A new 2D tessellation for angle problems: The polar diagram

The new approach we propose in this paper is a plane partition with similar features to those of the Voronoi Diagram, but the Euclidean minimum distance criterion is replaced for the minimal angle criterion. The result is a new tessellation of the plane in regions called Polar Diagram, in which every site is owner of a polar region as the locus of points with smallest polar angle respect to this site. We prove that polar diagrams, used as preprocessing, can be applied to many problems in Computational Geometry in order to speed up their processing times. Some of these applications are the convex hull, visibility problems, and path planning problems

The floodlight problem

Given three angles summing to 2, given n points in the plane and a tripartition k1 + k2 + k3 = n, we can tripartition the plane into three wedges of the given angles so that the i-th wedge contains ki of the points. This new result on dissecting point sets is used to prove that lights of specied angles not exceeding can be placed at n xed points in the plane to illuminate the entire plane if and only if the angles sum to at least 2. We give O(n log n) algorithms for both these problems

Proper Coloring of Geometric Hypergraphs

We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions

On the number of touching pairs in a set of planar curves

Given a set of planar curves (Jordan arcs), each pair of which meets -- either crosses or touches -- exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has $O(t^2n)$ touchings, where $t$ is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.Comment: 14 pages, 7 figure

Embedding Graphs into Embedded Graphs

A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed. In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings. To the best of our knowledge an analogous result was previously known essentially only when G is a cycle

Inserting one edge into a simple drawing is hard

A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G + e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment s, it can be decided in polynomial time whether there exists a pseudocircle Fs extending s for which A ¿ {Fs} is again an arrangement of pseudocircles.Peer ReviewedPostprint (published version

ConTesse: Accurate Occluding Contours for Subdivision Surfaces

International audienceThis paper proposes a method for computing the visible occluding contours of subdivision surfaces. The paper first introduces new theory for contour visibility of smooth surfaces. Necessary and sufficient conditions are introduced for when a sampled occluding contour is valid, that is, when it may be assigned consistent visibility. Previous methods do not guarantee these conditions, which helps explain why smooth contour visibility has been such a challenging problem in the past. The paper then proposes an algorithm that, given a subdivision surface, finds sampled contours satisfying these conditions, and then generates a new triangle mesh matching the given occluding contours. The contours of the output triangle mesh may then be rendered with standard non-photorealistic rendering algorithms, using the mesh for visibility computation. The method can be applied to any triangle mesh, by treating it as the base mesh of a subdivision surface

Automated Aerial Refueling Position Estimation Using a Scanning LiDAR

This research examines the application of using a scanning Light Detection and Ranging(LiDAR) to perform Automated Aerial Refueling(AAR). Specifically this thesis presents two algorithms to determine the relative position between the tanker and receiver aircraft. These two algorithms require a model of the tanker aircraft and the relative attitude between the aircraft. The first algorithm fits the measurements to the model of the aircraft using a modified Iterative Closest Point (ICP) algorithm. The second algorithm uses the model to predict LiDAR scans and compare them to actual measurements while perturbing the estimated location of the tanker. Each algorithm was tested with simulated LiDAR data before real data became available from test flights. The data collected from this test ight was used to determine the accuracy of the two algorithms with real LiDAR data. After correcting for modeling errors the accuracy of each algorithm is about a Mean Radial Spherical Error of 40cm

Descriptive geometry

This student’s book is developed for the first-year students of engineering majors in order to improve their skills at independent work and to provide their study process by additional material. This book is a combination of a textbook and a workbook, that makes her appropriate for classwork. It can also be intended for distance learning case as it contains theoretical material on the descriptive geometry as the first and the fundamental branch for the academic course “Engineering graphics”. In addition, the algorithms given here enable students to solve similar problems. The specific distinction from the other books is in providing students with professional linguacultural knowledge on the subject