Guarding curvilinear art galleries with vertex or point guards

One of the earliest and most well known problems in computational geometry is the so-called art gallery problem. The goal is to compute the minimum possible number guards placed on the vertices of a simple polygon in such a way that they cover the interior of the polygon. In this paper we consider the problem of guarding an art gallery which is modeled as a polygon with curvilinear walls. Our main focus is on polygons the edges of which are convex arcs pointing towards the exterior or interior of the polygon (but not both), named piecewise-convex and piecewise-concave polygons. We prove that, in the case of piecewise-convex polygons, if we only allow vertex guards, $\lfloor\frac{4n}{7}\rfloor-1$ guards are sometimes necessary, and $\lfloor\frac{2n}{3}\rfloor$ guards are always sufficient. Moreover, an $O(n\log{}n)$ time and O(n) space algorithm is described that produces a vertex guarding set of size at most $\lfloor\frac{2n}{3}\rfloor$. When we allow point guards the afore-mentioned lower bound drops down to $\lfloor\frac{n}{2}\rfloor$. In the special case of monotone piecewise-convex polygons we can show that $\lfloor\frac{n}{2}\rfloor$ vertex guards are always sufficient and sometimes necessary; these bounds remain valid even if we allow point guards. In the case of piecewise-concave polygons, we show that $2n-4$ point guards are always sufficient and sometimes necessary, whereas it might not be possible to guard such polygons by vertex guards. We conclude with bounds for other types of curvilinear polygons and future work.

💡 Research Summary

The paper extends the classic art‑gallery problem, traditionally defined for simple polygons with straight edges, to polygons whose boundaries consist of curved arcs. The authors focus on two natural families of such curvilinear polygons: piecewise‑convex polygons, where each edge is a convex arc that bulges either outward or inward (but not both), and piecewise‑concave polygons, where each edge is a convex arc that bulges toward both sides of its tangent line. For each family they study two types of guards: vertex guards (guards placed at polygon vertices) and point guards (guards placed anywhere inside or on the boundary).

Piecewise‑convex polygons
A piecewise‑convex polygon may be locally convex but globally non‑convex. The authors first construct a polygonal approximation: they insert auxiliary vertices along each non‑linear convex arc so that the resulting shape is a linear polygon with the same combinatorial structure as the original. On this approximation they compute a constrained triangulation and apply Fisk’s three‑coloring method. Selecting the smallest color class yields a vertex guard set of size at most ⌊2n/3⌋, where n is the number of original vertices. The auxiliary vertices are then mapped back to the original polygon, preserving visibility, which gives a vertex guard set for the curved polygon of the same size. The whole procedure runs in O(n log n) time and O(n) space, assuming the arcs are algebraic curves of constant degree (so basic geometric predicates are O(1)).

To complement the upper bound, the authors exhibit a family of piecewise‑convex polygons that require ⌊4n/7⌋ − 1 vertex guards, establishing a non‑trivial lower bound that is larger than the classic ⌊n/3⌋ bound for straight‑edge polygons. When point guards are allowed, the lower bound drops to ⌊n/2⌋, because a single point can see across curved edges that a vertex cannot.

Monotone piecewise‑convex polygons
If the polygon is monotone with respect to some line L (every line orthogonal to L meets the polygon at most twice), the authors prove a tighter result: ⌊n/2⌋ + 1 vertex guards are always sufficient, and ⌊n/2⌋ point guards are sufficient and sometimes necessary. An O(n)‑time algorithm finds L, merges two sorted lists of vertices on each side of L, and selects alternating vertices to form the guard set. This demonstrates that monotonicity dramatically reduces the guard number for curved polygons, unlike the situation for straight‑edge monotone polygons where the bound remains ⌊n/3⌋.

Piecewise‑concave polygons
For piecewise‑concave polygons, vertex guards may be insufficient; the authors give examples where no finite set of vertex guards can cover the interior. Consequently they focus on point guards. They construct an interior locally‑concave polygon Q that is “maximally tangent” to the original polygon’s edges. Guarding Q reduces to guarding a set of pairwise‑disjoint faces, each of which needs exactly two point guards (the vertices of a triangulation). This yields an upper bound of 2n − 4 point guards. They also present configurations that require exactly 2n − 4 point guards, establishing the bound as tight.

Algorithmic contributions
The central algorithmic contribution is the O(n log n) procedure for piecewise‑convex polygons, which can be viewed as a natural generalization of Fisk’s linear‑time algorithm for straight‑edge polygons. The key steps are: (1) polygonal approximation by inserting auxiliary vertices; (2) constrained triangulation of the approximation; (3) three‑coloring and selection of the smallest color class; (4) mapping the selected vertices back to the original curved polygon. The authors discuss how each step respects the curvature and why visibility is preserved.

Additional results and future work
The paper also sketches bounds for locally convex polygons, monotone locally convex polygons, and completely general curvilinear polygons, though these are less tight. The authors assume arcs are algebraic curves of constant degree to keep primitive operations constant‑time; relaxing this assumption is left as an open problem. They suggest extensions to higher‑dimensional curved polyhedra, dynamic settings where arcs move, and approximation algorithms for minimizing the number of guards when exact computation is NP‑hard (as in the linear case).

Overall significance
This work is the first systematic study of art‑gallery guarding for polygons with curved edges. It shows that many classic combinatorial techniques (triangulation, three‑coloring) can be adapted to the curved setting, while also revealing new phenomena: the guard numbers for piecewise‑convex polygons are higher than for straight‑edge polygons, monotonicity has a stronger effect, and piecewise‑concave polygons may be unguardable by vertices alone. The tight upper and lower bounds, together with an efficient algorithm, provide a solid foundation for further research in curvilinear visibility and its applications in robotics, computer graphics, and sensor placement.