We address the problem of covering a target segment using a finite set of guards placed on a source segment within a simple polygon , assuming weak visibility between the target and source. Without geometric constraints, may be infinite, as shown by prior hardness results. To overcome this, we introduce the line aspect ratio (AR), defined as the ratio of the long width (LW) to the short width (SW) of . These widths are determined by parallel lines tangent to convex vertices outside (LW) and reflex vertices inside (SW), respectively. Under the assumption that AR is constant or polynomial in (the polygon's complexity), we prove that a finite guard set always exists, with size bounded by (AR). This AR-based framework generalizes some previous assumptions, encompassing a broader class of polygons. Our result establishes a framework guaranteeing finite solutions for segment guarding under practical and intuitive geometric constraints.
Avis et. al. [
The combinatorial conditions for mutual visibility between segments have been extensively studied.
Sack et. al. [
Guard placement optimization has been explored in related contexts. Tóth [
The study of visibility within simple polygons has advanced significantly, particularly with specialized
cases like sliding cameras, reflections, and structured segment visibility. Biedl et al. explored
slidingcamera guards, providing approximation algorithms and demonstrating the NP-hardness of
slidingcamera coverage in polygons with holes [
The inherent dificulty of unrestricted guarding has motivated the introduction of geometric
constraints. Bonnet et. al. [
Existing research has laid a strong foundation in:
• Eficient computation of weak visibility polygons [
Consider two line segments: a target segment and a source segment . The visibility between these segments may fall into one of three cases: 1. and are completely visible (CVP). 2. At least one point of or is invisible to the other segment. 3. and are partially visible (i.e., every point on one segment is visible to at least one point on the other, but not necessarily all points). From now on we refer to this case as the target is weakly visible from the source.
This work focuses on the third case. We aim to find a finite and polynomial set source segment such that: of points on the
To ensure a finite size for , certain assumptions about are necessary. This section introduces these assumptions. Specifically, we present an algorithm for determining the points of and demonstrate that, under Assumption 1, the algorithm yields a finite set whose size is polynomial in . Assumption 1 (Line Aspect Ratio (Our assumption)). For a simple polygon , the long width ( ) is the maximum distance between two parallel lines tangent to convex vertices of , on the outside of without intersecting its interior. The short width ( ) is the minimum distance between two such parallel line segments tangent to the reflex vertices in the interior of and constrained by the polygon’s boundary. The line aspect ratio is:
where is the complexity of .
• Constant line aspect ratio: ARline = (1) • Polynomial line aspect ratio: ARline = poly() Assumption 2 (Disk Aspect Ratio). For a simple polygon , the long diameter () is the diameter of the smallest enclosing circle tangent to the boundary. The short diameter () is the diameter of the largest inscribed circle tangent to the boundary. The disk aspect ratio is:
• Constant disk aspect ratio: ARdisk = (1) • Polynomial disk aspect ratio: ARdisk = poly()
For consistency, we use AR = ARline to denote the line aspect ratio in subsequent discussions.
Theorem 1. Under the line aspect ratio assumption (Definition 1), there exists a finite set that: on such || is bounded by AR Proof. The size of is determined by AR = / : 1. The slicing approach (Section 4.1) decomposes into visibility intervals. 2. By Lemma 1, the points in cover the target. 3. Observation 3 establishes that each interval has length ≥ .
4. Since has maximum length , the number of intervals is ≤ = AR. Thus || is finite and bounded by AR.
In the continue we present the slicing algorithm in Subsection 4.1, and Subsection 4.2 covers the lemmas and observations and their proofs. Section 5 provides a final discussion.
This subsection covers the slicing algorithm that splits a given source segment () by some middle points so that the union visibility of the set of all these points including the endpoints of the source segment covers an entire given target segment (). Without lost of generality, we already suppose that the given source and target segments are weakly visible.
We start the slicing algorithm by defining two specific reflex vertices and their computing approach.
Since the target is weakly visible from the source, consider the visibility of those points on the source whose view of the target is obstructed by some reflex vertices of . For each point on the source, its visibility can be blocked by at most two reflex vertices. However, these two reflex vertices may difer for diferent points on the source. For a precise definition of these reflex vertices, refer to Definition 1. Definition 1. Consider two reflex vertices: LBV, denoting the Left Blocking Vertex, and RBV, representing the Right Blocking Vertex. These reflex vertices are defined with respect to a specific point on the source. For a point on the source, imagine standing at , positioned between and , while observing . Assume that lies to the left and lies to the right of . There exists a single reflex vertex on each side of such that LBV and RBV are uniquely determined by (see Observation 2).
LBV (if it exists) is the reflex vertex where the line segment LBV intersects with and lies entirely inside , passes through at least one reflex vertex ( LBV), and has the exterior of on the left side of LBV. If multiple reflex vertices lie on a single line crossing LBV, the closest reflex vertex to along that line defines LBV .
The same strategy defines RBV , except that the exterior of lies on the right side of RBV.
We already know that and are weakly visible. Consider the line and run a sweeping algorithm on the reflex vertices of to obtain a line that meets the requirements of Definition 1. That is a line that lies on at least one reflex vertex passing and holds other reflex vertices of on its left side. Note that if multiple reflex vertices lie on this line, the closest reflex vertex to along that line defines LBV.
Using the same sweeping algorithm on the other side with an opposite direction obtains RBV. 4.1.2. Computing points on Denoting as 0 and as 0, we will perform the iterations described below to compute a sequence of points and on . This process continues until an iteration ≥ 1 is reached where lies to the right of . We will demonstrate that, assuming has a bounded line aspect ratio, the number of iterations has a polynomial upper bound (Lemma 2). Furthermore, when lies to the right of , the target will be covered by the set of points {, | 0 ≤ ≤ } (denoted as ) (see Lemma 1).
Iterations of the algorithm after computing LBV and RBV vertices Consider two lines: the line intersecting and LBV , and the line intersection and RBV , ≥ 0.
The line crossing LBV intersect the target on . The line crossing RBV intersects the target on .
In each iteration 0 ≤ < , compute and .
Consider / as a middle point on where places at the left side of . Assuming as the source compute LBV and RBV vertices for and points.
Draw the line crossing and LBV . The intersection of this line with creates a point denoted as +1.
Draw the line crossing and RBV . The intersection of this line with creates a point denoted as +1.
If +1 lies to the left of +1, set = + 1 and repeat the iteration procedure. Otherwise (that +1 lies to the right of (or if they lie on one point) +1), we reach the th iteration and the slicing algorithm stops since the target is covered (Lemma 3 reveals that in the ℎ iteration the target gets covered successfully).
In case one of the points LBV, RBV does not exist, the corresponding lines do not exist as well. If it is an or points the point in that iteration can see the rest of the target. If it is a point it can see rest of the source so the next point on the next point on the source is or itself. So, the algorithm has already reached a position where the points can see the entire target and the slicing algorithm terminates.
Coverage of y points u ty0 tx2 ty1 tx1
Coverage of x points ty2 tx0 v x x1 y2 x2 y1 y 4.1.3. Results of the slicing algorithm Lemma 1 indicates that the set obtained by the slicing algorithm covers the target. Lemma 2 we know that under the cases of Assumption 1 || remains polynomial in .
Observation 1. Given two segments a source and inside a simple polygon , Assumption 2 cannot guarantee a finite set of points on the source to cover the target.
Proof. See Figure.2. Based on Lemma 4 of [
a d
LD = |ac| target Ip ℓ
SD
SW
p source c
Observation 2. Given a segment as the source and a target segment that is partially visible to , consider a midpoint on the source, denoted as . There exists a unique reflex vertex obstructing the visibility of , denoted by LBV. This vertex is unique for . Specifically, if we stand on with to our left, LBV blocks the visibility of from the left side (if it exists). Similarly, RBV is unique (if it exists) and blocks the visibility of from the right side of .
Proof. We have to prove that LBV and RBV are unique reflex vertices on each side for a specific point on . Suppose considering the condition of the lemma both of these reflex vertices exist.
Without lost of generality consider LBV. On the contrary, suppose it is not unique. For proof, let’s consider another reflex vertex denoted by ̸= LBV, which could potentially obstruct the visibility of (a point on ) not to see some part of the target from the left side.
See Figure.3. To begin, we show that the visibility of cannot be obstructed by any other reflex vertex aside from LBV. Suppose, for the sake of contradiction, that there exists a reflex vertex on the left side of LBV, which obstructs the visibility of , preventing it from seeing a portion of the target. In such a scenario, the line crossing should holds LBV on its left side. Otherwise, defines LBV itself.
The same analysis reveals that RBV is unique for on the other side.
Observation 3. Any point on the source sees an interval which < , where comes from Assumption 1.
Proof. A point moving from to on the source, the half-lines through LBV and RBV are divergent. So, the interval created on the target are larger than distance between the parallel lines crossing LBV and RBV.
Lemma 1. Given two weakly visible segments, (the source) and (the target), inside a simple polygon , the slicing algorithm described in Subsection 4.1 generates a set of points on , denoted by , that collectively cover the entire target.
Proof. Without loss of generality, consider . Any point sees the target between two lines: one crossing LBV and the other crossing RBV . Note that the target is weakly visible to the source. Thus, = 0 must see , and the visibility of the points progressively aims to cover the target from to . The reflex vertices that block the visibility of the series from seeing a part of the target are the
LBVq lrv lrv
RBVq x q y v LBV vertices. In each iteration , the line crossing LBV determines +1 (see Figure.4), and +1 can see the target starting from +1 . Therefore, the visibility of the points on the target are connected. Thus, the target is visible to from to +1 . This process continues until the iteration stops, either when sees or when reaching a point where the remaining portion of the target has already been covered by points from the previous iterations.
Lemma 2. The output of the slicing algorithm (as detailed in Subsection 4.1) produces a finite set of points, , on the source. Under Assumption 1, the size of is polynomial in , where represents the complexity of .
Proof. Without loss of generality, we present the proof considering only the points labeled .
First, observe that RBV+1 and LBV must be the same reflex vertex. If they were diferent, +1 would be able to see a point closer to on the target, and LBV would not be obstructing its visibility.
Thus, in each iteration, sees the target between two lines: one crossing LBV and the other crossing RBV . From Observation 3, we know that the number of points on the source obtained by the slicing algorithm is upper bounded by the ratio of to .
Lemma 3. Given two weakly visible segments, the source and the target , inside a simple polygon , the slicing algorithm presented in Subsection 4.1 completes its execution in the ℎ iteration, where lies to the left of . In this iteration, the target segment is guaranteed to be fully covered. Proof. We only have to prove that lies to the left of (or on) . This means that the coverage of the target from the left meets the coverage from the right side.
u LBVy0
LBVx0
RBVy0 x = x0
y = y0 (a)
v RBVx0
P
Without loss of generality, suppose the visibility of both and is obstructed by reflex vertices. Consider the triangle formed by the points , , and LBV . Since is to the left of and has visibility of some part of the target, must lie to the left of the line passing through LBV . Similarly, analyzing the triangle formed by , , and RBV reveals that lies to the right of the line passing through RBV . Given that LBV lies to the left of the interior of and RBV lies to the right of the interior of , the lines passing through LBV and RBV must intersect, placing their intersection on the target. Consequently, lies to the left of on the target.
In case and coincide at a single point, the LBV and RBV of that point are distinct. The lines passing through (or ) and LBV, and through (or ) and RBV, intersect the target at diferent points such that lies to the left of .
Lemma 4. For two parallel lines 1 and 2 at distance tangent to reflex vertices in a simple polygon , the strip between them contains a region of area Ω(2).
Proof. Let 1 and 2 be two parallel lines at distance , tangent to reflex vertices of a simple polygon . Since 1 and 2 are tangent, the polygon must touch both lines, ensuring that spans the strip between them.
The region of within this strip must occupy an area dictated by the strip’s width . The minimal configuration occurs when forms a parallelogram spanning the strip, with base and height both equal to , yielding an area of 2. In degenerate cases, the polygon may form a triangular region within the strip, with area 22 .
Since both cases satisfy the lower bound of Ω(2), the result follows.
To be more precise: 1. The tangent lines create a “corridor” of width . v y u txj
tyj tx0 LBVxj
RBVyj x x1 yj xj y1 2. By the isoperimetric inequality, the minimal-area shape fitting this corridor is a rectangle or parallelogram. 3. The polygon must occupy at least the area of a parallelogram with: Base = , and Height = (ensuring area 2) 4. In degenerate cases, the region may be a triangle with area 2/2, preserving Ω(2).
Our work addresses a core problem in segment guarding by ensuring finite solutions without relying on restrictive stability assumptions. We introduce the novel concept of the line aspect ratio (AR), a geometric parameter quantifying the anisotropy of reflex vertices. This framework guarantees finite guard sets () with a size bounded by (AR).
1. Optimality of ||: Is || = Θ(AR) optimal? 2. Dynamic Guarding: Can the guard set adapt to polygon deformation while maintaining bounded size? 3. Extension to 3D: How can the AR framework be generalized for guarding problems in threedimensional environments? The AR assumption aligns with realistic geometric constraints observed in various domains: 1. Architectural Layouts: Reflex vertices often form corridors with bounded anisotropy, such as in building floor plans [19]. 2. Geographic Meshing: Terrain models exhibit moderate variations in width and feature distribution [20]. 3. Sensor Networks: Eficient sensor deployment frequently leverages regions with bounded aspect ratios [21]. 4. Robotic Navigation: Structured environments simplify visibility reasoning for autonomous systems [22].
This framework bridges theoretical advances and practical applications, providing a robust foundation for future research in visibility and guarding problems
The author has not employed any Generative AI tools. International Symposium on Computational Geometry (SoCG’17) 77 (2017) 20:1–20:15. URL: https://hal.science/hal-01994349v1. doi:10.4230/LIPIcs.SoCG.2017.20. [19] Y. Schwartzburg, M. Pauly, High-resolution topological tools for immersive architectural design, Computer-Aided Design 55 (2014) 45–57. Demonstrates bounded aspect ratios in architectural feature decomposition. [20] E. Puppo, Variable resolution terrain surfaces, Proc. CG International (1997) 81–90. Shows natural terrains have bounded width ratios in mesh simplification. [21] B. Wang, K. C. Chua, Coverage in hybrid mobile sensor networks, in: IEEE MASS, 2007, pp. 1–8.
Uses aspect-ratio constraints for sensor placement optimization. [22] S. M. Lavalle, Probabilistic roadmaps for path planning in high-dimensional configuration spaces, IEEE Transactions on Robotics 12 (2004) 566–580. Assumes bounded environment anisotropy for eficient visibility sampling.