for Jordan curves.

Common Mistakes

Mistake: Assuming a figure-eight shape is a Jordan curve because it is closed.

Correction: A figure-eight crosses itself at the center, violating the simplicity condition. A Jordan curve must be both closed and non-self-intersecting.