Let $M=(E,s,\rho )$ be a discrete surface. The dual surface for $M$ is then the discrete surface

$M^{\ast }=(E,s^{-1}\circ \rho ,\rho )$.

Thus $M^{\ast }$ has the same set $E$ of oriented edges as $M$. Also the reversal  $\rho$ of edges is the same. The faces of $M^{\ast }$ are the cycles of $s^{-1}\circ \rho =\rho \circ (s\circ \rho {)}^{-1}\circ {\rho }^{-1}$, which means they are the images under $\rho$ of the vertices of $M$. The vertices of $M^{\ast }$ are the cycles of $s^{-1}\circ \rho \circ \rho =s^{-1}$ and therefore are the faces of $M$. To say it briefly: $M^{\ast }$ is a surface where the vertices and faces of $M$ have interchanged roles.

To draw $M^{\ast }$ one can draw a new (“dual”) vertex in the middle of every face of $M$ and connect dual vertices whenever the corresponding faces of $M$ are adjacent.

For every oriented edge $e$ we can draw an arrow from the center of face $\text{left}(e)$ to the center of face $\text{right}(e)$. So we rotate the arrow  representing an oriented edge of $M$ in a clockwise fashion to draw the corresponding edge of $M^{\ast }$. We think of this as just another graphical representation for the same oriented edge $e$.

A number written on an oriented edge can mean two different things: First, it could describe the value of a 1-form  $\omega$ on $e$. For example, in case $\omega =df$ the number signals the difference of function values of $f$ at the end points. Second, it could describe the value of a dual 1-form  $\eta$ on $e$. For example, it could stand for the difference of the values of a dual o-form on the two sides of $e$. It is clear that such a dual 0-form can also be seen as a 0-form on the dual surface $M^{\ast }$, justifying our notation ${\mathrm{\Omega }}_0(M^{\ast })$. The same holds for ${\mathrm{\Omega }}_1(M^{\ast })$.