A MATHEMATICAL SPACE ODYSSEY Solid Geometry in the 21St Century
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A Parallelepiped Based Approach
3D Modelling Using Geometric Constraints: A Parallelepiped Based Approach Marta Wilczkowiak, Edmond Boyer, and Peter Sturm MOVI–GRAVIR–INRIA Rhˆone-Alpes, 38330 Montbonnot, France, [email protected], http://www.inrialpes.fr/movi/people/Surname Abstract. In this paper, efficient and generic tools for calibration and 3D reconstruction are presented. These tools exploit geometric con- straints frequently present in man-made environments and allow cam- era calibration as well as scene structure to be estimated with a small amount of user interactions and little a priori knowledge. The proposed approach is based on primitives that naturally characterize rigidity con- straints: parallelepipeds. It has been shown previously that the intrinsic metric characteristics of a parallelepiped are dual to the intrinsic charac- teristics of a perspective camera. Here, we generalize this idea by taking into account additional redundancies between multiple images of multiple parallelepipeds. We propose a method for the estimation of camera and scene parameters that bears strongsimilarities with some self-calibration approaches. Takinginto account prior knowledgeon scene primitives or cameras, leads to simpler equations than for standard self-calibration, and is expected to improve results, as well as to allow structure and mo- tion recovery in situations that are otherwise under-constrained. These principles are illustrated by experimental calibration results and several reconstructions from uncalibrated images. 1 Introduction This paper is about using partial information on camera parameters and scene structure, to simplify and enhance structure from motion and (self-) calibration. We are especially interested in reconstructing man-made environments for which constraints on the scene structure are usually easy to provide. -
Area, Volume and Surface Area
The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 YEARS 810 Area, Volume and Surface Area (Measurement and Geometry: Module 11) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 810 {4} A guide for teachers AREA, VOLUME AND SURFACE AREA ASSUMED KNOWLEDGE • Knowledge of the areas of rectangles, triangles, circles and composite figures. • The definitions of a parallelogram and a rhombus. • Familiarity with the basic properties of parallel lines. • Familiarity with the volume of a rectangular prism. • Basic knowledge of congruence and similarity. • Since some formulas will be involved, the students will need some experience with substitution and also with the distributive law. -
Geometry Around Us an Introduction to Non-Euclidean Geometry
Geometry Around Us An Introduction to Non-Euclidean Geometry Gaurish Korpal (gaurish4math.wordpress.com) National Institute of Science Education and Research, Bhubaneswar 23 April 2016 Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 1 / 15 Theorem In any triangle, trisector lines intersect at three points, that are vertices of an equilateral triangle. A. Bogomolny, Morley's Theorem: Proof by R. J. Webster, Interactive Mathematics Miscellany and Puzzle, http://www.cut-the-knot.org/triangle/Morley/Webster.shtml Morley's Miracle Frank Morley (USA) discovered a theorem about triangle in 1899, approximately 2000 years after first theorems about triangle were published by Euclid. Euclid (Egypt) is referred as father of Plane Geometry, often called Euclidean Geometry. Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 2 / 15 Morley's Miracle Frank Morley (USA) discovered a theorem about triangle in 1899, approximately 2000 years after first theorems about triangle were published by Euclid. Euclid (Egypt) is referred as father of Plane Geometry, often called Euclidean Geometry. Theorem In any triangle, trisector lines intersect at three points, that are vertices of an equilateral triangle. A. Bogomolny, Morley's Theorem: Proof by R. J. Webster, Interactive Mathematics Miscellany and Puzzle, http://www.cut-the-knot.org/triangle/Morley/Webster.shtml Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 2 / 15 Morley's original proof was (published in 1924) stemmed from his results on algebraic curves tangent to a given number of lines. It arose from the consideration of cardioids [(x 2 + y 2 − a2)2 = 4a2((x − a)2 + y 2)]. -
Fibonacci Number
Fibonacci number From Wikipedia, the free encyclopedia • Have questions? Find out how to ask questions and get answers. • • Learn more about citing Wikipedia • Jump to: navigation, search A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, … , are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, ... (Sometimes this sequence is considered to start at F1 = 1, but in this article it is regarded as beginning with F0=0.) The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India. [1] [2] • [edit] Origins The Fibonacci numbers first appeared, under the name mātrāmeru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandah-shāstra, the Art of Prosody, 450 or 200 BC). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well-known text on these. -
Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
What Lies Between Rigidity and Flexibility of Structures
S A J _ 2011 _ 3 _ original scientific article approval date 03 05 2011 UDK BROJEVI: 514.113.5 ; 614.073.5.042 ID BROJ: 184974860 WHAT LIES BETWEEN RIGIDITY AND FLEXIBILITY OF STRUCTURES A B S T R A C T The borderline between continuous flexibility and rigidity of structures like polyhedra or frameworks is not strict. There can be different levels of infinitesimal flexibility. This article presents the mathematical background and some examples of structures which under particular conditions are flexible or almost flexible and otherwise rigid. Hellmuth Stachel 102 KEY WORDS Vienna University of Technology - Institute of Discrete Mathematics and Geometry RIGIDITY FLEXIBILITY INFINITESIMAL FLEXIBILITY FLEXIBLE POLYHEDRA SNAPPING POLYHEDRA KOKOTSAKIS MESHES ORIGAMI MECHANISMS S A J _ 2011 _ 3 _ INTRODUCTION A framework or a polyhedron will be called “rigid” when the edge lengths determine its planar or spatial shape uniquely; under the term “shape” we mean its spatial form – apart from movements in space. More generally and under inclusion of smooth or piecewise linear surfaces, a structure is called rigid when its intrinsic metric defines its spatial shape uniquely. In this sense, the intrinsic metric of a polyhedron is defined by its net (unfolding), i.e., the coplanar set of faces with identified pairs of edges originating from the same edge of the spatial form. After cutting out this net from paper or cardboard, a paper- or cardboard-model of this polyhedron can be built in the usual way. Does such a net really define the shape of a polyhedron uniquely? Think of a cube where one face is replaced by a four-sided pyramid with a small height. -
A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems*
A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems* Petr Timokhin 1[0000-0002-0718-1436], Mikhail Mikhaylyuk2[0000-0002-7793-080X], and Klim Panteley3[0000-0001-9281-2396] Federal State Institution «Scientific Research Institute for System Analysis of the Russian Academy of Sciences», Moscow, Russia 1 [email protected], 2 [email protected], 3 [email protected] Abstract. The paper proposes a new technology of creating panoramic video with a 360-degree view based on virtual environment projection on regular do- decahedron. The key idea consists in constructing of inner dodecahedron surface (observed by the viewer) composed of virtual environment snapshots obtained by twelve identical virtual cameras. A method to calculate such cameras’ projec- tion and orientation parameters based on “golden rectangles” geometry as well as a method to calculate snapshots position around the observer ensuring synthe- sis of continuous 360-panorama are developed. The technology and the methods were implemented in software complex and tested on the task of virtual observing the Earth from space. The research findings can be applied in virtual environment systems, video simulators, scientific visualization, virtual laboratories, etc. Keywords: Visualization, Virtual Environment, Regular Dodecahedron, Video 360, Panoramic Mapping, GPU. 1 Introduction In modern human activities the importance of the visualization of virtual prototypes of real objects and phenomena is increasingly grown. In particular, it’s especially required for training specialists in space and aviation industries, medicine, nuclear industry and other areas, where the mistake cost is extremely high [1-3]. The effectiveness of work- ing with virtual prototypes is significantly increased if an observer experiences a feeling of being present in virtual environment. -
Cones, Pyramids and Spheres
The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 12 CONES, PYRAMIDS AND SPHERES A guide for teachers - Years 9–10 June 2011 YEARS 910 Cones, Pyramids and Spheres (Measurement and Geometry : Module 12) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. 2011. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 12 CONES, PYRAMIDS AND SPHERES A guide for teachers - Years 9–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 910 {4} A guide for teachers CONES, PYRAMIDS AND SPHERES ASSUMED KNOWLEDGE • Familiarity with calculating the areas of the standard plane figures including circles. • Familiarity with calculating the volume of a prism and a cylinder. • Familiarity with calculating the surface area of a prism. • Facility with visualizing and sketching simple three‑dimensional shapes. • Facility with using Pythagoras’ theorem. • Facility with rounding numbers to a given number of decimal places or significant figures. -
Thesis Final Copy V11
“VIENS A LA MAISON" MOROCCAN HOSPITALITY, A CONTEMPORARY VIEW by Anita Schwartz A Thesis Submitted to the Faculty of The Dorothy F. Schmidt College of Arts & Letters in Partial Fulfillment of the Requirements for the Degree of Master of Art in Teaching Art Florida Atlantic University Boca Raton, Florida May 2011 "VIENS A LA MAlSO " MOROCCAN HOSPITALITY, A CONTEMPORARY VIEW by Anita Schwartz This thesis was prepared under the direction of the candidate's thesis advisor, Angela Dieosola, Department of Visual Arts and Art History, and has been approved by the members of her supervisory committee. It was submitted to the faculty ofthc Dorothy F. Schmidt College of Arts and Letters and was accepted in partial fulfillment of the requirements for the degree ofMaster ofArts in Teaching Art. SUPERVISORY COMMIITEE: • ~~ Angela Dicosola, M.F.A. Thesis Advisor 13nw..Le~ Bonnie Seeman, M.F.A. !lu.oa.twJ4..,;" ffi.wrv Susannah Louise Brown, Ph.D. Linda Johnson, M.F.A. Chair, Department of Visual Arts and Art History .-dJh; -ZLQ_~ Manjunath Pendakur, Ph.D. Dean, Dorothy F. Schmidt College ofArts & Letters 4"jz.v" 'ZP// Date Dean. Graduate Collcj;Ze ii ACKNOWLEDGEMENTS I would like to thank the members of my committee, Professor John McCoy, Dr. Susannah Louise Brown, Professor Bonnie Seeman, and a special thanks to my committee chair, Professor Angela Dicosola. Your tireless support and wise counsel was invaluable in the realization of this thesis documentation. Thank you for your guidance, inspiration, motivation, support, and friendship throughout this process. To Karen Feller, Dr. Stephen E. Thompson, Helena Levine and my colleagues at Donna Klein Jewish Academy High School for providing support, encouragement and for always inspiring me to be the best art teacher I could be. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
Analytic Geometry
STATISTIC ANALYTIC GEOMETRY SESSION 3 STATISTIC SESSION 3 Session 3 Analytic Geometry Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres Point, Line, Plane and Solid A Point has no dimensions, only position A Line is one-dimensional A Plane is two dimensional (2D) A Solid is three-dimensional (3D) Plane Geometry Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper). 2D Shapes Activity: Sorting Shapes Triangles Right Angled Triangles Interactive Triangles Quadrilaterals (Rhombus, Parallelogram, etc) Rectangle, Rhombus, Square, Parallelogram, Trapezoid and Kite Interactive Quadrilaterals Shapes Freeplay Perimeter Area Area of Plane Shapes Area Calculation Tool Area of Polygon by Drawing Activity: Garden Area General Drawing Tool Polygons A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons. Here are some more: Pentagon Pentagra m Hexagon Properties of Regular Polygons Diagonals of Polygons Interactive Polygons The Circle Circle Pi Circle Sector and Segment Circle Area by Sectors Annulus Activity: Dropping a Coin onto a Grid Circle Theorems (Advanced Topic) Symbols There are many special symbols used in Geometry. Here is a short reference for you: -
Solids, Nets, and Cross Sections
SOLIDS, NETS, AND CROSS SECTIONS Polyhedra: Exercises: 1. The figure below shows a tetrahedron with the faces ∆ABD,,∆∆ ABC ACD and ∆ BCD . A The edges are AB,,,, AC AD DB BC and DC . The vertices are A, B, C and D, D B C 2. The given figure is a pentahedron with faces K ∆KLM,∆∆∆ KMN , KNO , KLO and . LMNO . The edges are KL,,,,,, KM KN KO LM MN NO and LO . O L The vertices are K, L, M, N, and O. N M Prisms: Exercises (page-4): 1. The right regular octagonal prism has 8 lateral faces, so totally it has 10 faces. It has 24 edges, 8 of them are lateral edges. The prism has 16 vertices. 2. The figure below illustrates an oblique regular triangular prism. It has 5 faces (3 of them are lateral faces), 3 lateral edges and totally 9 edges. The prism has 6 vertices. Pyramids Exercises (page-6): 1. The pentagonal pyramid has 5 lateral faces and thus 6 faces in total. It has 5 lateral edges, 10 edges and 6 vertices. 2. The hexagonal pyramid has 6 lateral faces and 7 faces in total. It has 6 lateral edges, 12 edges and 7 vertices. Nets Exercises: 1. Regular right hexagonal pyramid 2. Rectangular prism 3. Regular right octagonal prism 4. Regular oblique hexagonal prism 5. and so on… 6. Right Cylinder 7. Right Cone Cross sections: Exercises (page-15): 1. i) In the cross section parallel to the base, the cross section is a regular hexagon of the same size as the base.