# Solving Geometric Problems with the Rotating Calipers *

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Solving Geometric Problems with the Rotating Calipers * Godfried Toussaint School of Computer Science McGill University Montreal, Quebec, Canada ABSTRACT Shamos [1] recently showed that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be used on several convex polygons simultaneously. We then show that these generalizations allow us to obtain simple O(n) algorithms for solving a variety of problems defined on convex polygons. Such problems include (1) finding the minimum-area rectangle enclosing a polygon, (2) computing the maximum distance between two polygons, (3) performing the vector-sum of two polygons, (4) merging polygons in a convex hull finding algorithms, and (5) finding the critical support lines between two polygons. Finding the critical support lines, in turn, leads to obtaining solutions to several additional problems concerned with visibility, collision, avoidance, range fitting, linear separability, and computing the Grenander distance between sets. 1. Introduction Let P = (p 1, p 2,..., p n ) be a convex polygon with n vertices in standard form, i.e., the vertices are specified according to cartesian coordinates in a clockwise order and no three consecutive vertices are colinear. We assume the reader is familiar with [1]. In [1] Shamos presents a very simple algorithm for computing the diameter of P. The diameter is the greatest distance between parallel lines of support of P. A line L is a line of support of P if the interior of P lies completely to one side of L. We assume here that L is directed such that P lies to the right of L. Figure 1 illustrates two parallel lines of support. A pair of vertices, p j is an antipodal pair if it admits parallel lines of support. The algorithm of Shamos [1] generates all O(n) antipodal pairs of vertices and selects the pair with largest distance as the diameter-pair. The procedure resembles rotating a pair of dynamically adjustable calipers once around the polygon. Consider Figure 1. To initialize the algorithm a direction such as the x-axis is chosen and the two antipodal vertices and p j can be found in O(n) time. To generate the next antipodal pair we consider the angles that the lines of support at and p j make with edges +1 and p j p j+1, respectively. Let angle θ j < θ i. Then we rotate the lines of support by an angle θj, and p j+1, becomes the next antipodal pair. This process is continued until we come full circle to the starting position. In the event that θ j = θ i three new antipodal pairs are generated. * Published in Proceedings of IEEE MELECON 83, Athens, Greece, May

2 p j p j θ j +2 p j+1 θ j p j+2 θ i +1 p k θ k θ i θ l Fig. 1 In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used on one polygon or one pair of calipers can be used on several polygons. We then show that these generalizations provide simple O(n) solutions to a variety of geometrical problems defined on convex polygons. Such problems include the minimum-area enclosing rectangle, the maximum distance between sets, the vector sum of two convex polygons, merging convex polygons, and finding the critical support lines of linearly separable sets. The last problem, in turn, has applications to problems concerning visibility, collision avoidance, range fitting, linear separability, and computing the Grenander distance between sets. 2. The Smallest-Area Enclosing Rectangle This problem has received attention recently in the image processing literature and has applications in certain packing and optimal layout problems [2] as well as automatic tariffing in goodstraffic [3]. Freeman and Shapira [2] prove the following crucial theorem for solving this problem Theorem 2.1: The rectangle of minimum area enclosing a convex polygon has a side collinear with one of the edges of the polygon. The algorithm presented in [2] constructs a rectangle in O(n) time for each edge of P and selects the smallest of these for a total running time of O(n 2 ). This problem can be solved in O(n) time using two pairs of calipers orthogonal to each other. Let L s ( ) denote the directed line of support of the polygon at vertex such that P is to the right of the line. Let L(, p j ) denote the line through and p j. The first step consists of finding the vertices with the minimum and maximum x and y coordinates. Let these vertices be denoted by, p k, p l, and p j, respectively, and refer to Figure 2. We next construct L s (p j ) and L s (p l ) as the first set of calipers in the x direction, and L s ( ), L s (p k ) as the second set of calipers in a direction orthogonal p l Fig

3 P θ i q j+1 Q φ j q j Fig. 3 to that of the first set. All this can be done in O(n) time. As in Shamos diameter algorithm we now have four, instead of two, angles to consider θ i, θ j, θ k and θ l. Let θ i = min{θ i, θ j, θ k, θ l }. We rotate the four lines of support by an angle θ i, L(,+1 ) forms the base line of the rectangle associated with edge +1 and the corners of the rectangle can be computed easily in O(1) time from the coordinates of, +1, p j, p k and p l. We now have a new set of angles and the procedure is repeated until we scan the entire polygon. The area of each rectangle can be computed in constant time in this way resulting in a total running time of O(n). Another O(n) algorithm that implements this idea using a data structure known as a star is described in [4]. 3. The Maximum Distance Between Two Convex Polygons Let P = (p 1, p 2,..., p n ) and Q = (q 1, q 2,..., q n ) be two convex polygons. The maximum distance between P and Q, denoted by d max (P,Q), is defined as d max (P,Q) = max {d(, q j )} i,j = 1, 2,..., n, i, j where d(, q j ) is the euclidean distance between and q j. This distance measure has applications in cluster analysis [5]. A rather complicated O(n) algorithm for this problem appears in [6]. However, a very simple solution can be obtained by using a pair of calipers as in Figure 3. In Figure 3 the parallel lines of support L s (q j ) and L s ( ) have opposite directions and thus and q j are an antipodal pair between the sets P and Q. The two lines of support define two angles θ i and φ j, and the algorithm proceeds as in the diameter problem of Shamos [1]. Note that d max (P,Q) diameter (P Q) in general and thus we cannot use the diameter algorithm on P Q to solve this problem. For further details the reader is referred to [7]

4 +2 +1 P q j+1 θ i Q φ j q j Fig The Vector Sum of Two Convex Polygons Consider two convex polygons P and Q. Given a point r = (x r, y r ) ε P and a point s = (x s, y s ) ε Q, the vector sum of r and s, denoted by r s is a point on the plane t = (x r + x s, y r + y s ). The vector sum of the two sets P and Q, denoted as P Q is the set consisting of all the elements obtained by adding every point in Q to every point in P. Vector sums of polygons and polyhedra have applications in collision avoidance problems [8]. The following theorems make the problem computable. Theorem 4.1: P Q is a convex polygon. Theorem 4.2: P Q has no more than 2n vertices. Theorem 4.3: The vertices of P Q are vector sums of the vertices of P and Q. These theorems suggest the following algorithms for computing P Q. First compute q j, for i, j = 1, 2,..., n to obtain n 2 candidates for the vertices of P Q. Then apply an O(n log n) convex hull algorithm to the candidates. The total running time of such an algorithm is O(n 2 log n). We now show that P Q can be computed in O(n) time using the rotating calipers. Two vertices ε P and q j ε Q that admit parallel lines of support in the same direction as illustrated in Figure 4 will be referred to as a co-podal pair. The following theorem allows us to search only copodal pairs of P and Q in constructing P Q. Theorem 4.4: The vertices of P Q are vector sums of co-podal pairs of P and Q

5 θi +2 φ j q j+1 P q j Q -1 q j-1 Fig. 5 Finally, theorem 4.5 allows us to use the rotating calipers to construct P Q while searching the co-podal pairs of vertices. Theorem 4.5: Let z k = q j denote the vertex of P Q being considered. Then the succeeding vertex z k+1 = q j+1 if φ j < θ i, z k+1 = +1 q j if θ i < φ j, and z k+1 = +1 q j+1 if θ i = φ j. Thus, each vertex of P Q can be constructed in O(1) time after an O(n) initialization step, and since there are at most 2n such vertices, O(n) time suffices to compute P Q. 5. Merging Convex Hulls A typical divide-and-conquer approach to finding the convex hull of a set of n points on the plane consists of sorting the points along the x axis and subsequently merging bigger and bigger convex polygons until one final convex polygon is obtained [9]. Performing the merge in linear time will guarantee an O(n log n) upper bound on the complexity of the entire process. Merging two convex polygons P, Q consists of essentially finding two pairs of vertices, p j and q k, q l such that the new edges q k and q l p j, together with the two outer chains q k, q k+1,..., q l and p j, p j+1,..., form the convex hull of P Q. An edge such as q k is called a bridge and the vertices making up a bridge (such as and q k ) are referred to as bridge points. While O(n) algorithms exist for finding the bridges of two disjoint convex polygons [9], we show here that the bridges can also be computed very simply with the rotating calipers

6 +1 P -1-2 q j-1 q j-2 q j Q q j+1 Fig. 6 The following theorem leads to the desired algorithm. Theorem 5.1: Two vertices ε P and q j ε Q are bridge points if, and only if, they form a co-podal pair and the vertices -1, +1, q j-1, q j+1 all lie on the same side of L(, q j ). For example, in Figure 5 and q j are co-podal but q j+1 lies above L(, q j ). Hence q j is not a bridge. A simple algorithm for finding the bridges now becomes clear. As the co-podal pairs are being generated during caliper rotation we merely test the four adjacent vertices of the co-podal vertices to determine if they lie on the same side of the line collinear with the co-podal vertices, and we stop when two bridges have been found. Thus we can determine whether a co-podal pair is a bridge in O(1) time and the entire algorithm runs in O(n) time. 6. Finding Critical Support Lines Given two disjoint convex polygons P and Q a critical support line is a line L(, q j ) such that it is a line of support for P at, for Q at q j, and such that P and Q lie on opposite sides of L(, q j ). Critical support (CS) lines have applications in a variety of problems. 6.1 Visibility - 6 -

7 A typical problem in two-dimensional graphics consists of computing all visibility lists of a set of objects for a tour or a path taken on the plane by an observer [10]. The first step of an algorithm for solving this task consists of partitioning the plane into regions R i such that the visibility list is the same for an observer stationed anywhere in some fixed region. For two convex polygons the two CS lines partition the plane into the required visibility regions. 6.2 Collision avoidance Visibility and collision avoidance problems are closely related [11]. Given two convex polygons P and Q we may ask whether Q can be translated by an arbitrary amount in a specified direction without colliding with P. The CS lines provide an answer to this question. 6.3 Range fitting and linear separability Both of these problems involve finding a line that separates two convex polygons [12]. The critical support lines provide one solution to these problems. Consider Figure 6, where L(, q j ) and L(-2,q j-2 ) are the two CS lines. Denote their intersection by l*. We can choose as our separating line that line that goes through l* and bisects angle l*q j The Grenander distance Given two disjoint convex polygons P and Q there are many ways of defining the distance between P and Q. One method already discussed is d max (P,Q). Grenander [13] uses a distance measure based on CS lines. Let LE(, p j ) denote the sum of the edge lengths of the polygonal chain, +1,..., p j-1, p j and refer to Figure 6. The distance between P and Q would be d sep (P, Q) = d(, q j ) + d(-2, q j-2 ) - LE(-2, ) - LE(q j-2, q j ). Clearly the computation of d ses dominated by the computation of CS lines. 6.5 Computing the CS lines Theorem 6.1: Two vertices ε P and q j ε Q determine a critical support line if, and only if, they form an antipodal pair and +1, -1 lie on one side of L(, q j ) while q j-1, q j+1 lie on the other side of L(, q j ). The above theorem allows us to proceed as for finding bridge points. We can determine if an antipodal pair is a CS line pair in O(1) time for a total running time O(n). Thus all the problems mentioned in section 6 can also be solved simply in O(n) time using the rotating calipers. 7. Conclusion In section 3 the problem of computing d max (P, Q) was solved with the rotating calipers. One naturally considers the alternate problem of computing d min (P, Q) = min {d(, p j )} i, j = 1, 2,..., n. i, j - 7 -

8 It turns out however that the pair of vertices determining d min, surprisingly, is neither a co-podal nor an antipodal pair and thus the techniques used with success on d max fail on d min. Finding an O(n) algorithm for the latter problem remains an open question. The field of computational geometry is in need of general principles and methodologies that can be used to solve large classes of problems. Shamos [1] established that the Voronoi diagram is one such general structure that can be used to solve a variety of geometric problems efficiently. The results of this paper would indicate that the rotating calipers are another general tool for solving geometric problems. 8. References [1] M.I. Shamos, Computational geometry, Ph.D. thesis, Yale University, [2] H. Freeman and R. Shapira, Determining the minimum-area encasing rectangle for an arbitrary closed curve, Comm. A.C.M., Vol. 18, July 1975, pp [3] F.C.A. Groen et al., The smallest box around a package, Tech. Report, Delft University of Technology. [4] G.T. Toussaint, Pattern recognition and geometrical complexity, Proc. Fifth International Conference on Pattern Recognition, Miami Beach, December 1980, pp [5] R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis, Wiley, New York, [6] B.K Bhattacharya and G.T. Toussaint, Efficient algorithms for computing the maximum distance between two finite planar sets, Journal of Algorithms, in press. [7] G.T. Toussaint and J.A. McAlear, A simple O(n log n) algorithm for finding the maximum distance between two finite planar sets, Pattern Recognition Letters, Vol. 1, October 1982, pp [8] T. Lozano-Perez, An algorithm for planning collision-free paths among polyhedral obstacles, Comm. ACM, Vol. 22, 1979, pp [9] F.P. Preparata and S. Hong, Convex hulls of finite sets of points in two and three dimensions, Comm. ACM, Vol. 20, 1977, pp [10] H. Edelsbrunner et al., Graphics in flatland: a case study, Tech. Rept., University of Waterloo, CS-82-25, August [11] L.J. Guibas and F.F. Yao, On translating a set of rectangles, Proc. of the Twelfth Annual ACM Symposium on Theory of Computing, Los Angeles, April 1980, pp [12] J. O Rourke, An on-line algorithm for fitting straight lines between data ranges, Comm. ACM, Vol. 24, September 1981, pp [13] U. Grenander, Pattern Synthesis, Springer-Verlag, New York,

### Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS

1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps

### Binary Space Partitions

Title: Binary Space Partitions Name: Adrian Dumitrescu 1, Csaba D. Tóth 2,3 Affil./Addr. 1: Computer Science, Univ. of Wisconsin Milwaukee, Milwaukee, WI, USA Affil./Addr. 2: Mathematics, California State

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### The Essentials of CAGD

The Essentials of CAGD Chapter 2: Lines and Planes Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/essentials-cagd c 2000 Farin & Hansford

### Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer

### Tutorial: 3D Pipe Junction Using Hexa Meshing

Tutorial: 3D Pipe Junction Using Hexa Meshing Introduction In this tutorial, you will generate a mesh for a three-dimensional pipe junction. After checking the quality of the first mesh, you will create

### Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11

Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter

### x 2 f 2 e 1 e 4 e 3 e 2 q f 4 x 4 f 3 x 3

Karl-Franzens-Universitat Graz & Technische Universitat Graz SPEZIALFORSCHUNGSBEREICH F 003 OPTIMIERUNG und KONTROLLE Projektbereich DISKRETE OPTIMIERUNG O. Aichholzer F. Aurenhammer R. Hainz New Results

### 3D Distance from a Point to a Triangle

3D Distance from a Point to a Triangle Mark W. Jones Technical Report CSR-5-95 Department of Computer Science, University of Wales Swansea February 1995 Abstract In this technical report, two different

### ABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).

Degrees of Freedom Versus Dimension for Containment Orders Noga Alon 1 Department of Mathematics Tel Aviv University Ramat Aviv 69978, Israel Edward R. Scheinerman 2 Department of Mathematical Sciences

### GeoGebra. 10 lessons. Gerrit Stols

GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter

### Volume matching with application in medical treatment planning

Volume matching with application in medical treatment planning Yam Ki Cheung and Ovidiu Daescu and Steven Kirtzic and Lech Papiez Department of Computer Science The University of Texas at Dallas Richardson,

### We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model

CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant

### Minkowski Sum of Polytopes Defined by Their Vertices

Minkowski Sum of Polytopes Defined by Their Vertices Vincent Delos, Denis Teissandier To cite this version: Vincent Delos, Denis Teissandier. Minkowski Sum of Polytopes Defined by Their Vertices. Journal

### FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM

International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 34-48 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT

### CS 534: Computer Vision 3D Model-based recognition

CS 534: Computer Vision 3D Model-based recognition Ahmed Elgammal Dept of Computer Science CS 534 3D Model-based Vision - 1 High Level Vision Object Recognition: What it means? Two main recognition tasks:!

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### Visualization of General Defined Space Data

International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 013 Visualization of General Defined Space Data John R Rankin La Trobe University, Australia Abstract A new algorithm

### 28 Closest-Point Problems -------------------------------------------------------------------

28 Closest-Point Problems ------------------------------------------------------------------- Geometric problems involving points on the plane usually involve implicit or explicit treatment of distances

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.

An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points

### Solutions to Homework 6

Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example

### Segmentation of building models from dense 3D point-clouds

Segmentation of building models from dense 3D point-clouds Joachim Bauer, Konrad Karner, Konrad Schindler, Andreas Klaus, Christopher Zach VRVis Research Center for Virtual Reality and Visualization, Institute

### Catalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.

7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni- Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,

### New Distributed Algorithm for Connected Dominating Set in Wireless Ad Hoc Networks

0-7695-1435-9/0 \$17.00 (c) 00 IEEE 1 Proceedings of the 35th Annual Hawaii International Conference on System Sciences (HICSS-35 0) 0-7695-1435-9/0 \$17.00 00 IEEE Proceedings of the 35th Hawaii International

### The Container Selection Problem

The Container Selection Problem Viswanath Nagarajan 1, Kanthi K. Sarpatwar 2, Baruch Schieber 2, Hadas Shachnai 3, and Joel L. Wolf 2 1 University of Michigan, Ann Arbor, MI, USA viswa@umich.edu 2 IBM

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### Collision Detection Algorithms for Motion Planning

Collision Detection Algorithms for Motion Planning P. Jiménez F. Thomas C. Torras This is the sixth chapter of the book: Robot Motion Planning and Control Jean-Paul Laumond (Editor) Laboratoire d Analye

### Topological Data Analysis Applications to Computer Vision

Topological Data Analysis Applications to Computer Vision Vitaliy Kurlin, http://kurlin.org Microsoft Research Cambridge and Durham University, UK Topological Data Analysis quantifies topological structures

### 521493S Computer Graphics. Exercise 2 & course schedule change

521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 16-18 in TS128 Question 2.1 Given two nonparallel,

### ORIENTATIONS. Contents

ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit n-simplex which

### SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM

SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the

### Multi-layer Structure of Data Center Based on Steiner Triple System

Journal of Computational Information Systems 9: 11 (2013) 4371 4378 Available at http://www.jofcis.com Multi-layer Structure of Data Center Based on Steiner Triple System Jianfei ZHANG 1, Zhiyi FANG 1,

### Mean Value Coordinates

Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its

### The number of generalized balanced lines

The number of generalized balanced lines David Orden Pedro Ramos Gelasio Salazar Abstract Let S be a set of r red points and b = r + 2δ blue points in general position in the plane, with δ 0. A line l

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

### Ronald Graham: Laying the Foundations of Online Optimization

Documenta Math. 239 Ronald Graham: Laying the Foundations of Online Optimization Susanne Albers Abstract. This chapter highlights fundamental contributions made by Ron Graham in the area of online optimization.

### An Iterative Image Registration Technique with an Application to Stereo Vision

An Iterative Image Registration Technique with an Application to Stereo Vision Bruce D. Lucas Takeo Kanade Computer Science Department Carnegie-Mellon University Pittsburgh, Pennsylvania 15213 Abstract

### CHAPTER 1 Splines and B-splines an Introduction

CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the

### A PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT END-EFFECTORS

A PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT END-EFFECTORS Sébastien Briot, Ilian A. Bonev Department of Automated Manufacturing Engineering École de technologie supérieure (ÉTS), Montreal,

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### THE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE. Sample Solutions for Specimen Test 2

THE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE Sample Solutions for Specimen Test. A. The vector from P (, ) to Q (8, ) is PQ =(6, 6). If the point R lies on PQ such

### Fast and Robust Normal Estimation for Point Clouds with Sharp Features

1/37 Fast and Robust Normal Estimation for Point Clouds with Sharp Features Alexandre Boulch & Renaud Marlet University Paris-Est, LIGM (UMR CNRS), Ecole des Ponts ParisTech Symposium on Geometry Processing

### ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

### Figure 2.1: Center of mass of four points.

Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

### Monash University Clayton s School of Information Technology CSE3313 Computer Graphics Sample Exam Questions 2007

Monash University Clayton s School of Information Technology CSE3313 Computer Graphics Questions 2007 INSTRUCTIONS: Answer all questions. Spend approximately 1 minute per mark. Question 1 30 Marks Total

### Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions

Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.

### http://school-maths.com Gerrit Stols

For more info and downloads go to: http://school-maths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It

### A Linear Online 4-DSS Algorithm

A Linear Online 4-DSS Algorithm V. Kovalevsky 1990: algorithm K1990 We consider the recognition of 4-DSSs on the frontier of a region in the 2D cellular grid. Algorithm K1990 is one of the simplest (implementation)

### Nimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff

Nimble Algorithms for Cloud Computing Ravi Kannan, Santosh Vempala and David Woodruff Cloud computing Data is distributed arbitrarily on many servers Parallel algorithms: time Streaming algorithms: sublinear

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### Surface bundles over S 1, the Thurston norm, and the Whitehead link

Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In

### PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS

PUZZLES WITH POLYHEDRA AND PERMUTATION GROUPS JORGE REZENDE. Introduction Consider a polyhedron. For example, a platonic, an arquemidean, or a dual of an arquemidean polyhedron. Construct flat polygonal

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Estimating the Average Value of a Function

Estimating the Average Value of a Function Problem: Determine the average value of the function f(x) over the interval [a, b]. Strategy: Choose sample points a = x 0 < x 1 < x 2 < < x n 1 < x n = b and

### Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates

Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates Donatas Elvikis, Horst W. Hamacher, Marcel T. Kalsch Department of Mathematics, University of Kaiserslautern, Kaiserslautern,

### 3D Viewing. Chapter 7. Projections. 3D clipping. OpenGL viewing functions and clipping planes

3D Viewing Chapter 7 Projections 3D clipping OpenGL viewing functions and clipping planes 1 Projections Parallel Perspective Coordinates are transformed along parallel lines Relative sizes are preserved

### Fixed Point Theorems in Topology and Geometry

Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler

### MapReduce Approach to Collective Classification for Networks

MapReduce Approach to Collective Classification for Networks Wojciech Indyk 1, Tomasz Kajdanowicz 1, Przemyslaw Kazienko 1, and Slawomir Plamowski 1 Wroclaw University of Technology, Wroclaw, Poland Faculty

### B4 Computational Geometry

3CG 2006 / B4 Computational Geometry David Murray david.murray@eng.o.ac.uk www.robots.o.ac.uk/ dwm/courses/3cg Michaelmas 2006 3CG 2006 2 / Overview Computational geometry is concerned with the derivation

### Computer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)

Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include

### Categorical Data Visualization and Clustering Using Subjective Factors

Categorical Data Visualization and Clustering Using Subjective Factors Chia-Hui Chang and Zhi-Kai Ding Department of Computer Science and Information Engineering, National Central University, Chung-Li,

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### Studia Scientiarum Mathematicarum Hungarica 41 (2), 243 266 (2004)

Studia Scientiarum Mathematicarum Hungarica 4 (), 43 66 (004) PLANAR POINT SETS WITH A SMALL NUMBER OF EMPTY CONVEX POLYGONS I. BÁRÁNY and P. VALTR Communicated by G. Fejes Tóth Abstract A subset A of

### A fast and robust algorithm to count topologically persistent holes in noisy clouds

A fast and robust algorithm to count topologically persistent holes in noisy clouds Vitaliy Kurlin Durham University Department of Mathematical Sciences, Durham, DH1 3LE, United Kingdom vitaliy.kurlin@gmail.com,

### Treemaps with bounded aspect ratio

technische universiteit eindhoven Department of Mathematics and Computer Science Master s Thesis Treemaps with bounded aspect ratio by Vincent van der Weele Supervisor dr. B. Speckmann Eindhoven, July

### Using Data-Oblivious Algorithms for Private Cloud Storage Access. Michael T. Goodrich Dept. of Computer Science

Using Data-Oblivious Algorithms for Private Cloud Storage Access Michael T. Goodrich Dept. of Computer Science Privacy in the Cloud Alice owns a large data set, which she outsources to an honest-but-curious

### 1. Graphing Linear Inequalities

Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means

### Further Steps: Geometry Beyond High School. Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu

Further Steps: Geometry Beyond High School Catherine A. Gorini Maharishi University of Management Fairfield, IA cgorini@mum.edu Geometry the study of shapes, their properties, and the spaces containing

### For example, estimate the population of the United States as 3 times 10⁸ and the

CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number

### Machining processes simulation with the use of design and visualization technologies in a virtual environment

PLM: Assessing the industrial relevance 271 Machining processes simulation with the use of design and visualization technologies in a virtual environment Bilalis Nikolaos Technical University of Crete

### Hierarchical Data Visualization

Hierarchical Data Visualization 1 Hierarchical Data Hierarchical data emphasize the subordinate or membership relations between data items. Organizational Chart Classifications / Taxonomies (Species and

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### 12.5 Equations of Lines and Planes

Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

### WHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? 1. Introduction

WHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? DAVIDE P. CERVONE, WILLIAM V. GEHRLEIN, AND WILLIAM S. ZWICKER Abstract. Consider an election in which each of the n voters casts a vote consisting of

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Notes on Elastic and Inelastic Collisions

Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just

### Math for Game Programmers: Dual Numbers. Gino van den Bergen gino@dtecta.com

Math for Game Programmers: Dual Numbers Gino van den Bergen gino@dtecta.com Introduction Dual numbers extend real numbers, similar to complex numbers. Complex numbers adjoin an element i, for which i 2

### Objectives After completing this section, you should be able to:

Chapter 5 Section 1 Lesson Angle Measure Objectives After completing this section, you should be able to: Use the most common conventions to position and measure angles on the plane. Demonstrate an understanding

### Circle Object Recognition Based on Monocular Vision for Home Security Robot

Journal of Applied Science and Engineering, Vol. 16, No. 3, pp. 261 268 (2013) DOI: 10.6180/jase.2013.16.3.05 Circle Object Recognition Based on Monocular Vision for Home Security Robot Shih-An Li, Ching-Chang

### Star and convex regular polyhedra by Origami.

Star and convex regular polyhedra by Origami. Build polyhedra by Origami.] Marcel Morales Alice Morales 2009 E D I T I O N M O R A L E S Polyhedron by Origami I) Table of convex regular Polyhedra... 4

### Data Exploration Data Visualization

Data Exploration Data Visualization What is data exploration? A preliminary exploration of the data to better understand its characteristics. Key motivations of data exploration include Helping to select

### Parametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015

Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point

Université du Québec en Outaouais, Canada 1/46 Outline Intro Known Ad hoc GRN 1 Introduction 2 Networks with known topology 3 Ad hoc networks 4 Geometric radio networks 2/46 Outline Intro Known Ad hoc

### Metrics on SO(3) and Inverse Kinematics

Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction

### Voronoi Treemaps in D3

Voronoi Treemaps in D3 Peter Henry University of Washington phenry@gmail.com Paul Vines University of Washington paul.l.vines@gmail.com ABSTRACT Voronoi treemaps are an alternative to traditional rectangular

### CS 4204 Computer Graphics

CS 4204 Computer Graphics 3D views and projection Adapted from notes by Yong Cao 1 Overview of 3D rendering Modeling: *Define object in local coordinates *Place object in world coordinates (modeling transformation)

### B2.53-R3: COMPUTER GRAPHICS. NOTE: 1. There are TWO PARTS in this Module/Paper. PART ONE contains FOUR questions and PART TWO contains FIVE questions.

B2.53-R3: COMPUTER GRAPHICS NOTE: 1. There are TWO PARTS in this Module/Paper. PART ONE contains FOUR questions and PART TWO contains FIVE questions. 2. PART ONE is to be answered in the TEAR-OFF ANSWER

### 9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

### Sum of Degrees of Vertices Theorem

Sum of Degrees of Vertices Theorem Theorem (Sum of Degrees of Vertices Theorem) Suppose a graph has n vertices with degrees d 1, d 2, d 3,...,d n. Add together all degrees to get a new number d 1 + d 2

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two