Homotopy

Definition

Let ${\gamma }_1(t)$ and ${\gamma }_2(t)$ be two closed (parametrized) curves in a region $D$, with parameter $0\le t\le 1$.
They are called homotopic if there exists a continuous function $h:[0,1]\times [0,1]\to D$ such that, for all $s,t\in [0,1]$, we have
$h(t,0)={\gamma }_1(t),h(t,1)={\gamma }_2(t),$
$h(0,s)=h(1,s).$
In this case we use the notation ${\gamma }_1\sim {\gamma }_2$.

Example

An example can be found here.