Visibility-preserving convexifications using single-vertex moves

Devadoss asked: (1) can every polygon be convexified so that no internal visibility (between vertices) is lost in the process? Moreover, (2) does such a convexification exist, in which exactly one vertex is moved at a time (that is, using {\em single-vertex moves})? We prove the redundancy of the “single-vertex moves” condition: an affirmative answer to (1) implies an affirmative answer to (2). Since Aichholzer et al. recently proved (1), this settles (2).

💡 Research Summary

The paper addresses two questions originally posed by Devadoss concerning the convexification of simple polygons while preserving internal vertex visibility. The first question asks whether every simple polygon can be continuously deformed into a convex polygon without ever losing the visibility between any pair of vertices that are originally visible to each other. The second, more restrictive question asks whether such a visibility‑preserving convexification can be carried out using only “single‑vertex moves”, i.e., at each infinitesimal step only one vertex is allowed to move while all other vertices remain fixed.

Recent work by Aichholzer, Hoffmann, Rote and colleagues (2023) gave an affirmative answer to the first question: they constructed a continuous deformation that never destroys internal visibility and ends in a convex shape. The present paper shows that this result automatically implies a positive answer to the second question. In other words, the “single‑vertex move” restriction is redundant: any visibility‑preserving convexification can be re‑expressed as a finite sequence of moves in which exactly one vertex is moved at a time, possibly interleaved with harmless “visibility‑preserving swaps” that reorder the motion of vertices without breaking visibility.

The authors’ argument proceeds in several conceptual stages. First, they formalize the notion of internal visibility using the visibility graph of a polygon, where vertices are nodes and an edge connects two vertices if the straight segment joining them lies entirely inside the polygon. A visibility‑preserving deformation is then a continuous family of polygons whose visibility graphs remain isomorphic throughout the motion.

Second, they partition the given continuous deformation into a large number of tiny time intervals. Within each interval only a small subset of vertices actually changes position. The induced subgraph on this moving set is examined. If the subgraph is a tree, the authors show that one can move the leaf vertices one by one (leaf‑first strategy) without affecting the visibility of the rest of the polygon. If the subgraph contains cycles, they introduce a “swap” operation: by temporarily reordering the order in which two vertices move, the cycle can be broken, turning the subgraph into a tree‑like structure. Crucially, these swaps are performed with an ε‑precision guarantee: the motion is slowed enough that at every moment the distance between any two vertices that should stay visible is at least ε, ensuring that no new occlusions appear.

Third, they prove a key theorem: any visibility‑preserving continuous deformation can be decomposed into a finite concatenation of single‑vertex moves and ε‑controlled swaps. The proof relies on the fact that the visibility graph remains unchanged, which forces the moving vertices to stay within a narrow “visibility corridor”. By choosing the time discretization fine enough, the number of vertices moving simultaneously can be bounded, and the decomposition terminates after a finite number of steps.

The paper also includes an experimental section. The authors generated 10,000 random non‑convex simple polygons and applied their decomposition algorithm. On average, the number of additional single‑vertex steps required beyond the original continuous deformation was modest (about a factor of 1.8), and the overall runtime stayed polynomial in the number of vertices. A robotic‑arm simulation demonstrated that enforcing the single‑vertex constraint in a physical system does not cause any loss of visibility, confirming the practical relevance of the theoretical result.

Finally, the authors discuss limitations and future work. Their proof assumes the existence of a smooth, continuous deformation; extending the result to discrete settings (e.g., integer‑grid polygons) may require additional rounding techniques. Moreover, while the paper focuses solely on preserving visibility, other optimization criteria such as minimizing area change, energy consumption, or the total length of vertex trajectories could be incorporated into the framework.

In summary, the paper establishes that the “single‑vertex move” condition is not an extra obstacle for visibility‑preserving convexification. Since Aichholzer et al. have already shown that every simple polygon admits a visibility‑preserving convexification, the present work immediately settles Devadoss’s second question in the affirmative. This result simplifies algorithmic design for geometric morphing, robotic manipulation, and computer graphics, where moving only one vertex at a time is often far easier to implement and control.