Related papers: A sufficient condition for visibility paths in sim…
A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum…
An O(n) test for polygon convexity is stated and proved. It is also proved that the test is minimal in a certain exact sense.
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field $k$ of characteristic $p>0$ and $B \subset X(k)$ is a finite, possibly empty, set of points. The Newton polygon of a degree $p$ Galois cover of…
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…
A shadow of a geometric object $A$ in a given direction $v$ is the orthogonal projection of $A$ on the hyperplane orthogonal to $v$. We show that any topological embedding of a circle into Euclidean $d$-space can have at most two shadows…
For two non-congruent regular polygons of the same type, the method of finding the points in the plane at the equal distances to the vertices, is established. The existence of two points with this property is proved for two polygons with a…
Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its…
We consider the problem of finding an optimal piecewise linear path (polygonal line) connecting two given points with the possibility of making n turns at some points (the absolute value of each turn angle does not exceed a prescribed…
In this article, we determine the existing condition of cylinders in smooth minimal geometrically rational surfaces over a perfect field. Furthermore, we show that for any birational map between smooth projective surfaces, one contains a…
There are many practical applications that require simplification of polylines. Some of the goals are to reduce the amount of information necessary to store, improve processing time, or simplify editing. The simplification is usually done…
Given a polygon and a visibility range, the Myopic Watchman Problem with Discrete Vision (MWPDV) asks for a closed path P and a set of scan points S, such that (i) every point of the polygon is within visibility range of a scan point; and…
We address the question of whether a point inside a domain bounded by a simple closed arc spline is circularly visible from a specified arc from the boundary. We provide a simple and numerically stable linear time algorithm that solves this…
We present a method for proving the existence of solutions to a class of one dimensional variational problems. The method is demonstrated by two examples of optimal interpolation problems which are motivated by engineering applications. In…
We prove that computing a shortest monotone path to the optimum of a linear program over a simple polytope is NP-hard, thus resolving a 2022 open question of De Loera, Kafer, and Sanit\`a. As a consequence, finding a shortest sequence of…
This article considers two variants of a shortest path problem for a car-like robot visiting a set of waypoints. The sequence of waypoints to be visited is specified in the first variant while the robot is allowed to visit the waypoints in…
$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}}…
Contour polygonal approximation is a simplified representation of a contour by line segments, so that the main characteristics of the contour remain in a small number of line segments. This paper presents a novel method for polygonal…
The circuits of a polyhedron are a superset of its edge directions. Circuit walks, a sequence of steps along circuits, generalize edge walks and are "short" if they have few steps or small total length. Both interpretations of short are…
Detecting polygons defined by a set of line segments in a plane is an important step in analyzing vector drawings. This paper presents an approach combining several algorithms to detect basic polygons from arbitrary line segments. The…