Orientation

Another feature required by many applications is orientation. Orientation can be defined as a binary function with two arguments $\mathbf{a}$ and $\mathbf{c}$ which are elements of the cell complex and have the dimension $n$ and $n+1$ . The binary function returns whether the elements are oriented consistently or not. Usually the values $+1$ and $-1$ are used for the result of the orientation function $\mathcal{O}$ .

Figure 2.5: An edge and its two incident vertices can be used for the evaluation of the orientation function $\mathcal{O}$ .

For instance, an edge can be oriented in a manner that an oriented path is given which begins on one boundary vertex $\mathbf{v}_1$ and ends in the other boundary vertex $\mathbf{v}_2$ . Figure 2.5 shows an edge with an internal orientation and two boundary vertices of the edge. The orientation function is passed the edge and the source vertex and yields $-1$ . In contrast, when passed the edge and its sink vertex, the orientation function yields $+1$ .

$$\mathcal{O} ( \mathbf{e}, \mathbf{v}_1 ) = 1$$ (25)

$$\mathcal{O} ( \mathbf{e}, \mathbf{v}_2 ) = -1$$ (26)

This orientation function can be used in any simulation which makes use of a graph comprising oriented edges. It is important that it can be determined in which direction edge-related quantities, such as voltages in electronic circuits, are measured. It can be observed that the actual direction itself usually does not play any role, i.e. that the methods are independent from the orientation of the edges. However, once an edge orientation is chosen for all edges, the results are only valid with respect to this orientation. Other edge orientations result in another vector of solutions, whereas the obtained result remains unchanged. The orientation function is also available for elements of higher dimension as far as the dimension of two passed elements differs by one.