Carving Polytopes with Saws in 3D

TL;DR

This work studies carving 3D polytopes using two cut models: half-plane cuts and bounded-length ray sweeps. It provides a formal carveability notion, a complete characterization for half-plane carveability, and two decision algorithms with tight running times: a deterministic $O(n^2)$ method and a Las Vegas randomized $O(n^{3/2+\varepsilon})$ method, together with an $O(n^5)$ algorithm for ray sweeps that can output an explicit carving when feasible. The core techniques rely on face-plane projections, convex-hull tangents, and Clarkson–Shor sampling to efficiently manage geometric interactions across all faces. The results advance the theory of 3D cutting with practical, implementable procedures and illuminate open directions for subquadratic Half-Plane carving and optimized ray-sweep carving, with potential applications in computerized sculpture, manufacturability analysis, and CAD tooling.

Abstract

We investigate the problem of carving an $n$-face triangulated three-dimensional polytope using a tool to make cuts modelled by either a half-plane or sweeps from an infinite ray. In the case of half-planes cuts, we present a deterministic algorithm running in $O(n^2)$ time and a randomized algorithm running in $O(n^{3/2+\varepsilon})$ expected time for any $\varepsilon>0$. In the case of cuts defined by sweeps of infinite rays, we present an algorithm running in $O(n^5)$ time.

Figures (7)

• Figure 1: (Left.) A polytope containing a cavity that can be carved with ray sweeps but not half-planes. (Right.) A polytope that can be carved with rays, but with exactly one face that cannot be carved by half-plane cuts.
• Figure 2: Non-convex polytopes with holes that can be carved using half-planes.
• Figure 3: The tangents of $v$ and the halfplanes $H^\uparrow_i(v)$ and $H^\downarrow_i(v)$. The figure is drawn on the plane $L_T$.
• Figure 4: A construction for family of polytopes carveable using half-planes (to increase the size, add more pillars). For a member of this family with $n$ facial triangles, the algorithm described in \ref{['thm:quad-half-plane']} takes $\Theta(n^2)$ time.
• Figure 5: A polytope for which all faces are externally visible that cannot be cut with ray sweeps, since ray sweeps require bounded length. In particular, the shaded face would require a ray sweep of infinite length (i.e., a space-filling curve).

Theorems & Definitions (13)

• Theorem 2.1
• proof
• Theorem 2.4
• proof
• Lemma 2.5
• proof
• Corollary 2.6
• Theorem 3.1
• proof
• Theorem 3.2