Drawing the diagram

For a given $n,k\in\mathbb{Z}^{+}$, $k>2$, the preceding sections tell us which gnomon $n$ will be placed in in a $k$-polygonal diagram, and in which spot along the gnomon.

To complete the diagram, we need to also determine how much the gnomon is bent at the particular position that $n$ finds itself at. Because the gnomon is a layer in a regular $k$-gon, it bends in increments of the base angle $alpha_{k}$ that is given by Equation (8).

As we draw a gnomon, we’ll start at the left-most position, which for a given $n$ can be expressed by $(x_{0}(n,k),y_{0}(n,k)$ as defined in (9).

Starting at the initial point $(x_{0}(n,k),y_{0}(n,k)$ for a given $n$, we want to move along the gnomon layer up to the gnomon depth of $n$ ($\text{gdepth}(n)$), bending along the gnomon as required. This process is expressed by the sums in the definitions of $x(n,k)$ and $y(n,k)$ of (10).