DOCSLIB.ORG

Hyperboloid Structure 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 parametric representation .......................................................................................................... 6 Symmetries of a hyperboloid ...................................................................................................................... 6 2.3 Benefits: .................................................................................................................................................... 6 2.4 Disadvantages: ...................................................................................................................................... 7 2.5 Parameters: ............................................................................................................................................... 7 2.6 Design:....................................................................................................................................................... 7 Design in GSA Software: .............................................................................................................................. 7 2.6 Construction Techniques: ..................................................................................................................... 9 3. SURVEY ..........................................................................................................................................................10 3.1 List of existing hyperboloid structures ....................................................................................................10 4. MODEL DESIGN .............................................................................................................................................12 4.1 Materials & Quantity ..............................................................................................................................12 4.2 Costing of Model .....................................................................................................................................12 4.3 Estimation of Full Scale model ................................................................................................................12 4.4 Design Drawings ......................................................................................................................................12 5. MODEL MAKING ............................................................................................... Error! Bookmark not defined. 5.1 Photographs of making / crafting model [Stage by Stage] ........................ Error! Bookmark not defined. 6. FINAL MODEL ................................................................................................... Error! Bookmark not defined. 6.1 Model Photographs ................................................................................... Error! Bookmark not defined. 7. REFERENCE ....................................................................................................... Error! Bookmark not defined. Web Reference ................................................................................................ Error! Bookmark not defined. Research Papers ............................................................................................... Error! Bookmark not defined. Thesis / Books .................................................................................................. Error! Bookmark not defined. 8. CREDIT .............................................................................................................. Error! Bookmark not defined. Created by: ....................................................................................................... Error! Bookmark not defined. Guided by: ........................................................................................................ Error! Bookmark not defined. HYPERBOLOID STRUCTURE 3 1.INTRODUCTION 1.1 Concept In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. A hyperboloid is a surface that may be obtained from a paraboloid of revolution by deforming it by means of directional scaling, or more generally, of an affine transformation. A hyperboloid is a quadric surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a centre of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has also three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry. Hyperboloid of two sheets Hyperboloid of one sheet conical surface in between Given a hyperboloid, if one chooses a Cartesian coordinate system whose axes are axes of symmetry of the hyperboloid, and origin is the center of symmetry of the hyperboloid, then the hyperboloid may be defined by one of the two following equations: or Both of these surfaces are asymptotic to the cone of equation One has a hyperboloid of revolution if and only ifa2=b2. Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis. There are two kinds of hyperboloids. In the first case (+1 in the right-hand side of the equation), one has a one-sheet hyperboloid, also called hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussian curvature at every point. This implies that the tangent plane at any point intersect the hyperboloid into two lines, and thus that the one-sheet hyperboloid is a doubly ruled surface. In the second case (−1 in the right-hand side of the equation), one has a two-sheet hyperboloid, also called elliptic hyperboloid. The surface has two connected components, and a positive Gaussian curvature at every point. Thus the surface is convex in the sense that the tangent plane at every point intersects the surface only in this point. HYPERBOLOID STRUCTURE 4 2. HYPERBOLOID STRUCTURE 2.1 Parametric representations Cartesian coordinates for the hyperboloids can be defined, similar to spherical coordinates, keeping the azimuth angle θ ∈ [0, 2π), but changing inclination v into hyperbolic trigonometric functions: One-surface hyperboloid: v ∈ (−∞, ∞) Two-surface hyperboloid: v ∈ [0, ∞) Hyperboloid of one sheet: generation by a rotating 2.2 Properties of a hyperboloid of one sheetLines on the surface hyperbola (top) and line (bottom: red or blue) A hyperboloid of one sheet contains two pencils of lines. It is a doubly ruled surface. If the hyperboloid has the equation then the lines. are contained in the surface.In case a=b he hyperboloid is a surface of revolution and can be generated by rotating one of the two lines + - g0 or g0 which are skew to the rotation axis (see picture). The more common generation of a hyperboloid of revolution is rotating a hyperbola around its semi-minor axis (see picture). Remark: A hyperboloid of two sheets is protectively equivalent to a hyperbolic paraboloid. Hyperboloid of one sheet: plane sections Plane sections For simplicity the plane sections of the unit hyperboloid with 2 2 2 equation H1: x +y -z =1 are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too. A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects H1 in an ellipse, A plane with a slope equal to 1 containing the origin intersects H1 in a pair of parallel lines, A plane with a slope equal 1 not containing the origin intersects H1in a parabola, A tangential plane intersects H1 in a pair of intersecting lines, A non-tangential plane with a slope greater than 1 intersects H1 in a hyperbola. Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case. HYPERBOLOID STRUCTURE 5 Properties of a hyperboloid of two sheets The hyperboloid of two sheets does not contain
Load more

Recommended publications
  • Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia
    Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia Brief information on some classes of the surfaces which cylinders, cones and ortoid ruled surfaces with a constant were not picked out into the special section in the encyclo- distribution parameter possess this property. Other properties pedia is presented at the part “Surfaces”, where rather known of these surfaces are considered as well. groups of the surfaces are given. It is known, that the Plücker conoid carries two-para- At this section, the less known surfaces are noted. For metrical family of ellipses. The straight lines, perpendicular some reason or other, the authors could not look through to the planes of these ellipses and passing through their some primary sources and that is why these surfaces were centers, form the right congruence which is an algebraic not included in the basic contents of the encyclopedia. In the congruence of the4th order of the 2nd class. This congru- basis contents of the book, the authors did not include the ence attracted attention of D. Palman [8] who studied its surfaces that are very interesting with mathematical point of properties. Taking into account, that on the Plücker conoid, view but having pure cognitive interest and imagined with ∞2 of conic cross-sections are disposed, O. Bottema [9] difficultly in real engineering and architectural structures. examined the congruence of the normals to the planes of Non-orientable surfaces may be represented as kinematics these conic cross-sections passed through their centers and surfaces with ruled or curvilinear generatrixes and may be prescribed a number of the properties of a congruence of given on a picture.
    [Show full text]
  • Chapter 11. Three Dimensional Analytic Geometry and Vectors
    Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1.
    [Show full text]
  • Calculus & Analytic Geometry
    TQS 126 Spring 2008 Quinn Calculus & Analytic Geometry III Quadratic Equations in 3-D Match each function to its graph 1. 9x2 + 36y2 +4z2 = 36 2. 4x2 +9y2 4z2 =0 − 3. 36x2 +9y2 4z2 = 36 − 4. 4x2 9y2 4z2 = 36 − − 5. 9x2 +4y2 6z =0 − 6. 9x2 4y2 6z =0 − − 7. 4x2 + y2 +4z2 4y 4z +36=0 − − 8. 4x2 + y2 +4z2 4y 4z 36=0 − − − cone • ellipsoid • elliptic paraboloid • hyperbolic paraboloid • hyperboloid of one sheet • hyperboloid of two sheets 24 TQS 126 Spring 2008 Quinn Calculus & Analytic Geometry III Parametric Equations (§10.1) and Vector Functions (§13.1) Definition. If x and y are given as continuous function x = f(t) y = g(t) over an interval of t-values, then the set of points (x, y)=(f(t),g(t)) defined by these equation is a parametric curve (sometimes called aplane curve). The equations are parametric equations for the curve. Often we think of parametric curves as describing the movement of a particle in a plane over time. Examples. x = 2cos t x = et 0 t π 1 t e y = 3sin t ≤ ≤ y = ln t ≤ ≤ Can we find parameterizations of known curves? the line segment circle x2 + y2 =1 from (1, 3) to (5, 1) Why restrict ourselves to only moving through planes? Why not space? And why not use our nifty vector notation? 25 Definition. If x, y, and z are given as continuous functions x = f(t) y = g(t) z = h(t) over an interval of t-values, then the set of points (x,y,z)= (f(t),g(t), h(t)) defined by these equation is a parametric curve (sometimes called a space curve).
    [Show full text]
  • Extension of Eigenvalue Problems on Gauss Map of Ruled Surfaces
    S S symmetry Article Extension of Eigenvalue Problems on Gauss Map of Ruled Surfaces Miekyung Choi 1 and Young Ho Kim 2,* 1 Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Korea; [email protected] 2 Department of Mathematics, Kyungpook National University, Daegu 41566, Korea * Correspondence: [email protected]; Tel.: +82-53-950-5888 Received: 20 September 2018; Accepted: 12 October 2018; Published: 16 October 2018 Abstract: A finite-type immersion or smooth map is a nice tool to classify submanifolds of Euclidean space, which comes from the eigenvalue problem of immersion. The notion of generalized 1-type is a natural generalization of 1-type in the usual sense and pointwise 1-type. We classify ruled surfaces with a generalized 1-type Gauss map as part of a plane, a circular cylinder, a cylinder over a base curve of an infinite type, a helicoid, a right cone and a conical surface of G-type. Keywords: ruled surface; pointwise 1-type Gauss map; generalized 1-type Gauss map; conical surface of G-type 1. Introduction Nash’s embedding theorem enables us to study Riemannian manifolds extensively by regarding a Riemannian manifold as a submanifold of Euclidean space with sufficiently high codimension. By means of such a setting, we can have rich geometric information from the intrinsic and extrinsic properties of submanifolds of Euclidean space. Inspired by the degree of algebraic varieties, B.-Y. Chen introduced the notion of order and type of submanifolds of Euclidean space. Furthermore, he developed the theory of finite-type submanifolds and estimated the total mean curvature of compact submanifolds of Euclidean space in the late 1970s ([1]).
    [Show full text]
  • A Cochleoid Cone Udc 514.1=111
    FACTA UNIVERSITATIS Series: Architecture and Civil Engineering Vol. 9, No 3, 2011, pp. 501 - 509 DOI: 10.2298/FUACE1103501N CONE WHOSE DIRECTRIX IS A CYLINDRICAL HELIX AND THE VERTEX OF THE DIRECTRIX IS – A COCHLEOID CONE UDC 514.1=111 Vladan Nikolić*, Sonja Krasić, Olivera Nikolić University of Niš, The Faculty of Civil Engineering and Architecture, Serbia * [email protected] Abstract. The paper treated a cone with a cylindrical helix as a directrix and the vertex on it. Characteristic elements of a surface formed in such way and the basis are identified, and characteristic flat intersections of planes are classified. Also considered is the potential of practical application of such cone in architecture and design. Key words: cocleoid cones, vertex on directrix, cylindrical helix directrix. 1. INTRODUCTION Cone is a deriving singly curved rectilinear surface. A randomly chosen point A on the directrix d1 will, along with the directirx d2 determine totally defined conical surface k, figure 1. If directirx d3 penetrates through this conical surface in the point P, then the connection line AP, regarding that it intersects all three directrices (d1, d2 and d3), will be the generatrix of rectilinear surface. If the directrix d3 penetrates through the men- tioned conical surface in two, three or more points, then through point A will pass two, three or more generatrices of the rectilinear surface.[7] Directrix of a rectilinear surface can be any planar or spatial curve. By changing the form and mutual position of the directrices, various type of rectilinear surfaces can be obtained. If the directrices d1 and d2 intersect, and the intersection point is designated with A, then the top of the created surface will occur on the directrix d2, figure 2.
    [Show full text]
  • Structural Forms 1
    Key principles The hyperboloid of revolution is a Surface and may be generated by revolving a Hyperbola about its conjugate axis. The outline of the elevation will be a Hyperbola. When the conjugate axis is vertical all horizontal sections are circles. The horizontal section at the mid point of the conjugate axis is known as the Throat. The diagram shows the incomplete construction for drawing a hyperbola in a rectangle. (a) Draw the outline of the both branches of the double hyperbola in the rectangle. The diagram shows the incomplete Elevation of a hyperboloid of revolution. (a) Determine the position of the throat circle in elevation. (b) Draw the outline of the both branches of the double hyperbola in elevation. DESIGN & COMMUNICATION GRAPHICS Structural forms 1 NAME: ______________________________ DATE: _____________ The diagram shows the plan and incomplete elevation of an object based on the hyperboloid of revolution. The focal points and transverse axis of the hyperbola are also shown. (a) Using the given information draw the outline of the elevation.. F The diagram shows the axis, focal points and transverse axis of a double hyperbola. (a) Draw the outline of both branches of the double hyperbola. (b) The difference between the focal distances for any point on a double hyperbola is constant and equal to the length of the transverse axis. (c) Indicate this principle on the drawing below. DESIGN & COMMUNICATION GRAPHICS Structural forms 2 NAME: ______________________________ DATE: _____________ Key principles The diagram shows the plan and incomplete elevation of a hyperboloid of revolution. The hyperboloid of revolution may also be generated by revolving one skew line about another.
    [Show full text]
  • 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.
    [Show full text]
  • Bruxelles, 2015 Art and Literature Scientific and Analytical Journal
    Art and Literature Scientific and Analytical Journal Texts 4.2015 Bruxelles, 2015 EDITORIAL BOARD Chief editor Burganova M. A. Bowlt John Ellis (USA) — Doctor of Science, Professor of Slavic Languages and Literatures in University of Southern California; Burganov A. N. (Russia) — Doctor of Science, Professor of Stroganoff Moscow State Art Industrial University, Full-member of Russia Academy of Arts, National Artist of Russia, member of the Dissertation Council of Stroganoff Moscow State Art Industrial University; Burganova M. A. (Russia) — Doctor of Science, Professor of Stroganoff Moscow State Art Industrial University, Full-member of Russia Academy of Arts, Honored Artist of Russia, member of the Dissertation Council of Stroganoff Moscow State Art Industrial University, editor-in-chief; Glanc Tomáš (Germany) — Doctor of Science of The Research Institute of East European University of Bremen (Germany), and assistant professor of The Charles University (Czech Republic); Kazarian Armen (Russia) — Architectural historian, Doctor of Fine Arts in The State Institute of Art History, Advisor in Academy of Architecture and Construction Sciences; Kravetsky A. G. (Russia) — Candidate of Sciences, research associate of Russian Language Institute of the Russian Academy of Sciences; Lavrentyev Alexander N. (Russia) — Doctor of Arts, Professor of Stroganoff Moscow State Art Industrial University and Moscow State University of Printing Arts; Alessandro De Magistris (Italy) — PhD, Full-Professor of History of Architecture Politecnico di Milano Department of Architecture and Urban Studies; Misler Nicoletta (Italy) — Professor of Modern East European Art at the Istituto Universitario Orientale, Naples; Pavlova I. B. (Russia) — Candidate of Sciences, Senior Researcher of Institute of World Literature of the Russian Academy of Sciences; ISSN 2294-8902 © TEXTS, 2015 Pletneva A.
    [Show full text]
  • Quadric Surfaces
    Quadric Surfaces Six basic types of quadric surfaces: • ellipsoid • cone • elliptic paraboloid • hyperboloid of one sheet • hyperboloid of two sheets • hyperbolic paraboloid (A) (B) (C) (D) (E) (F) 1. For each surface, describe the traces of the surface in x = k, y = k, and z = k. Then pick the term from the list above which seems to most accurately describe the surface (we haven't learned any of these terms yet, but you should be able to make a good educated guess), and pick the correct picture of the surface. x2 y2 (a) − = z. 9 16 1 • Traces in x = k: parabolas • Traces in y = k: parabolas • Traces in z = k: hyperbolas (possibly a pair of lines) y2 k2 Solution. The trace in x = k of the surface is z = − 16 + 9 , which is a downward-opening parabola. x2 k2 The trace in y = k of the surface is z = 9 − 16 , which is an upward-opening parabola. x2 y2 The trace in z = k of the surface is 9 − 16 = k, which is a hyperbola if k 6= 0 and a pair of lines (a degenerate hyperbola) if k = 0. This surface is called a hyperbolic paraboloid , and it looks like picture (E) . It is also sometimes called a saddle. x2 y2 z2 (b) + + = 1. 4 25 9 • Traces in x = k: ellipses (possibly a point) or nothing • Traces in y = k: ellipses (possibly a point) or nothing • Traces in z = k: ellipses (possibly a point) or nothing y2 z2 k2 k2 Solution. The trace in x = k of the surface is 25 + 9 = 1− 4 .
    [Show full text]
  • (Anti-)De Sitter Space-Time
    Geometry of (Anti-)De Sitter space-time Author: Ricard Monge Calvo. Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain. Advisor: Dr. Jaume Garriga Abstract: This work is an introduction to the De Sitter and Anti-de Sitter spacetimes, as the maximally symmetric constant curvature spacetimes with positive and negative Ricci curvature scalar R. We discuss their causal properties and the characterization of their geodesics, and look at p;q the spaces embedded in flat R spacetimes with an additional dimension. We conclude that the geodesics in these spaces can be regarded as intersections with planes going through the origin of the embedding space, and comment on the consequences. I. INTRODUCTION In the case of dS4, introducing the coordinates (T; χ, θ; φ) given by: Einstein's general relativity postulates that spacetime T T is a differential (Lorentzian) manifold of dimension 4, X0 = a sinh X~ = a cosh ~n (4) a a whose Ricci curvature tensor is determined by its mass- energy contents, according to the equations: where X~ = X1;X2;X3;X4 and ~n = ( cos χ, sin χ cos θ, sin χ sin θ cos φ, sin χ sin θ sin φ) with T 2 (−∞; 1), 0 ≤ 1 8πG χ ≤ π, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π, then the line element Rµλ − Rgµλ + Λgµλ = 4 Tµλ (1) 2 c is: where Rµλ is the Ricci curvature tensor, R te Ricci scalar T ds2 = −dT 2 + a2 cosh2 [dχ2 + sin2 χ dΩ2] (5) curvature, gµλ the metric tensor, Λ the cosmological con- a 2 stant, G the universal gravitational constant, c the speed of light in vacuum and Tµλ the energy-momentum ten- where the surfaces of constant time dT = 0 have metric 2 2 2 2 sor.
    [Show full text]
  • Wire Structure Image-Based 3D Reconstruction Aided by Deep Learning
    Research Collection Conference Paper Wire structure image-based 3d reconstruction aided by deep learning Author(s): Kniaz, Vladimir V.; Zheltov, Sergey Yu.; Remondino, Fabio; Knyaz, Vladimir A.; Gruen, Armin Publication Date: 2020 Permanent Link: https://doi.org/10.3929/ethz-b-000442466 Originally published in: International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLIII- B2-2020, http://doi.org/10.5194/isprs-archives-XLIII-B2-2020-435-2020 Rights / License: Creative Commons Attribution 4.0 International This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLIII-B2-2020, 2020 XXIV ISPRS Congress (2020 edition) WIRE STRUCTURE IMAGE-BASED 3D RECONSTRUCTION AIDED BY DEEP LEARNING Vladimir V. Kniaz1,2,∗ Sergey Yu. Zheltov1, Fabio Remondino3, Vladimir A. Knyaz1,2, Artyom Bordodymov1, Armin Gruen4 1 State Res. Institute of Aviation Systems (GosNIIAS), 125319, 7, Victorenko str., Moscow, Russia – (vl.kniaz, zhl, knyaz, bordodymov)@gosniias.ru 2 Moscow Institute of Physics and Technology (MIPT), Dolgoprudny, Russia 3 Bruno Kessler Foundation (FBK), Trento, Italy – [email protected] 4 ETH Zurich, Switzerland – [email protected] Commission II, WG II/8 KEY WORDS: structure from motion, wire structures 3D reconstruction, segmentation, deep learning, Shukhov Radio tower ABSTRACT: Objects and structures realized by connecting and bending wires are common in modern architecture, furniture design, metal sculpting, etc. The 3D reconstruction of such objects with traditional range- or image-based methods is very difficult and poses challenges due to their unique characteristics such as repeated structures, slim elements, holes, lack of features, self-occlusions, etc.
    [Show full text]
  • 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].
    [Show full text]