SIMPLE CLOSED CURVES. SIMPLY AND MULTIPLY-CONNECTED REGIONS

A simple closed curve is a curve which does not intersect itself anywhere. Mathematically, a curve in the $xy$ plane is defined by the parametric equations $x = \phi(t),\ y = \psi(t)$ where $\phi$ and $\psi$ are single-valued and continuous in an interval $t_1 \le t \le t_2$ . If $\phi(t_1) = \phi(t_2)$ and $\psi(t_1) = \psi(t_2)$ , the curve is said to be closed. If $\phi(u) = \phi(v)$ and $\psi(u) = \psi(v)$ only when $u = v$ (except in the special case where $u = t_1$ and $v = t_2$ ), the curve is closed and does not intersect itself and so is a simple closed curve. We shall also assume, unless otherwise stated, that $\phi$ and $\psi$ are piecewise differentiable in $t_1 \le t \le t_2$ .

If a plane region has the property that any closed curve in it can be continuously shrunk to a point without leaving the region, then the region is called simply-connected, otherwise it is called multiply-connected.

As the parameter $t$ varies from $t_1$ to $t_2$ , the plane curve is described in a certain sense or direction. For curves in the $xy$ plane, we arbitrarily describe this direction as positive or negative according as a person traversing the curve in this direction with his head pointing in the positive $z$ direction has the region enclosed by the curve always toward his left or right respectively. If we look down upon a simple closed curve in the $xy$ plane, this amounts to saying that traversal of the curve in the counterclockwise direction is taken as positive while traversal in the clockwise direction is taken as negative.