A Contribution to Triangulation Algorithms for Simple Polygons
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Geometry Honors Mid-Year Exam Terms and Definitions Blue Class 1
Geometry Honors Mid-Year Exam Terms and Definitions Blue Class 1. Acute angle: Angle whose measure is greater than 0° and less than 90°. 2. Adjacent angles: Two angles that have a common side and a common vertex. 3. Alternate interior angles: A pair of angles in the interior of a figure formed by two lines and a transversal, lying on alternate sides of the transversal and having different vertices. 4. Altitude: Perpendicular segment from a vertex of a triangle to the opposite side or the line containing the opposite side. 5. Angle: A figure formed by two rays with a common endpoint. 6. Angle bisector: Ray that divides an angle into two congruent angles and bisects the angle. 7. Base Angles: Two angles not included in the legs of an isosceles triangle. 8. Bisect: To divide a segment or an angle into two congruent parts. 9. Coincide: To lie on top of the other. A line can coincide another line. 10. Collinear: Lying on the same line. 11. Complimentary: Two angle’s whose sum is 90°. 12. Concave Polygon: Polygon in which at least one interior angle measures more than 180° (at least one segment connecting two vertices is outside the polygon). 13. Conclusion: A result of summary of all the work that has been completed. The part of a conditional statement that occurs after the word “then”. 14. Congruent parts: Two or more parts that only have the same measure. In CPCTC, the parts of the congruent triangles are congruent. 15. Congruent triangles: Two triangles are congruent if and only if all of their corresponding parts are congruent. -
Chapter 6: Things to Know
MAT 222 Chapter 6: Things To Know Section 6.1 Polygons Objectives Vocabulary 1. Define and Name Polygons. • polygon 2. Find the Sum of the Measures of the Interior • vertex Angles of a Quadrilateral. • n-gon • concave polygon • convex polygon • quadrilateral • regular polygon • diagonal • equilateral polygon • equiangular polygon Polygon Definition A figure is a polygon if it meets the following three conditions: 1. 2. 3. The endpoints of the sides of a polygon are called the ________________________ (Singular form: _______________ ). Polygons must be named by listing all of the vertices in order. Write two different ways of naming the polygon to the right below: _______________________ , _______________________ Example Identifying Polygons Identify the polygons. If not a polygon, state why. a. b. c. d. e. MAT 222 Chapter 6 Things To Know Number of Sides Name of Polygon 3 4 5 6 7 8 9 10 12 n Definitions In general, a polygon with n sides is called an __________________________. A polygon is __________________________ if no line containing a side contains a point within the interior of the polygon. Otherwise, a polygon is _________________________________. Example Identifying Convex and Concave Polygons. Identify the polygons. If not a polygon, state why. a. b. c. Definition An ________________________________________ is a polygon with all sides congruent. An ________________________________________ is a polygon with all angles congruent. A _________________________________________ is a polygon that is both equilateral and equiangular. MAT 222 Chapter 6 Things To Know Example Identifying Regular Polygons Determine if each polygon is regular or not. Explain your reasoning. a. b. c. Definition A segment joining to nonconsecutive vertices of a convex polygon is called a _______________________________ of the polygon. -
The Graphics Gems Series a Collection of Practical Techniques for the Computer Graphics Programmer
GRAPHICS GEMS IV This is a volume in The Graphics Gems Series A Collection of Practical Techniques for the Computer Graphics Programmer Series Editor Andrew Glassner Xerox Palo Alto Research Center Palo Alto, California GRAPHICS GEMS IV Edited by Paul S. Heckbert Computer Science Department Carnegie Mellon University Pittsburgh, Pennsylvania AP PROFESSIONAL Boston San Diego New York London Sydney Tokyo Toronto This book is printed on acid-free paper © Copyright © 1994 by Academic Press, Inc. All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. AP PROFESSIONAL 955 Massachusetts Avenue, Cambridge, MA 02139 An imprint of ACADEMIC PRESS, INC. A Division of HARCOURT BRACE & COMPANY United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Graphics gems IV / edited by Paul S. Heckbert. p. cm. - (The Graphics gems series) Includes bibliographical references and index. ISBN 0-12-336156-7 (with Macintosh disk). —ISBN 0-12-336155-9 (with IBM disk) 1. Computer graphics. I. Heckbert, Paul S., 1958— II. Title: Graphics gems 4. III. Title: Graphics gems four. IV. Series. T385.G6974 1994 006.6'6-dc20 93-46995 CIP Printed in the United States of America 94 95 96 97 MV 9 8 7 6 5 4 3 2 1 Contents Author Index ix Foreword by Andrew Glassner xi Preface xv About the Cover xvii I. Polygons and Polyhedra 1 1.1. -
Classifying Polygons
Elementary Classifying Polygons Common Core: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. (5.G.B.3) Classify two-dimensional figures in a hierarchy based on properties. (5.G.B.4) Objectives: 1) Students will learn vocabulary related to polygons. 2) Students will use that vocabulary to classify polygons. Materials: Classifying Polygons worksheet Internet access (for looking up definitions) Procedure: 1) Students search the Internet for the definitions and record them on the Classifying Polygons worksheet. 2) Students share and compare their definitions since they may find alternative definitions. 3) Introduce or review prefixes and suffixes. 4) Students fill in the Prefixes section, and share answers. 5) Students classify the polygons found on page 2 of the worksheet. 6) With time remaining, have students explore some extension questions: *Can a polygon be regular and concave? Show or explain your reasoning. *Can a triangle be concave? Show or explain your reasoning. *Could we simplify the definition of regular to just… All sides congruent? or All angles congruent? Show or explain your reasoning. *Can you construct a pentagon with 5 congruent angles but is not considered regular? Show or explain your reasoning. *Can you construct a pentagon with 5 congruent sides but is not considered regular? Show or explain your reasoning. Notes to Teacher: I have my students search for these answers and definitions online, however I am sure that some math textbook glossaries -
PESIT Bangalore South Campus Hosur Road, 1Km Before Electronic City, Bengaluru -100 Department of Computer Science and Engineering
USN 1 P E PESIT Bangalore South Campus Hosur road, 1km before Electronic City, Bengaluru -100 Department of Computer Science and Engineering INTERNAL ASSESSMENT TEST – 2 Solution Date : 03-04-18 Max Marks: 40 Subject & Code : Computer Graphics and Visualization (15CS62) Section: VI CSE A,B,C Name of faculty: Dr.Sarasvathi V / Ms. Evlin Time: 08.30 - 10.00AM Note: Answer FIVE full Questions 1 Explain the scan line polygon fill algorithm with necessary diagram. 8 Determine the intersection positions of the boundaries. For each scanline that crosses the polygon, edge-intersection are sorted from left to right, then pixel positions, b/w including each intersection pair is filled. Solving pair of simultaneous linear equations. Whenever a scan line passes through a vertex, it intersects two polygon edges at that point. Scan line y’- even number of edges – two pair correctly identify interior pixels Scanline y- five edges. Here we must count vertex intersection as one point. For scanline y, the two edges sharing an intersection vertex are on opposite sides of the scan line.For scan line y’, the two intersecting edges are both above the scan line. vertex that has adjoining edges on opposite sides of an intersecting scan line should be counted as just one. Trace around clockwise or counter clockwise and observing the relative changes in y values. VI CSE A, B &C USN 1 P E PESIT Bangalore South Campus Hosur road, 1km before Electronic City, Bengaluru -100 Department of Computer Science and Engineering Adjustment to vertex intersection count shorten some polygon edges to split those vertices that should be counted as one intersection. -
Unit 8 – Geometry QUADRILATERALS
Unit 8 – Geometry QUADRILATERALS NAME _____________________________ Period ____________ 1 Geometry Chapter 8 – Quadrilaterals ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1.____ (8-1) Angles of Polygons – Day 1- Pages 407-408 13-16, 20-22, 27-32, 35-43 odd 2. ____ (8-2) Parallelograms – Day 1- Pages 415 16-31, 37-39 3. ____ (8-3) Test for Parallelograms – Day 1- Pages 421-422 13-23 odd, 25 -31 odd 4. ____ (8-4) Rectangles – Day 1- Pages 428-429 10, 11, 13, 16-26, 30-32, 36 5. ____ (8-5) Rhombi and Squares – Day 1 – Pages 434-435 12-19, 20, 22, 26 - 31 6.____ (8-6) Trapezoids – Day 1– Pages 10, 13-19, 22-25 7. _____ Chapter 8 Review 2 (Reminder!) A little background… Polygon is the generic term for _____________________________________________. Depending on the number, the first part of the word - “Poly” - is replaced by a prefix. The prefix used is from Greek. The Greek term for 5 is Penta, so a 5-sided figure is called a _____________. We can draw figures with as many sides as we want, but most of us don’t remember all that Greek, so when the number is over 12, or if we are talking about a general polygon, many mathematicians call the figure an “n-gon.” So a figure with 46 sides would be called a “46-gon.” Vocabulary – Types of Polygons Regular - ______________________________________________________________ _____________________________________________________________________ Irregular – _____________________________________________________________ _____________________________________________________________________ Equiangular - ___________________________________________________________ Equilateral - ____________________________________________________________ Convex - a straight line drawn through a convex polygon crosses at most two sides . -
Exploring Topics of the Art Gallery Problem
The College of Wooster Open Works Senior Independent Study Theses 2019 Exploring Topics of the Art Gallery Problem Megan Vuich The College of Wooster, [email protected] Follow this and additional works at: https://openworks.wooster.edu/independentstudy Recommended Citation Vuich, Megan, "Exploring Topics of the Art Gallery Problem" (2019). Senior Independent Study Theses. Paper 8534. This Senior Independent Study Thesis Exemplar is brought to you by Open Works, a service of The College of Wooster Libraries. It has been accepted for inclusion in Senior Independent Study Theses by an authorized administrator of Open Works. For more information, please contact [email protected]. © Copyright 2019 Megan Vuich Exploring Topics of the Art Gallery Problem Independent Study Thesis Presented in Partial Fulfillment of the Requirements for the Degree Bachelor of Arts in the Department of Mathematics and Computer Science at The College of Wooster by Megan Vuich The College of Wooster 2019 Advised by: Dr. Robert Kelvey Abstract Created in the 1970’s, the Art Gallery Problem seeks to answer the question of how many security guards are necessary to fully survey the floor plan of any building. These floor plans are modeled by polygons, with guards represented by points inside these shapes. Shortly after the creation of the problem, it was theorized that for guards whose positions were limited to the polygon’s j n k vertices, 3 guards are sufficient to watch any type of polygon, where n is the number of the polygon’s vertices. Two proofs accompanied this theorem, drawing from concepts of computational geometry and graph theory. -
Solutions Key 6 Polygons and Quadrilaterals ARE YOU READY? PAGE 377 19
CHAPTER Solutions Key 6 Polygons and Quadrilaterals ARE YOU READY? PAGE 377 19. F (counterexample: with side lengths 5, 6, 10); if a triangle is an acute triangle, then it is a scalene 1. F 2. B triangle; F (counterexample: any equilateral 3. A 4. D triangle). 5. E 6-1 PROPERTIES AND ATTRIBUTES OF 6. Use Sum Thm. POLYGONS, PAGES 382–388 x ° + 42° + 32° = 180° x ° = 180° - 42° - 32° CHECK IT OUT! x ° = 106° 1a. not a polygon b. polygon, nonagon 7. Use Sum Thm. x ° + 53° + 90° = 180° c. not a polygon x ° = 180° - 53° - 90° 2a. regular, convex b. irregular, concave x ° = 37° 3a. Think: Use Polygon ∠ Sum Thm. 8. Use Sum Thm. (n - 2)180° x ° + x ° + 32° = 180° (15 - 2)180° 2 x ° = 180° - 34° 2340° 2 x ° = 146° b. (10 - 2)180° = 1440° x ° = 73° m∠1 + m∠2 + … + m∠10 = 1440° 9. Use Sum Thm. 10m∠1 = 1440° 2 x ° + x ° + 57° = 180° m∠1 = 144° 3 x ° = 180° - 57° 4a. Think: Use Polygon Ext. ∠ Sum Thm. 3 x ° = 123° m∠1 + m∠2 + … + m∠12 = 360° x ° = 41° 12m∠1 = 360° 10. By Lin. Pair Thm., 11. By Alt. Ext. Thm., m∠1 = 30° m∠1 + 56 = 180 m∠2 = 101° b. 4r + 7r + 5r + 8r = 360 m∠1 = 124° By Lin. Pair Thm., 24r = 360 By Vert. Thm., m∠1 + m∠2 = 180 r = 15 m∠2 = 56° m∠1 + 101 = 180 By Corr. Post., m∠1 = 79° 5. By Polygon Ext. ∠ Sum Thm., sum of ext. ∠ m∠3 = m∠1 = 124° Since ⊥ m, m n measures is 360°. -
Angles of Polygons
5.3 Angles of Polygons How can you fi nd a formula for the sum of STATES the angle measures of any polygon? STANDARDS MA.8.G.2.3 1 ACTIVITY: The Sum of the Angle Measures of a Polygon Work with a partner. Find the sum of the angle measures of each polygon with n sides. a. Sample: Quadrilateral: n = 4 A Draw a line that divides the quadrilateral into two triangles. B Because the sum of the angle F measures of each triangle is 180°, the sum of the angle measures of the quadrilateral is 360°. C D E (A + B + C ) + (D + E + F ) = 180° + 180° = 360° b. Pentagon: n = 5 c. Hexagon: n = 6 d. Heptagon: n = 7 e. Octagon: n = 8 196 Chapter 5 Angles and Similarity 2 ACTIVITY: The Sum of the Angle Measures of a Polygon Work with a partner. a. Use the table to organize your results from Activity 1. Sides, n 345678 Angle Sum, S b. Plot the points in the table in a S coordinate plane. 1080 900 c. Write a linear equation that relates S to n. 720 d. What is the domain of the function? 540 Explain your reasoning. 360 180 e. Use the function to fi nd the sum of 1 2 3 4 5 6 7 8 n the angle measures of a polygon −180 with 10 sides. −360 3 ACTIVITY: The Sum of the Angle Measures of a Polygon Work with a partner. A polygon is convex if the line segment connecting any two vertices lies entirely inside Convex the polygon. -
AN EXPOSITION of MONSKY's THEOREM Contents 1. Introduction
AN EXPOSITION OF MONSKY'S THEOREM WILLIAM SABLAN Abstract. Since the 1970s, the problem of dividing a polygon into triangles of equal area has been a surprisingly difficult yet rich field of study. This paper gives an exposition of some of the combinatorial and number theoretic ideas used in this field. Specifically, this paper will examine how these methods are used to prove Monsky's theorem which states only an even number of triangles of equal area can divide a square. Contents 1. Introduction 1 2. The p-adic Valuation and Absolute Value 2 3. Sperner's Lemma 3 4. Proof of Monsky's Theorem 4 Acknowledgments 7 References 7 1. Introduction If one tried to divide a square into triangles of equal area, one would see in Figure 1 that an even number of such triangles would work, but could an odd number work? Fred Richman [3], a professor at New Mexico State, considered using this question in an exam for graduate students in 1965. After being unable to find any reference or answer to this question, he decided to ask it in the American Mathematical Monthly. n Figure 1. Dissection of squares into an even number of triangles After 5 years, Paul Monsky [4] finally found an answer when he proved that only an even number of triangles of equal area could divide a square, which later came to be known as Monsky's theorem. Formally, the theorem states: Theorem 1.1 (Monsky [4]). Let S be a square in the plane. If a triangulation of S into m triangles of equal area is given, then m is even. -
CSE 5319-001 (Computational Geometry) SYLLABUS
CSE 5319-001 (Computational Geometry) SYLLABUS Spring 2012: TR 11:00-12:20, ERB 129 Instructor: Bob Weems, Associate Professor, http://ranger.uta.edu/~weems Office: 627 ERB, 817/272-2337 ([email protected]) Hours: TR 12:30-1:50 PM and by appointment (please email by 8:30 AM) Prerequisite: Advanced Algorithms (CSE 5311) Objective: Ability to apply and expand geometric techniques in computing. Outcomes: 1. Exposure to algorithms and data structures for geometric problems. 2. Exposure to techniques for addressing degenerate cases. 3. Exposure to randomization as a tool for developing geometric algorithms. 4. Experience using CGAL with C++/STL. Textbooks: M. de Berg et.al., Computational Geometry: Algorithms and Applications, 3rd ed., Springer-Verlag, 2000. https://libproxy.uta.edu/login?url=http://www.springerlink.com/content/k18243 S.L. Devadoss and J. O’Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. References: Adobe Systems Inc., PostScript Language Tutorial and Cookbook, Addison-Wesley, 1985. (http://Www-cdf.fnal.gov/offline/PostScript/BLUEBOOK.PDF) B. Casselman, Mathematical Illustrations: A Manual of Geometry and PostScript, Springer-Verlag, 2005. (http://www.math.ubc.ca/~cass/graphics/manual) CGAL User and Reference Manual (http://www.cgal.org/Manual) T. Cormen, et.al., Introduction to Algorithms, 3rd ed., MIT Press, 2009. E.D. Demaine and J. O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, 2007. (occasionally) J. O’Rourke, Art Gallery Theorems and Algorithms, Oxford Univ. Press, 1987. (http://maven.smith.edu/~orourke/books/ArtGalleryTheorems/art.html, occasionally) J. O’Rourke, Computational Geometry in C, 2nd ed., Cambridge Univ. -
Optimal Higher Order Delaunay Triangulations of Polygons*
Optimal Higher Order Delaunay Triangulations of Polygons Rodrigo I. Silveira and Marc van Kreveld Department of Information and Computing Sciences Utrecht University, 3508 TB Utrecht, The Netherlands {rodrigo,marc}@cs.uu.nl Abstract. This paper presents an algorithm to triangulate polygons optimally using order-k Delaunay triangulations, for a number of qual- ity measures. The algorithm uses properties of higher order Delaunay triangulations to improve the O(n3) running time required for normal triangulations to O(k2n log k + kn log n) expected time, where n is the number of vertices of the polygon. An extension to polygons with points inside is also presented, allowing to compute an optimal triangulation of a polygon with h ≥ 1 components inside in O(kn log n)+O(k)h+2n expected time. Furthermore, through experimental results we show that, in practice, it can be used to triangulate point sets optimally for small values of k. This represents the first practical result on optimization of higher order Delaunay triangulations for k>1. 1 Introduction One of the best studied topics in computational geometry is the triangulation. When the input is a point set P , it is defined as a subdivision of the plane whose bounded faces are triangles and whose vertices are the points of P .When the input is a polygon, the goal is to decompose it into triangles by drawing diagonals. Triangulations have applications in a large number of fields, including com- puter graphics, multivariate analysis, mesh generation, and terrain modeling. Since for a given point set or polygon, many triangulations exist, it is possible to try to find one that is the best according to some criterion that measures some property of the triangulation.