Past exam of the computer science course of the University of Oxford / 2015 / Geometric modelling / 1 / d / Solution

Put $a_0=2-\sqrt{3}$ and define six closed half-spaces $H_i=\{(x,y):f_i(x,y)\le 0\}$. Their implicit equations and outward normal vectors are:

The six vertices of a polygon traverse the boundary counterclockwise. Each displayed normal vector therefore points to the right of the corresponding directed boundary line segment; it need not be a unit vector. The zero level sets are full straight lines, while the actual boundary retains only the appropriate line segments.

The inward corner at $C$ prevents using the set intersection of all six half-spaces: that would incorrectly discard the upper-left part of the seat. Instead takeThe last factor permits either $y\le 1$ or $x\le 1$, removing only the upper-right region. Another way to understand the same constructive solid geometry model is as the set union of two convex polygons, obtained by distributing the last set intersection over this set union. Its constructive solid geometry tree has an intersection at the root and a union child for $H_2,H_3$.