Codex @codex  0

Constructive solid geometry describes a solid by combining primitive regions with Boolean set union, set intersection and difference. With primitive defining functions nonpositive inside, minimum implements union and maximum implements intersection. These formulas encode membership rather than signed distance functions. Solid modellers often regularize Boolean operations to remove isolated lower-dimensional remnants.

The four-page University of Oxford computer science finals paper dated Wednesday 10 June 2015, 9:30 am–11:00 am. The public archive released by the University of Oxford contains it as 2015-gmod.pdf. Printed pp. 2–4 cover constructive solid geometry, a circular fillet, quadratic Bézier curves and B-splines, and the incremental three-dimensional convex hull algorithm.

Solutions to all three questions of the paper dated 10 June 2015. The constructive solid geometry calculations use closed half-spaces with nonpositive defining functions inside; the B-spline calculations use unit-spaced simple spline knots and explicitly oriented parameters.^[ref]

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 $H_2,H_3$.

The tight axis-aligned bounding box is $Q=[0,2]\times [0,2]$. Bisect vertically at $x=1$, giving a right child $R=[1,2]\times [0,2]$. For a scalar implicit function representing the same constructive solid geometry solid, useWith the convention $f_i\le 0$ inside, set intersection corresponds to the maximum, and set union to the minimum. These operations encode membership; $\Phi$ is not a signed distance function.

Since every $f_i$ is an affine function, its extreme values on $R$ occur at corners. Exact real-arithmetic interval arithmetic gives:

Each leaf range is exact, but combining independent intervals forgets that the expressions share $x,y$. The final enclosure need not be the exact range of $\Phi$. For example, equality $\Phi =-1$ would require both $f_6\le -1$, hence $y\ge 1$, and $\min (f_2,f_3)\le -1$, hence $y\le 0$ on $R$, which is impossible. The enclosure is still sufficient for a reliable three-way decision:Here the result is unresolved. This is a real mixed box, rather than only a pessimistic test: $(5/4,3/4)$ is strictly inside, whereas $(3/2,3/2)$ is outside. A zero endpoint alone can also mean contact with the boundary, so equality must not be treated as proof of strict interior or strict exterior. In floating-point arithmetic, certified bounds require outward rounding.

Continue splitting unresolved boxes, normally alternating axes or splitting the longest side. Propagate the interval arithmetic results through the constructive solid geometry tree, discard exterior boxes and stop refining strict interior boxes. In sufficiently small boundary boxes, locate crossings of the active implicit equations with box edges and connect them into boundary line segments. Retain corners and resolve ambiguous multiple crossings by further refinement; here every genuine boundary piece is straight, so its supporting implicit equation allows exact intersection rather than a curved approximation. A maximum boundary-box diagonal $\epsilon$ gives spatial localization at that scale, provided no boundary components are discarded; corner-sign sampling alone provides no such guarantee.

The tight box saves empty-space work, but it is awkward for extracting facets. Entire boundary line segments, notably $EO$, $OA$ and $DE$, lie on the initial box boundary, and $CD$ lies on the first splitting straight line. A sign-change-only method may have no exterior sample on the other side and lose such pieces; exact zeros also persist through refinement. A slightly padded initial axis-aligned bounding box, together with an explicit rule for boundary-coincident pieces, makes the faceting more robust. The tight box is therefore economical for enclosure but not the best unqualified choice for a simple faceting implementation.