Taneli Luotoniemi H Y P E R Sp a C Ial Inte R L a C E Ta N E L I L Uo T O N Ie M I
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CS 543: Computer Graphics Lecture 7 (Part I): Projection Emmanuel
CS 543: Computer Graphics Lecture 7 (Part I): Projection Emmanuel Agu 3D Viewing and View Volume n Recall: 3D viewing set up Projection Transformation n View volume can have different shapes (different looks) n Different types of projection: parallel, perspective, orthographic, etc n Important to control n Projection type: perspective or orthographic, etc. n Field of view and image aspect ratio n Near and far clipping planes Perspective Projection n Similar to real world n Characterized by object foreshortening n Objects appear larger if they are closer to camera n Need: n Projection center n Projection plane n Projection: Connecting the object to the projection center camera projection plane Projection? Projectors Object in 3 space Projected image VRP COP Orthographic Projection n No foreshortening effect – distance from camera does not matter n The projection center is at infinite n Projection calculation – just drop z coordinates Field of View n Determine how much of the world is taken into the picture n Larger field of view = smaller object projection size center of projection field of view (view angle) y y z q z x Near and Far Clipping Planes n Only objects between near and far planes are drawn n Near plane + far plane + field of view = Viewing Frustum Near plane Far plane y z x Viewing Frustrum n 3D counterpart of 2D world clip window n Objects outside the frustum are clipped Near plane Far plane y z x Viewing Frustum Projection Transformation n In OpenGL: n Set the matrix mode to GL_PROJECTION n Perspective projection: use • gluPerspective(fovy, -
Viewing in 3D
Viewing in 3D Viewing in 3D Foley & Van Dam, Chapter 6 • Transformation Pipeline • Viewing Plane • Viewing Coordinate System • Projections • Orthographic • Perspective OpenGL Transformation Pipeline Viewing Coordinate System Homogeneous coordinates in World System zw world yw ModelViewModelView Matrix Matrix xw Tractor Viewing System Viewer Coordinates System ProjectionProjection Matrix Matrix Clip y Coordinates v Front- xv ClippingClipping Wheel System P0 zv ViewportViewport Transformation Transformation ne pla ing Window Coordinates View Specifying the Viewing Coordinates Specifying the Viewing Coordinates • Viewing Coordinates system, [xv, yv, zv], describes 3D objects with respect to a viewer zw y v P v xv •A viewing plane (projection plane) is set up N P0 zv perpendicular to zv and aligned with (xv,yv) yw xw ne pla ing • In order to specify a viewing plane we have View to specify: •P0=(x0,y0,z0) is the point where a camera is located •a vector N normal to the plane • P is a point to look-at •N=(P-P)/|P -P| is the view-plane normal vector •a viewing-up vector V 0 0 •V=zw is the view up vector, whose projection onto • a point on the viewing plane the view-plane is directed up Viewing Coordinate System Projections V u N z N ; x ; y z u x • Viewing 3D objects on a 2D display requires a v v V u N v v v mapping from 3D to 2D • The transformation M, from world-coordinate into viewing-coordinates is: • A projection is formed by the intersection of certain lines (projectors) with the view plane 1 2 3 ª x v x v x v 0 º ª 1 0 0 x 0 º « » « -
Combinatorics: the Fine Art of Counting
Combinatorics: The Fine Art of Counting Week Three Solutions Note: in these notes multiplication is assumed to take precedence over division, so 4!/2!2! = 4!/(2!*2!), and binomial coefficients are written horizontally: (4 2) denotes “4 choose 2” or 4!/2!2! Appetizers 1. How many 5 letter words have at least one double letter, i.e. two consecutive letters that are the same? 265 – 26*254 = 1,725,126 2. A Venn diagram is drawn using three circular regions of radius 1 with their centers all distance 1 from each other. What is the area of the intersection of all three regions? What is the area of their union? Let A, B, and C be the three circular regions. A∪B∪C can be seen to be equal to the union of three 60° sectors of radius 1, call them X, Y, and Z. The intersection of any 2 or 3 of these is an equilateral triangle with side length 1. Applying PIE, , we have |XUYUZ| = |X| + |Y| + |Z| - (|X∪Y| + |Y∪Z| + |Z∪X|) + |X∪Y∪Z| which is equal to 3*3*π/6 – 3*√3/4 + √3/4 = π/2 - √3/2. The intersection of any two of A, B, or C has area 2π/3 - √3/2, as shown in class. Applying PIE, |AUBUC| = |A| + |B| + |C| - (|A∪B| + |B∪C| + |C∪A|) + |A∪B∪B| equal to 3π - 3*(2π/3 - √3/2) + (π/2 - √3/2) = 3π/2 + √3 3. A diagonal of polygon is any line segment between vertices which is not an edge of the polygon. -
Implementation of Projections
CS488 Implementation of projections Luc RENAMBOT 1 3D Graphics • Convert a set of polygons in a 3D world into an image on a 2D screen • After theoretical view • Implementation 2 Transformations P(X,Y,Z) 3D Object Coordinates Modeling Transformation 3D World Coordinates Viewing Transformation 3D Camera Coordinates Projection Transformation 2D Screen Coordinates Window-to-Viewport Transformation 2D Image Coordinates P’(X’,Y’) 3 3D Rendering Pipeline 3D Geometric Primitives Modeling Transform into 3D world coordinate system Transformation Lighting Illuminate according to lighting and reflectance Viewing Transform into 3D camera coordinate system Transformation Projection Transform into 2D camera coordinate system Transformation Clipping Clip primitives outside camera’s view Scan Draw pixels (including texturing, hidden surface, etc.) Conversion Image 4 Orthographic Projection 5 Perspective Projection B F 6 Viewing Reference Coordinate system 7 Projection Reference Point Projection Reference Point (PRP) Center of Window (CW) View Reference Point (VRP) View-Plane Normal (VPN) 8 Implementation • Lots of Matrices • Orthographic matrix • Perspective matrix • 3D World → Normalize to the canonical view volume → Clip against canonical view volume → Project onto projection plane → Translate into viewport 9 Canonical View Volumes • Used because easy to clip against and calculate intersections • Strategies: convert view volumes into “easy” canonical view volumes • Transformations called Npar and Nper 10 Parallel Canonical Volume X or Y Defined by 6 planes -
Almost-Magic Stars a Magic Pentagram (5-Pointed Star), We Now Know, Must Have 5 Lines Summing to an Equal Value
MAGIC SQUARE LEXICON: ILLUSTRATED 192 224 97 1 33 65 256 160 96 64 129 225 193 161 32 128 2 98 223 191 8 104 217 185 190 222 99 3 159 255 66 34 153 249 72 40 35 67 254 158 226 130 63 95 232 136 57 89 94 62 131 227 127 31 162 194 121 25 168 200 195 163 30 126 4 100 221 189 9 105 216 184 10 106 215 183 11 107 214 182 188 220 101 5 157 253 68 36 152 248 73 41 151 247 74 42 150 246 75 43 37 69 252 156 228 132 61 93 233 137 56 88 234 138 55 87 235 139 54 86 92 60 133 229 125 29 164 196 120 24 169 201 119 23 170 202 118 22 171 203 197 165 28 124 6 102 219 187 12 108 213 181 13 109 212 180 14 110 211 179 15 111 210 178 16 112 209 177 186 218 103 7 155 251 70 38 149 245 76 44 148 244 77 45 147 243 78 46 146 242 79 47 145 241 80 48 39 71 250 154 230 134 59 91 236 140 53 85 237 141 52 84 238 142 51 83 239 143 50 82 240 144 49 81 90 58 135 231 123 27 166 198 117 21 172 204 116 20 173 205 115 19 174 206 114 18 175 207 113 17 176 208 199 167 26 122 All rows, columns, and 14 main diagonals sum correctly in proportion to length M AGIC SQUAR E LEX ICON : Illustrated 1 1 4 8 512 4 18 11 20 12 1024 16 1 1 24 21 1 9 10 2 128 256 32 2048 7 3 13 23 19 1 1 4096 64 1 64 4096 16 17 25 5 2 14 28 81 1 1 1 14 6 15 8 22 2 32 52 69 2048 256 128 2 57 26 40 36 77 10 1 1 1 65 7 51 16 1024 22 39 62 512 8 473 18 32 6 47 70 44 58 21 48 71 4 59 45 19 74 67 16 3 33 53 1 27 41 55 8 29 49 79 66 15 10 15 37 63 23 27 78 11 34 9 2 61 24 38 14 23 25 12 35 76 8 26 20 50 64 9 22 12 3 13 43 60 31 75 17 7 21 72 5 46 11 16 5 4 42 56 25 24 17 80 13 30 20 18 1 54 68 6 19 H. -
Victorian Network
Victorian Network Volume 7, Number 1 Summer 2016 Victorian Brain © Victorian Network Volume 7, Number 1 Summer 2016 www.victoriannetwork.org Guest Editor Sally Shuttleworth General Editor Sophie Duncan Founding Editor Katharina Boehm Editorial Board Megan Anderluh Sarah Crofton Rosalyn Gregory Tammy Ho Lai-Ming Sarah Hook Alison Moulds Heidi Weig Victorian Network is funded by the Arts & Humanities Research Council and supported by King’s College London. Victorian Network Volume 7, Number 1 (Summer 2016) TABLE OF CONTENTS GUEST EDITOR’S INTRODUCTION: VICTORIAN BRAIN 1 Sally Shuttleworth ARTICLES Lucid Daydreaming: Experience and Pathology in Charlotte Brontë 12 Timothy Gao Two Brains and a Tree: Defining the Material Bases 36 for Delusion and Reality in the Woodlanders Anna West ‘The Apotheosis of Voice’: Mesmerism as Mechanisation 61 in George Du Maurier’s Trilby Kristie A. Schlauraff Female Transcendence: Charles Howard Hinton 83 and Hyperspace Fiction Patricia Beesley The Hand and the Mind, the Man and the Monster 107 Kimberly Cox BOOK REVIEWS A Cultural History of the Senses in the Age of Empire, 137 Vol. 5, ed. Constance Classen (Bloomsbury, 2014) Ian Middlebrook Popular Fiction and Brain Science in the Late Nineteenth Century, 142 by Anne Stiles (Cambridge, 2011) Arden Hegele Thomas Hardy’s Brains: Psychology, Neurology, and Hardy’s Imagination, 148 by Suzanne Keen (Ohio State, 2014) Nicole Lobdell Victorian Network Volume 7, Number 1 (Summer 2016) The Poet’s Mind: The Psychology of Victorian Poetry 1830-1870, 153 by Gregory Tate (Oxford, 2012) Benjamin Westwood Theatre and Evolution from Ibsen to Beckett, 158 by Kirsten Shepherd-Barr (Columbia, 2015) Katharina Herold Victorian Network Volume 7, Number 1 (Summer 2016) Sally Shuttleworth 1 VICTORIAN BRAIN SALLY SHUTTLEWORTH, PROFESSOR OF ENGLISH (UNIVERSITY OF OXFORD) In April 1878 the first issue of Brain: A Journal of Neurology was published. -
THE ZEN of MAGIC SQUARES, CIRCLES, and STARS Also by Clifford A
THE ZEN OF MAGIC SQUARES, CIRCLES, AND STARS Also by Clifford A. Pickover The Alien IQ Test Black Holes: A Traveler’s Guide Chaos and Fractals Chaos in Wonderland Computers and the Imagination Computers, Pattern, Chaos, and Beauty Cryptorunes Dreaming the Future Fractal Horizons: The Future Use of Fractals Frontiers of Scientific Visualization (with Stuart Tewksbury) Future Health: Computers and Medicine in the 21st Century The Girl Who Gave Birth t o Rabbits Keys t o Infinity The Loom of God Mazes for the Mind: Computers and the Unexpected The Paradox of God and the Science of Omniscience The Pattern Book: Fractals, Art, and Nature The Science of Aliens Spider Legs (with Piers Anthony) Spiral Symmetry (with Istvan Hargittai) The Stars of Heaven Strange Brains and Genius Surfing Through Hyperspace Time: A Traveler’s Guide Visions of the Future Visualizing Biological Information Wonders of Numbers THE ZEN OF MAGIC SQUARES, CIRCLES, AND STARS An Exhibition of Surprising Structures across Dimensions Clifford A. Pickover Princeton University Press Princeton and Oxford Copyright © 2002 by Clifford A. Pickover Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data Pickover, Clifford A. The zen of magic squares, circles, and stars : an exhibition of surprising structures across dimensions / Clifford A. Pickover. p. cm Includes bibliographical references and index. ISBN 0-691-07041-5 (acid-free paper) 1. Magic squares. 2. Mathematical recreations. I. Title. QA165.P53 2002 511'.64—dc21 2001027848 British Library Cataloging-in-Publication Data is available This book has been composed in Baskerville BE and Gill Sans. -
Flatterland: the Play Based on Flatterland: Like Flatland Only More So
Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2012 Flatterland: The lP ay Kym Louie Harvey Mudd College Recommended Citation Louie, Kym, "Flatterland: The lP ay" (2012). HMC Senior Theses. 27. https://scholarship.claremont.edu/hmc_theses/27 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Flatterland: The Play based on Flatterland: Like Flatland Only More So by Ian Stewart Kym Louie Arthur Benjamin, Advisor Art Horowitz, Advisor Thomas Leabhart, Reader May, 2012 Department of Mathematics Copyright c 2012 Kym Louie. The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced ma- terials and notice is given that the copying is by permission of the author. To dis- seminate otherwise or to republish requires written permission from the author. Abstract This script is an adaptation of the popular science novel Flatterland: Like Flatland, Only More So by Ian Stewart. It breathes new life into mathemat- ical ideas and topics. By bringing math to the stage, this script presents concepts in a more friendly and accessible manner. This play is intended to generate new interest in and expose new topics to an audience of non- mathematicians. Preface I was first introduced to Flatterland by Sue Buckwalter while I was at Phillips Academy. -
Arxiv:1705.01294V1
Branes and Polytopes Luca Romano email address: [email protected] ABSTRACT We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the vertices of the k21, 2k1 and 1k2 families of uniform polytopes, respectively, while 3-branes correspond to the vertices of the rectified version of the 2k1 family. For 4-branes and higher rank solutions we find a general behavior. The interpretation of half- supersymmetric solutions as vertices of uniform polytopes reveals some intriguing aspects. One of the most relevant is a triality relation between 0-, 1- and 2-branes. arXiv:1705.01294v1 [hep-th] 3 May 2017 Contents Introduction 2 1 Coxeter Group and Weyl Group 3 1.1 WeylGroup........................................ 6 2 Branes in E11 7 3 Algebraic Structures Behind Half-Supersymmetric Branes 12 4 Branes ad Polytopes 15 Conclusions 27 A Polytopes 30 B Petrie Polygons 30 1 Introduction Since their discovery branes gained a prominent role in the analysis of M-theories and du- alities [1]. One of the most important class of branes consists in Dirichlet branes, or D-branes. D-branes appear in string theory as boundary terms for open strings with mixed Dirichlet-Neumann boundary conditions and, due to their tension, scaling with a negative power of the string cou- pling constant, they are non-perturbative objects [2]. -
15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P . -
A Non-Linear Take on the Flatland and Sphereland Films
MEDIA COLUMN Over-Sphere in the fourth dimension is voiced by Kate Mulgrew, who played Captain Janeway on Star Trek: Voyager, In addition to longer reviews for the Media Column, we invite and the minor character Captain Aero is voiced by Danica you to watch for and submit short snippets of instances of women McKellar, who studied mathematics at UCLA and is a force in mathematics in the media (WIMM Watch). Please submit to for encouraging girls to like and do math. In this day and the Media Column Editors: Sarah J. Greenwald, Appalachian age of excessive narcissism, I especially appreciated Tony State University, [email protected] and Alice Silverberg, Hale’s rendition of the King of Pointland; in a bonus inter- University of California, Irvine, [email protected]. view, Hale refers to this character as “just a sad emasculated point.” It’s great that such prominent celebrities were A Non-Linear Take on the enthusiastic about giving their all for some short animated educational math films. Flatland and Sphereland Class stratification was a major theme of Flatland. Films York reminds us in an actor interview that Abbott anticipated a number of political and sociological trends. Circles were Alice Silverberg superior. As stated in the Flatland film, “The more sides you have, the greater your angles, so the smarter you are.” Irregular I enjoyed the delightful 35 minute animated film polygons were viewed as monsters who could be “cured” by versions of the classic Flatland and its lesser known quasi- surgery, or executed. sequel Sphereland. They were released directly to DVD in Abbott’s Flatland society was blatantly sexist, though 2007 and 2012, respectively. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.