Concave Shapes

We originally latched onto the ideas behind the Escherization paper [Kaplan and Salesin2000], which introduced some ideas on how to do computer generated tessellations given some arbitrary shape. Given a shape, the described algorithm attempts to deform some ninety different isohedral shapes, which comprise the different classes of tilable shapes, into a reasonable approximation of the original shape. The different classes of isohedral polygons also are parameterized to determine the proper tiling pattern. While the Escherization method sounded applicable to the Cookie Cutter problem, we eventually decided that we would not be able to reimplement this algorithm within our time constraints. Moreover, the Escherization method clearly states that it works best for convex shapes, which only represented a subset of the total possible cookie shapes we potentially had to deal with.

However, we derived an important concept from the Escherization paper. The paper showed how the deformation of a regular polygon really equated to a ``give and take'' on its edges. A pull on one edge of the polygon would be complemented by a push on another edge. We believe that the convex and concave corners of the given shape represented the result of this sort of force and feedback equilibrium. Joining the complementing corners of two copies of a cookie shape should reduce the entropy. To take advantage of this, we needed to be able to measure the potential effects of joining a particular concave corner to a convex one or vice versa.

Our first real implementation of FFA matched convex and concave corners whose sum of angles were near $360\,^{\circ}$. We realized that this lead to situations where the arm lengths of one corner did not match the arm lengths of the other corner. Thus, the two cookie copies did not fit well with each other, and overall, the placement of all the cookie copies seems less than optimal.

Eventually, we struck upon another idea which runs along a similiar line of thinking behind the Slab Allocator [Bonwick1994] which minimizes the internal fragmentation in a hard drive file system. In short, the Slab Allocator allocates slabs of limited size to store files of certain sizes. By cleverly choosing the slab sizes, the Allocator can ensure speedy file access and only a certain portion of free space is wasted due to internal fragmentation. FFA follows suit by predetermining whether matching a pair of convex and concave corners will result in excessive internal fragmentation between the corners. Essentially, the FFA player compares the area of the triangle composed of the vertices from the convex corner and the other triangle formed from the vertices of the concave corner. Running this procedure over every pairwise combination of corners should result in a pair of convex and concave corners which have minimal wasted area when joined together.

Figure 1: Joining Concave Shapes with Labeled Corner Bisectors

Once FFA has determined which corners to join, we must compute the angle of rotation to bring corner, $C_i$, on to corner, $C_j$, where $i \ne j$.

$$b_i = \frac{C_i^p + C_i^n}{\vert C_i^p + C_i^n\vert}$$ (3)

$$\theta = cos^{-1}( -b_i \cdot b_j )$$ (4)

Eq. 3 represents how the bisector, $b_i$, is computed for corner, $C_i$, which has arm vectors, $C_i^p$ and $C_i^n$. Eq. 4 shows how FFA computes the angle of rotation between corners, $C_i$ and $C_j$. In Figure 1, if $C_3$ is matched with $C_6$, $\theta_{3,6} = 0$. Thus, a copy of the concave shape in Figure 1 will not have to rotate to join corners $C_3$ and $C_6$. Once the FFA has rotated $C_j$ towards $C_i$, FFA translates the cookie which contains $C_j$ such that both corners lie on top of one another. Finally, while the two cookie overlap, the cookie containing $C_j$ is translated away the other cookie in the direction of $b_i$.