An Optimal Algorithm for Finding the Kernel of a Polygon

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / DAAB-07-72-C-0259 [or DAAB-07-72-C-0592]National Science Foundation / NSF MCS-76-1732

On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

Generalized kernels of polygons under rotation

Given a set $\mathcal{O}$ of $k$ orientations in the plane, two points inside a simple polygon $P$ $\mathcal{O}$-see each other if there is an $\mathcal{O}$-staircase contained in $P$ that connects them. The $\mathcal{O}$-kernel of $P$ is the subset of points which $\mathcal{O}$-see all the other points in $P$. This work initiates the study of the computation and maintenance of the $\mathcal{O}$-${\rm Kernel}$ of a polygon $P$ as we rotate the set $\mathcal{O}$ by an angle $\theta$, denoted $\mathcal{O}$-${\rm Kernel}_{\theta}(P)$. In particular, we design efficient algorithms for (i) computing and maintaining $\{0^{o}\}$-${\rm Kernel}_{\theta}(P)$ while $\theta$ varies in $[-\frac{\pi}{2},\frac{\pi}{2})$, obtaining the angular intervals where the $\{0^{o}\}$-${\rm Kernel}_{\theta}(P)$ is not empty and (ii) for orthogonal polygons $P$, computing the orientation $\theta\in[0, \frac{\pi}{2})$ such that the area and/or the perimeter of the $\{0^{o},90^{o}\}$-${\rm Kernel}_{\theta}(P)$ are maximum or minimum. These results extend previous works by Gewali, Palios, Rawlins, Schuierer, and Wood.Comment: 12 pages, 4 figures, a version omitting some proofs appeared at the 34th European Workshop on Computational Geometry (EuroCG 2018

Kernel-based Construction Operators for Boolean Sum and Ruled Geometry

Boolean sum and ruling are two well-known construction operators for both parametric surfaces and trivariates. In many cases, the input freeform curves in IR2 or surfaces in IR3 are complex, and as a result, these construction operators might fail to build the parametric geometry so that it has a positive Jacobian throughout the domain. In this work, we focus on cases in which those constructors fail to build parametric geometries with a positive Jacobian throughout while the freeform input has a kernel point. We show that in the limit, for high enough degree raising or enough refinement, our construction scheme must succeed if a kernel exists. In practice, our experiments, on quadratic, cubic and quartic B´ezier and B-spline curves and surfaces show that for a reasonable degree raising and/or refinement, the vast majority of construction examples are successful

Maximizing the Minimum Angle with the Insertion of Steiner Vertices

We consider the problem of inserting a vertex inside a star-shaped input polygon at the location that maximizes the minimum angle in the resulting triangulation. An existing polynomial-time algorithm solves for the intersection of three polynomial surfaces (a prior paper indicates that these are eighth-degree polynomials) and computes the maxima of the curve of intersection of two such surfaces to solve the problem. We developed a similar technique through the geometric insight that at least two angles (typically, three) of the triangulation have to be identical at the optimal location. We combinatorially process the angles to compute the optimal location in each case. The worst-case complexity of the algorithm remains O(n 3 log n), but it is much easier to implement partly because our algorithm requires the solutions of an (at most) eighth-degree, univariate polynomial for each combination of the angles. We also modified the algorithm to lower the expected running time to O(n 2 ) using a recursive, randomized algorithm for LP-type problems. We extend the algorithm by imposing constraints on the location of the Steiner vertex and solving the constrained optimization problem in a similar manner. We also extend the algorithm to simultaneously insert two vertices by considering all possible topologies and ensuring that the necessary conditions for local maxima are satisfied

Generating Kernel Aware Polygons

Problems dealing with the generation of random polygons has important applications for evaluating the performance of algorithms on polygonal domain. We review existing algorithms for generating random polygons. We present an algorithm for generating polygons admitting visibility properties. In particular, we propose an algorithm for generating polygons admitting large size kernels. We also present experimental results on generating such polygons

Step into Computational Geometry Notebook III

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-79-C-0424National Science FoundationControl Data Corporatio

Sometimes Two Irrational Guards are Needed

In the art gallery problem, we are given a closed polygon P, with rational coordinates and an integer k. We are asked whether it is possible to find a set (of guards) G of size k such that any point p∈P is seen by a point in G. We say two points p, q see each other if the line segment pq is contained inside P. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur