Fundamentals of Geometric Construction Paul Zsombor-Murray
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Understanding Projection Systems
Understanding Projection Systems Understanding Projection Systems A Point: A point has no dimensions, a theoretical location that has neither length, width nor height. A point shows an exact location in space. It is important to understand that a point is not an object, but a position. We represent a point by placing a dot with a pencil. A Line: A line is a geometric object that has length and direction but no thickness. A line may be straight or curved. A line may be infinitely long. If a line has a definite length it is called a line segment or curve segment. A straight line is the shortest distance between two points which is known as the true length of the line. A line is named using letters to indicate its endpoints. B B A A AB - Straight Line Segment AB – Curved Line Segment A line may be seen as the locus of a point as it travels between two points. A B A line can graphically represent the intersection of two surfaces, the edge view of a surface, or the limiting element of a surface. B A Plane: A plane is a flat surface which is infinitely large with zero thickness. Just as a point generates a line, a line can generate a plane. A A portion of a plane is referred to as a lamina. A Plane may be defined in a number of different ways. - 1 - Understanding Projection Systems A plane may be defined by; (i) 3 non-linear points (ii) A line and a point (iii) Two intersecting lines (iv) Two Parallel Lines (The point can not lie on the line) Descriptive Geometry: refers to the representation of 3D objects in a 2D format using points, lines and planes. -
3D Viewing Week 8, Lecture 15
CS 536 Computer Graphics 3D Viewing Week 8, Lecture 15 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 1 Overview • 3D Viewing • 3D Projective Geometry • Mapping 3D worlds to 2D screens • Introduction and discussion of homework #4 Lecture Credits: Most pictures are from Foley/VanDam; Additional and extensive thanks also goes to those credited on individual slides 2 Pics/Math courtesy of Dave Mount @ UMD-CP 1994 Foley/VanDam/Finer/Huges/Phillips ICG Recall the 2D Problem • Objects exist in a 2D WCS • Objects clipped/transformed to viewport • Viewport transformed and drawn on 2D screen 3 Pics/Math courtesy of Dave Mount @ UMD-CP From 3D Virtual World to 2D Screen • Not unlike The Allegory of the Cave (Plato’s “Republic", Book VII) • Viewers see a 2D shadow of 3D world • How do we create this shadow? • How do we make it as realistic as possible? 4 Pics/Math courtesy of Dave Mount @ UMD-CP History of Linear Perspective • Renaissance artists – Alberti (1435) – Della Francesca (1470) – Da Vinci (1490) – Pélerin (1505) – Dürer (1525) Dürer: Measurement Instruction with Compass and Straight Edge http://www.handprint.com/HP/WCL/tech10.html 5 The 3D Problem: Using a Synthetic Camera • Think of 3D viewing as taking a photo: – Select Projection – Specify viewing parameters – Clip objects in 3D – Project the results onto the display and draw 6 1994 Foley/VanDam/Finer/Huges/Phillips ICG The 3D Problem: (Slightly) Alternate Approach • Think of 3D viewing as taking a photo: – Select Projection – Specify -
Stereographic Projection Patch Kessler, December 12, 2018
Stereographic Projection Patch Kessler, December 12, 2018 Stereographic projection is a way to flatten out the surface of a ball, such as the celestial sphere, which surrounds the earth and contains all the stars in the sky. The celestial sphere gives order out of chaos. It gives a way to predict star locations at different times, as well as the time of night from the stars. Here is a physical model of the celestial sphere from around 1480. 1480-1481, http://www.artfund.org/artwork/3600/spherical-astrolabe 1 According to legend, Ptolemy (AD 90 - AD 168) thought of stereographic projection after a donkey stepped on his model of the celestial sphere, making it flat. Applying stereographic projection to the celestial sphere results in a planar device called an astrolabe. 16th century islamic astrolabe The tips on the perforated upper plate correspond to stars, while points on the lower plate correspond to viewing directions. For instance, the point on the lower plate that the circles are shrinking towards corresponds to the viewing direction directly overhead (i.e., looking straight up). The stereographic projection of a point X on a sphere S is obtained by drawing a line from the north pole of the sphere through X, and continuing on to the plane G that the sphere is resting on. The projection of X is the point f(X) where this line hits the plane. E X S G f(X) 2 One of the most surprising things about stereographic projection is that circles get mapped to circles. P circle on the sphere E S circle in the plane G Circles that pass through the north pole get mapped to lines, however if you think of a line as an infinite radius circle, then you can say that circles get mapped to circles in all cases. -
Geometric Properties of Central Catadioptric Line Images
Geometric Properties of Central Catadioptric Line Images Jo˜aoP. Barreto and Helder Araujo Institute of Systems and Robotics Dept. of Electrical and Computer Engineering University of Coimbra Coimbra, Portugal {jpbar, helder}@isr.uc.pt http://www.isr.uc.pt/˜jpbar Abstract. It is highly desirable that an imaging system has a single effective viewpoint. Central catadioptric systems are imaging systems that use mirrors to enhance the field of view while keeping a unique center of projection. A general model for central catadioptric image formation has already been established. The present paper exploits this model to study the catadioptric projection of lines. The equations and geometric properties of general catadioptric line imaging are derived. We show that it is possible to determine the position of both the effec- tive viewpoint and the absolute conic in the catadioptric image plane from the images of three lines. It is also proved that it is possible to identify the type of catadioptric system and the position of the line at infinity without further infor- mation. A methodology for central catadioptric system calibration is proposed. Reconstruction aspects are discussed. Experimental results are presented. All the results presented are original and completely new. 1 Introduction Many applications in computer vision, such as surveillance and model acquisition for virtual reality, require that a large field of view is imaged. Visual control of motion can also benefit from enhanced fields of view [1,2,4]. One effective way to enhance the field of view of a camera is to use mirrors [5,6,7,8,9]. The general approach of combining mirrors with conventional imaging systems is referred to as catadioptric image formation [3]. -
U. S. Department of Agriculture Technical Release No
U. S. DEPARTMENT OF AGRICULTURE TECHNICAL RELEASE NO. 41 SO1 L CONSERVATION SERVICE GEOLOGY &INEERING DIVISION MARCH 1969 U. S. Department of Agriculture Technical Release No. 41 Soil Conservation Service Geology Engineering Division March 1969 GRAPHICAL SOLUTIONS OF GEOLOGIC PROBLEMS D. H. Hixson Geologist GRAPHICAL SOLUTIONS OF GEOLOGIC PROBLEMS Contents Page Introduction Scope Orthographic Projections Depth to a Dipping Bed Determine True Dip from One Apparent Dip and the Strike Determine True Dip from Two Apparent Dip Measurements at Same Point Three Point Problem Problems Involving Points, Lines, and Planes Problems Involving Points and Lines Shortest Distance between Two Non-Parallel, Non-Intersecting Lines Distance from a Point to a Plane Determine the Line of Intersection of Two Oblique Planes Displacement of a Vertical Fault Displacement of an Inclined Fault Stereographic Projection True Dip from Two Apparent Dips Apparent Dip from True Dip Line of Intersection of Two Oblique Planes Rotation of a Bed Rotation of a Fault Poles Rotation of a Bed Rotation of a Fault Vertical Drill Holes Inclined Drill Holes Combination Orthographic and Stereographic Technique References Figures Fig. 1 Orthographic Projection Fig. 2 Orthographic Projection Fig. 3 True Dip from Apparent Dip and Strike Fig. 4 True Dip from Two Apparent Dips Fig. 5 True Dip from Two Apparent Dips Fig. 6 True Dip from Two Apparent Dips Fig. 7 Three Point Problem Fig. 8 Three Point Problem Page Fig. Distance from a Point to a Line 17 Fig. Shortest Distance between Two Lines 19 Fig. Distance from a Point to a Plane 21 Fig. Nomenclature of Fault Displacement 23 Fig. -
National 4 & 5 Graphic Communication
Duncanrig Secondary School Department of Design, Engineering & Technology National 4 & 5 Graphic Communication - Revision Notes Contents Page 01 Exam Preparation and Techniques 02 - 03 The 3 P’s 04 British Standards Purpose, title blocks and scale 05 British Standards Line types & 3rd Angle Projection 06 - 08 British Standards Dimensioning 09 Drawing Types: Overview and Introduction 10 Drawing Types: Orthographic Views 11 Drawing Types: Sectional Views and Exploded Views 12 Sectional Drawing guide for answering questions 13 - 14 Drawing Types: Geometry 15 Answering true shape exam questions 16 A/C and A/F explained 17 - 18 Drawing Types: Pictorial drawings and exam questions 19 Interpreting/Reading Complex drawings 20 - 23 Building Drawings 24 - 27 Computer Terminology, Hardware Input, Output and Storage 28 - 29 Computer Software 30 Computer Aided Design Software 31 - 35 2D/3D CAD Features and Edits 36 - 37 Answering 3D CAD exam questions 38 CAD Assembly constraints 39 CAD Animation and Simulation 40 CAD illustration Techniques 41 - 42 Advantages and Limitations of CAD and Manual Techniques 43 Manual Graphics Techniques 44 - 49 DTP features and edits 50 - 52 DTP Elements and Principles 50 - 52 Colour Theory 53 - 55 Graphics Impact on Society 56 Graphs and Charts 1 Exam Preparation What makes up my grade in Graphic Communication? The exam has written questions to test Knowledge and Interpretation skills in Graphic Communication. A grade A, B, C or D is awarded at National 5. 33% of your course award is made up of the graphics assignment which you undertake in class over a period of 8 hours. The exam is worth 67%. -
Hyperboloid Structure 1
GYANMANJARI INSTITUTE OF TECHNOLOGY DEPARTMENT OF CIVIL ENGINEERING HYPERBOLOID STRUCTURE YOUR NAME/s HYPERBOLOID STRUCTURE 1 TOPIC TITLE HYPERBOLOID STRUCTURE By DAVE MAITRY [151290106010] [[email protected]] SOMPURA HEETARTH [151290106024] [[email protected]] GUIDED BY VIJAY PARMAR ASSISTANT PROFESSOR, CIVIL DEPARTMENT, GMIT. DEPARTMENT OF CIVIL ENGINEERING GYANMANJRI INSTITUTE OF TECHNOLOGY BHAVNAGAR GUJARAT TECHNOLOGICAL UNIVERSITY HYPERBOLOID STRUCTURE 2 TABLE OF CONTENTS 1.INTRODUCTION ................................................................................................................................................ 4 1.1 Concept ..................................................................................................................................................... 4 2. HYPERBOLOID STRUCTURE ............................................................................................................................. 5 2.1 Parametric representations ...................................................................................................................... 5 2.2 Properties of a hyperboloid of one sheet Lines on the surface ................................................................ 5 Plane sections .............................................................................................................................................. 5 Properties of a hyperboloid of two sheets .................................................................................................. 6 Common -
New Decompositions of the Displacement Gradient for Infinitesimal Strain
NEW DECOMPOSITIONS OF THE DISPLACEMENT GRADIENT FOR INFINITESIMAL STRAIN By P. BOULANGER De´partement de Mathe´matique, Universite´ Libre de Bruxelles, Belgium and M. HAYES*, M.R.I.A. Department of Mechanical Engineering, University College, Dublin [Received 24 July 2001. Read 16 March 2002. Published 31 December 2003.] ABSTRACT We consider, within the context of infinitesimal strain theory, the shears of pairs and triads of material line elements emanating from a point P in a body. The set of elements along the principal axes of strain at P is the only orthogonal set of unsheared elements. There is an infinite set of other unsheared triads. In general, if an unsheared pair is known, a third element completing an unsheared triad may be found. 1. Introduction We consider the three-dimensional infinitesimal strain at a point P in a body. If the dis- placement gradient at P is denoted by H, then the strain tensor e is defined by 2e=H+HT. We assume that e has eigenvalues ea, the principal strains, ordered e3>e2>e1. In general e is not positive definite, but a positive definite tensor E, defined by E=e+l1 (where l is suitably large: l+e1>0), may be associated with e [7]. The associated quadric, x . Ex=1, is an ellipsoid E (say), whose principal axes are the principal axes of strain. Because we assume that e3>e2>e1, it follows that E has two planes of central circular section (see, for instance, [2]). These planes are denoted by C+ and Cx. Previously it was shown [4] that, apart from two exceptions, for any infinitesimal material line element L lying in a plane P at the point P, there is just one other material line element Lk in P such that the angle between the elements along L and Lk is unchanged in the deformation, so that they are unsheared (in this case the element along Lk is said to be conjugate to the element along L—they form a conjugate or unsheared pair). -
THE GEOMETRY of MOVEMENT. [Juty?
4 76 THE GEOMETRY OF MOVEMENT. [Juty? THE GEOMETRY OF MOVEMENT. Geometrie der Bewegung in synlhetischer Darstellung. Von Dr. ARTHUR SCHOENFLIES. Leipzig, B. G. Teubner, 1886. 8vo, pp. vi + 194. La Géométrie du Mouvement. Exposé synthétique. Translated by CH. SPECKEL, Capitaine du Génie. Paris, Gauthier- Villars, 1893. 8vo, pp. vii + 292. PERHAPS apology is needed for noticing a book no longer in its infancy. But we feel that *' better late than never'' applies to acquaintance with a work which contains so much matter which was new at the time of writing and is not yet accessible in English, nor (we believe) well known. The main idea of the book is to consider a body in two or more positions relatively to another body, and thence as a limit case to discuss the instantaneous motion. Full ad vantage is taken of the duality arising from viewing things from the standpoint of the one body or the other. That we have found the book difficult is probably due to our early training ; but a few more figures and a few more de tails would have been welcome. The French translation is very reliable, and its value is increased by a good elementary account of complexes and congruences of lines (pp. 219-291), by G. Fouret. Our intention is, not to discuss the information in the book, but to select a few of the more salient theorems (omit ting such as are presumably familiar). When in this string of enunciations there seems occasion to interpolate a re mark, square brackets are used. §1. -
Engineering Drawing
LECTURE NOTES For Environmental Health Science Students Engineering Drawing Wuttet Taffesse, Laikemariam Kassa Haramaya University In collaboration with the Ethiopia Public Health Training Initiative, The Carter Center, the Ethiopia Ministry of Health, and the Ethiopia Ministry of Education 2005 Funded under USAID Cooperative Agreement No. 663-A-00-00-0358-00. Produced in collaboration with the Ethiopia Public Health Training Initiative, The Carter Center, the Ethiopia Ministry of Health, and the Ethiopia Ministry of Education. Important Guidelines for Printing and Photocopying Limited permission is granted free of charge to print or photocopy all pages of this publication for educational, not-for-profit use by health care workers, students or faculty. All copies must retain all author credits and copyright notices included in the original document. Under no circumstances is it permissible to sell or distribute on a commercial basis, or to claim authorship of, copies of material reproduced from this publication. ©2005 by Wuttet Taffesse, Laikemariam Kassa All rights reserved. Except as expressly provided above, no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission of the author or authors. This material is intended for educational use only by practicing health care workers or students and faculty in a health care field. PREFACE The problem faced today in the learning and teaching of engineering drawing for Environmental Health Sciences students in universities, colleges, health institutions, training of health center emanates primarily from the unavailability of text books that focus on the needs and scope of Ethiopian environmental students. -
Spherical and Hyperbolic Conics
Spherical and hyperbolic conics Ivan Izmestiev February 23, 2017 1 Introduction Most textbooks on classical geometry contain a chapter about conics. There are many well-known Euclidean, affine, and projective properties of conics. For broader modern presentations we can recommend the corresponding chapters in the textbook of Berger [2] and two recent books [1, 13] devoted exclusively to conics. For a detailed survey of the results and history before the 20th century, see the encyclopedia articles [6, 7]. At the same time, one rarely speaks about non-Euclidean conics. How- ever, they should not be seen as something exotic. These are projective conics in the presence of a Cayley{Klein metric. In other words, a non- 3 Euclidean conic is a pair of quadratic forms (Ω;S) on R , where Ω (the absolute) is non-degenerate. Geometrically, a spherical conic is the intersec- tion of the sphere with a quadratic cone; what are the metric properties of this curve with respect to the intrinsic metric of the sphere? In the Beltrami{ Cayley{Klein model of the hyperbolic plane, a conic is the intersection of an affine conic with the disk standing for the plane. Non-Euclidean conics share many properties with their Euclidean rela- tives. For example, the set of points on the sphere with a constant sum of distances from two given points is a spherical ellipse. The same is true in the hyperbolic plane. A reader interested in the bifocal properties of other hyperbolic conics can take a look at Theorem 6.10 or Figure 37. In the non-Euclidean geometry we have the polarity with respect to the absolute. -
Investigating Conics and Other Curves Dynamically James R. King
Investigating Conics and other Curves Dynamically James R. King, University of Washington Interactive software such as The Geometer's Sketchpad™ and Cabri Geometry™ allows one to manipulate dynamically the objects of elementary geometry, such as triangles and circles. Since the software will draw the trace of a moving object, one can also create and investigate many curves dynamically as well. These curves provide students with an interesting set of applications of elementary geometry and also serve as concrete examples of concepts studied later in calculus and differential geometry. The collection of curves that can be found in any handbook is very broad indeed. As examples of curves that can be studied with dynamic software, this paper will concentrate on the conic sections and curves related to them. Other examples can be found in [4]. Conics in Elementary Figures – the Parabola Even in elementary geometrical figures, conics and other curves may be found hiding in the background. In this figure, a point A and a line m are given; B is a point on line m. A circle is constructed through A so that m is tangent to the circle at B. This is done by constructing the center point O as the intersection of the perpendicular bisector p of AB and the line n through B perpendicular to m. Since this is dynamic geometry, it is natural to drag the given points A and B and see how O moves. If one drags A, then O simply moves along line n. But if one drags B, the center O traces a curve, a parabola.