Degeneracy of Angular Voronoi Diagram Hidetoshi Muta 1 and Kimikazu Kato 1,2 1 Department of Computer Science, University of Tokyo 2 Nihon Unisys, Ltd.

Slides:



Advertisements
Similar presentations
Singularity Theory and its Applications Dr Cathy Hobbs 30/01/09.
Advertisements

2-5 Proving Angles Congruent
KIM TAEHO PARK YOUNGMIN.  Curve Reconstruction problem.
By Groysman Maxim. Let S be a set of sites in the plane. Each point in the plane is influenced by each point of S. We would like to decompose the plane.
17. Computational Geometry Chapter 7 Voronoi Diagrams.
HW problem 1: Use Geometer’s Sketchpad to construct a rectangle whose side lengths are in the ratio of 2:1 without using the perpendicular, parallel, or.
Two - Dimensional Shapes
Geometry Vocabulary Test Ms. Ortiz. 1. If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
Polygons Two-dimensional shapes that have three or more sides made from straight lines. Examples: triangles squares rectangles.
Click to begin Click to begin Your Name Click here for Final Jeopardy Click here for Final Jeopardy
Presentation On Shapes Basic Summary: On Shapes By Priyank Shah.
Angles and Phenomena in the World
Geometry The strand of math that deals with measurement and comparing figures, both plane and solid .
Attributes A quality that is characteristic of someone or something.
Tests for Parallelograms Advanced Geometry Polygons Lesson 3.
Introduction Geometric figures can be graphed in the coordinate plane, as well as manipulated. However, before sliding and reflecting figures, the definitions.
A solid figure 3 dimensional figure.
Unit 5 Vocab By: Mika Hamady. Acute Angle Definition: An acute angle is an angle that is less then 90 degrees. Examples: 1.A 43 degree angle is acute.
Today we will be learning about
Warm-up 1/31/12 Given the number of sides of a regular polygon, Find: (a) the sum of the interior angles (b) one interior angle 1.5 sides2. 10 sides 3.What.
Section 8.4 Nack/Jones1 Section 8.4 Polyhedrons & Spheres.
Geometry Final Vocabulary. A ____________________ polygon is both equilateral and equiangular. regular.
SAT Prep. A.) Terminology and Notation Lines / Rays / Segments Angles – Classification Straight - 180° Vertical - = Circle – 360°
Coincidence of Voronoi Diagrams in a Quantum State Space Kimikazu Kato 1,2, Mayumi Oto 3,, Hiroshi Imai 1,4,5, Keiko Imai 6 Motivation Preliminaries Geometric.
CAD/Graphics 2013, Hong Kong Computation of Voronoi diagram of planar freeform closed convex curves using touching discs Bharath Ram Sundar and Ramanathan.
Unit 7 Polygons.
Definitions and Symbols © 2008 Mr. Brewer. A flat surface that never ends.
Distance fields. The program Multiple perspectives: euclidean, analytic, transformational Build algebraic skills Foreshadow future math concepts Build.
Quadrilaterals MATH 124. Quadrilaterals All quadrilaterals have four sides. All sides are line segments that connect at endpoints. The most widely accepted.
9 of 18 Introduction to medial axis transforms and their computation Outline DefinitionsMAS PropertiesMAS CAD modelsTJC The challenges for computingTJC.
5.2 Proving Quadrilaterals are Parallelograms Definition of Parallelogram: Both pairs of opposite sides are parallel.
Geometry Bingo. A parallelogram with four right angles. Opposite sides are parallel and congruent.
I go on and on in both directions What am I?. A line.
Attributes A quality that is characteristic of someone or something.
Quadrilaterals MA1G3d. Understand, use, and prove properties of and relationships among special quadrilaterals: parallelogram, rectangle, rhombus, square,
Voronoi Diagrams and a Numerical Estimation of a Quantum Channel Capacity 1,2 Kimikazu Kato, 3 Mayumi Oto, 1,4 Hiroshi Imai, and 5 Keiko Imai 1 Department.
Copyright © Ed2Net Learning, Inc.1 Quadrilaterals Grade 4.
Lesson 18Power Up DPage 114 Lines and Angles. Lines – No end, extends in both directions forever. Segments – Two endpoints, length can be measured. Lines.
Warm Up Classify each polygon. 1. a polygon with three congruent sides 2. a polygon with six congruent sides and six congruent angles 3. a polygon with.
7.2 Parallelograms. Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Consecutive angles Opposite angles.
LESSON 39: DIFFERENCE OF TWO SQUARES, PARALLELOGRAM PROOF, RHOMBUS.
Polygons A polygon is a plane shape with straight sides.
Vocabulary for the Common Core Sixth Grade.  base: The side of a polygon that is perpendicular to the altitude or height. Base of this triangle Height.
Lines, angles and polygons: Parallel lines and angles Triangles Quadrilaterals Angles in polygons Congruence Similarity.
Lines, angles and polygons: Parallel lines and angles Triangles Quadrilaterals Angles in polygons Congruence Similarity.
Homework Classifying Quadrilaterals Only ODD. Lesson 6.15 Quadrilaterals.
6 th grade Math Vocabulary Word, Definition, Model Emery UNIT 5: Area, Volume and Applications.
Unit 8 Part 2 Properties of Quadrilaterals Squares and Rhombi.
Measuring Angles 1-6. Angles  An angle is formed by two rays with the same endpoint. The rays are the sides of the angle. The endpoint is called the.
plane shape A shape in a plane that is formed by curves, line segments, or both. These are some plane figures 12.1.
Polygons Essential Question:
Go Math Grade 3 Chapter 12 Two-Dimensional Shapes
Plane and Space Curves Curvature-based Features
Properties of Geometric Shapes
Polygons with four sides
Do Now: What is the distance between (1, -3) and (5, -4)?
All sides have the same length and angles have the same measure.
Point-a location on a plane.
Introduction Geometric figures can be graphed in the coordinate plane, as well as manipulated. However, before sliding and reflecting figures, the definitions.
Warm Up Classify each polygon. 1. a polygon with three congruent sides
Warm Up Classify each polygon. 1. a polygon with three congruent sides
Linear Algebra Problem 3.4
Proving simple Geometric Properties by using coordinates of shapes
Classifying Quadrilaterals
On a Geometric Structure of Pure Multi-qubit Quantum States and Its Applicability to a Numerical Computation 1,2Kimikazu Kato, 3Mayumi Oto, 1,4Hiroshi.
Classifying Quadrilaterals
Voronoi Diagrams for Pure 1-qubit Quantum States
Presentation transcript:

Degeneracy of Angular Voronoi Diagram Hidetoshi Muta 1 and Kimikazu Kato 1,2 1 Department of Computer Science, University of Tokyo 2 Nihon Unisys, Ltd.

Angular Voronoi Diagram Introduced by Asano et al. in ISVD06 A tool to improve a polygon of triangular meshes Definition: For given line segments, the distance to determine the dominance of the regions is defined by a visual angle.

Equations of angular VD For given two line segments, as a boundary, there appear two equations which are the flip side of each other. Both equations are cubic (of degree three)

Why interested in the degeneracy of angular VD? It has a much more complicated structure than Euclidean VD It gives a hint for an extension of the existing complexity analysis for a general VD which regards its sites are in a general position It provides a good case study for computational robustness of a general VD

Degeneracy of Euclidean VD With some perturbation Or with some computational error More than four Voronoi sites are cocircular Complex crossing structure Voronoi edges meet at one point In theoretical context, they tend to avoid analysis of degeneracy, saying “assume the sites are in a general position” However, degeneracy takes special care in actual computation to achieve robustness

Computational complexity of algebraic VD Computational complexity of two dimensional VD whose boundaries are algebraic curves is shown to be [Halperin-Sharir 1994] It is proved by analyzing the structure of algebraic surfaces whose lower envelope is the VD Here again, it is assumed that the surfaces are “in a general position.” What happens in special cases?

Singular points of cubic curves NodeCuspIsolated point Singularities of cubic curves are classified into three types 図

Perturbation

Crossing at one pointCrossing at three points What wrong with robust computation? The number of intersecting points can drastically change with a perturbation

Degeneracy of angular VD For Euclidean VD, degeneracy is a concept of a position of multiple edges. However, for an angular VD, degeneracy is defined for a single edge. Degeneracy is defined as a curve which will change a topological position with a perturbation.

Classification of degeneracy Degenerate Non-Degenerate Degree three Degree two Degree one Singularity (node) Factorable (Circle x Line) Irreducible (Hyperbolic curve) Factorable (Line x Line) Never happens All AVD

Singularity (node) Factorable (Circle x Line) Irreducible ( Hyperbolic curve) Factorable (Line x Line) On same line On same line Same length Common endpoint Same length Parallel Same length, Parallel Diagonal lines cross vertically Same length with all endpoints in the same circle One line segment by the pair of endpoints is bisected vertically by the other Open! Degree 3Degree 2

Singularity of cubic curve It is proved that a node appears as a singularity of the boundary of an angular VD Whether other types of singularities (cusp and isolated point) appear or not is still open. (With some observation, we conjecture that they do not appear.)

Conclusion We classified the types of degeneracy of an angular Voronoi diagrams Classification of the sub-types of singular cubic curve case, i.e. whether a node is an only possible type of singularity, is still open. Our research shed light on degeneracy problem of a general Voronoi diagram w.r.t. an arbitrary distance.