DOCSLIB.ORG

Download PDF (337K)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Download PDF (337K) On the Curve of Constant Inclination, by TONOSUKE IWYATUKI in Sendai. In the following note it is designed to give a general method of finding the curve of constant inclination. Let us consider in general the curve of constant inclination on a given surface, formed by the rotation. of a curve z=g(as) lying in. the plane of rx, about the axis of z, where g(m) is a finite, one valued, con- tinuous, and differentiable function in an interval. If a particle moves with uniform velocity along the curve of constant inclinationon the surface, the orthogonalprojections of the particle,on the plane of my and the axis of z will move with uniform velocities Vain a and Vcos a respectively, where V and a axe the actual velocity of the particle and the. inclination of its direction to the axis of z. In the time-interval t the particle moves from the starting point A to the point P on the surface and its projectionson the plane of xy and the axis of r come to S and B, respectively; P lies on a circular section by a plane pass- ing through B and parallel to the plane of x, and S lies on the projection of the section on the plane of say. Let Q be the point of intersection of the circular section and the curve z=g(x) in the plane of zx, T a cor- responding point of Q in the plane of say; then T and S lie on the circle of centre 0 and of radius BQ. Put RQ:f(0R)f where f(x) is the inverse function of g(as). On the other hand, OR=Vcosa-t so that BQ=f(Vcos a-t)=OT. ON THE URVE OF CONSTANT INCLINATION 61 Then the equation of the circle (centre 0, radius T) is X+Y=ifWoosat)12. For. the sake of brevity we put Voos a=1, Vsin a-v, then OTfi(t), OR=t. if, in the small time interval M, T moves to T and S to S, then T, S lie on the circle (centre rad. T). Let us put T=x, so that x=f(t), TZ=ux, SS=88=v dt, and put SW=8a, W being the point of intersection of the axe and the radius OS. I n, the triangle SS'W, we have; neglectibg the tesirnals of higher orders than the second, Sae-I-8x2=aas2=e(by)2. In the. limit (da)2=v2(dt)2-(dx)2, whence (da)a=l v2 Jlt)ate; therefor WOO=do=T02 L.f(t)2I. and therefore 9=I.1 v2 V(01111 and moreover OS=rf(t). The polar equation to our curve is, therefore, given by the follow- ing relations (i) or (ii) 62 TORANANOS IWATSUKI: The sign of is taken positive or negative according as the ourvero=Ax) has the differentialcoefficient negative or positive, prom the result (ii), we see that the coordinatesare expressedonly in the terms of a, x, and the initial constant. Therefore the projection of the particle on. the plane, of my which moves along a curve of constant inclination on the given surface of revolution,describes the same locus whatever the velocity of the moving particle may be. In the case of right circular cone: x=nz, tga=m, whence 1/mv-nSd-1og= and r=nx, 11)=nrro; therefore ytlTti-n, leer. This representsa logarithmicspiral In order to find the intrinsicequation of the orthogonalprojection of the curve of constant inclination we proceed as follows: Since H=11V2-f (t) 2 as et=ft, the radius of curvature becomes and we have t- These give us the required intrinsic equation, the double,sign being so. determined that p is positive. This may be obtained by a direct geometrical consideration. If a particle moves. with a constant velocity along the curve in consideration, the orthogonal projection of the particle on the plane of xy moves equal distance in equal time, so that Os=coast. if 3t=coast. (1) This formula has, already been obtained by Bnschke, in a different manner, by using a formula in natural geometry. ON THE CURVE OF CONSTANT INCLINATION. 63 Thus the projections of the particle on the plane of y at and t-opt, lie on the circumference of the circle with centre S, radius S being the position of the moving particle, on our curve at the instant t On the other hand they lie also on the two circles (centre radius 01 and Centre radius OT) respectively, where T and T are the two positions on the axis of X at t+81 and t-31, respectively. Thus the two points S and S of the four points of intersection of the preceding circle and, the last two consecutive circles, must be the two consecutive positions of S on our curve. Now take OS as the axis of X and a stinaight line perpendicular to OS as that, of then the equations of the circle (centre S rad.) and the circles (Centre rad. T and centre rad. T) are respectively (X-OS)2+Y2=Ss2 (1) 12+Y2=(0S-TIT), (2) XZ+Ya=(OS+TT), (3) which become IX-f(4124-Y2=882, X2-Y2 f(t-8t)Z, X2+Y2=t+δt)2. If from these equations, we calculate the abseissae of S and S i.e. OL and OM; then, remembering SL=OL-x and MS=x-OM, we get finally EL=SL-MS 64 TORANOSUKE WATSUK: f(t+2)+r(94-e0 +r2 1f(t)8t12+1f(t+e2) f(t+61) 1f(t)8-28121(t) -8X, say, El, s2 being evanescent with t. Then we have f(t)f11(t)+f(t)12-(Ifi(t) On the other band 8X28Y2=881=v28t, (4) where ax and cSY denote MS' and MS" respectively; from which and and therefore which means lim LS GE-lim LOSE, O being the point of intersection of S"S and the circle. For ES=SM=SX, EO=S"M=aY, EL=SH=X; therefore and therefore lim L GSE=lim LS'GE=say. ON THE CURVE OF CONSTANT IAOLINATION. 65 Thus the radius of curvature is But from (4) we have and as eX differs from by a small qmntity of higher order than the second, we have Hence the required formulae are or where the double sign is to be taken as before. Ini the case of sphere 2+0=a2 , tga=m. Differentiating with respect to r. (1) Examples (1), (2), (3) and (4) have been already obtained by Enneper in the Mathematisohe Annalen, 19. The papers mentioned below may also be consulted. Pirondini: Sur quelquos lignes liees l'helice oylindrique, Crelle J. 121. Huth: Kurven von konstanter Steilheit, Jahrosberioht der Realschule zu Stollberg 1893. Appell: Sur une proprlet caractristique (lea h6lices, Archiv d. Math. 64. Jonas: Kurven konstanter Steilheit auf der Kugelflttoho, Archiv d. Math. 3. Beihe,Bd. 8. B1 as ehke: Allgemeine Sohraubenlinien, Monatshefto fair Math. and Physilc, 19. Jahrgang. Naito: Curve of constant Inclination, Tokyo Sitgaku-Butpurigakkwai Kizi IV. Nugel: Die Sechraubenlinien, Diss. Ialle, 1912. 86 TORANOSUKEIWATSUKI: whence 10=71 n2iCL7-(ms+11x2 Mr=8: therefore Therefore the curve is an fepioycloid. Ex. 2. In the case of spheroid, Hence we find p w nV, n2b4a-(ib2-a2)az(b2m2+a2), mz=s. Therefore Therefore the equation represents an epicycloid. Ex. 3. In the case of hyperboloid of revolution, The required equation may be obtained by putting -b2 in plat of b2 in the previous result. Therefore we have If bm2-a2>O, or I tg a I>a/b, the curve becomes a pseudocycloid; if b2m2-a2<O or I tg a (<a/b, thebecomescurve a hypocycloid; if b2m2-a20, the curve becomes a straight line. Hence according as the inclination of the curve is greater than, ON THE CXRVEOF CONSTANTINCLINATION. 67 equal to, or less than that of the generatingline of the asymptoticcone, the required curve is a pseudooyeloid,a straight line, or a hypocycloid. Ex. 4. In the case of f(z)=v/4z, xa=4dz, tga=m. We have p=V4dmaz-41m, rnz=s, map2=4d(ms-d), and the required curve is an involute of a circle..
Load more

Recommended publications
  • 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]
  • 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]
  • 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
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Handbook of Hydraulic Resistance
    7 / ° " ° ."t. /9EC-lr 4" / a. 11c) C / ý-i Ct Q 4 ~, LE. ldel'chik HANDBOOK OF HYDRAULIC. RESISTANCE Coefficients of Local Resistance and of Friction C LEEAR IN G H 0 U S E AND FOR FEDERAL SCIENTIFIC Novi 4 ?; TECHNICAL INFORMATION Hard~opMicrofiche I/ ~~OU~~9E @~3PP Translated from Russian •Published for the U.S. Atomic Energy Commission and the National Science Foundation, Washington, D.C. by the Israel Program for Scientific Translations Reproduced by NATIONAL TECHNICAL INFORMATION SERVICE. :Springfield, Va. 22151 : : - -T, --!, I.E.. IDELCCHIK HANDBOOK I OF HYDRAULIC RESISTANCE Coefficients of Local Resistance and of Friction (Spravochnik po gidravlicheskim soprotivleniyam. Koeffitsienty rnestnykh. soprotivlenii i soprotivleniya treniya) Gosudarstvennoe Energeticheskoe Izdatel'stvo Moskva-Leningrad 1960 Translated from Russian ,0 £ Israel Program for Scientific Translations Jerusalem 1966 i AEC-tr- 6630 Published Pursuant to an Agreement with THE U. S. ATOMIC ENERGY COMMISSION and THE NATIONAL SCIENCE FOUNDATION, WASHINGTON, D.C. Copyright (0 1966 Israel Program for Scientific Translations Ltd. IPST Cat. No. 1505 Translated by A. Barouch, M. Sc. Edited by D. Grunaer, P. E. and IPST Staff Printed in Jerusalem by S. Monson Price: $8.28 Available from the U.S. DEPARTMENT OF COMMERCE Clearinghouse for Federal Scientific and Technical Information Springfield, Va. 22151 x/S Table of Contents FOREWORD ............ vii Sect ion One. GENERAL INFORMATION AND RECOMMENDATIONS FOR USING THE HANDBOOK ......... 1 1-1. List of general symbols used th.rouglhout the book .1 1-2. General directions .......... ... .• .• .. .. .. ......I ......... 2 1-3. Properties of fluids ......... ................ 3 a. Specific gravity ......... ................ 3 b. Viscosity ............ ° , oo , • . .° .. .•.. .. ................ 4 1-4.
    [Show full text]
  • Three-Dimensional Location Estimation of Circular Features for Machine Vision
    624 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 8, NO. 5, OCTOBER 1992 Three-Dimensional Location Estimation of Circular Features for Machine Vision Reza Safaee-Rad, Ivo Tchoukanov, Member, IEEE, Kenneth Carless Smith, Fellow, IEEE, and Bensiyon Benhabib Abstruct- Estimation of 3-D information from 2-D image the 3-D coordinates are given in addition to the 2-D image coordinates is a fundamental problem both in machine vision coordinates, the problem is of the inverse type. In the first and computer vision. Circular features are the most common (direct) type, the objective is to estimate the 3-D location of quadratic-curved features that have been addressed for 3-D lo- cation estimation. In this paper, a closed-form analytical solution objects, landmarks, and features. This type of problem occurs to the problem of 3-D location estimation of circular features is in many areas: for example, in automatic assembly, tracking, presented. Two different cases are considered: 1) 3-D orientation and industrial metrology. In the second (inverse) type, the and 3-D position estimation of a circular feature when its radius objective is to estimate the relevant camera parameters: for is known, and 2) 3-D orientation and 3-D position estimation of a circular feature when its radius is not known. As well, a fixed camera, all the intrinsic and extrinsic parameters; for a extension of the developed method to 3-D quadratic features moving camera, only the extrinsic parameters, namely, its 3-D is addressed. Specifically, a closed-form analytical solution is location. This type of problem occurs in areas such as camera derived for 3-D position estimation of spherical features.
    [Show full text]
  • Differential Geometry of Three Dimensions
    DIFFERENTIAL GEOMETRY OF THREE DIMENSIONS DIFFERENTIAL GEOMETRY OF THREE DIMENSIONS By G. E. "WEATHERBURN, M.A., D.Sc., LL.D. EMERITUS PROFESSOR 07 MATHEMATICS UNIVERSITY OF WESTERN AUSTRALIA. VOLUME I CAMBRIDGE AT THE UNIVERSITY PRESS 1955 V, PUBLISHED BY THB SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS London Office Bentiey House, N.W. I American Branch New York Agents for Canada,, India, and Pakistan' Maximilian First Edition 1927 Reprinted 1931 1939 1947 1955 First printed in Great Britain at The University Press, Cambridge Eeprmted by Spottwwoode, Sattantyne <b Co., Ltd , Colchester PEEFAOE TO THE FOURTH IMPRESSION present impression is substantially a reprint of the original THEwork. Since the book was first published a few errors have been corrected, and one or two paragraphs rewritten. Among the friends and correspondents who kindly drew my attention to desirable changes were Mr A S. Ramsey of Magdaler-e College, Cambridge, who suggested the revision of 5, and the late R J. A Barnard of Melbourne University, whose mfluence was partly responsible for my initial interest in the subject. The demand for the book, since its first appearance twenty years ago, has justified the writer's belief in the need for such a vectonal treatment. By the use of vector methods the presentation of the subject is both simplified and condensed, and students are encouraged to reason geometrically rather than analytically. At a later stage some of these students will proceed to the study of multidimensional differential geometry and the tensor calculus. It is highly desirable that the study of the geometry of Euclidean 3-space should thus come first, and this can be undertaken with most students at an earlier stage by vector methods than by the Ricci calculus.
    [Show full text]
  • FORCES and Deforl\'IATIONS in CIRCULAR RINGS
    Appendix FORCES AND DEFORl\'IATIONS IN CIRCULAR RINGS Shells of revolution are frequently connected with circular rings, to which they transmit forces and moments. The theory of stresses and ,deformations of such rings is part of the theory of structures. While a. few formulas are found almost everywhere, it is not easy to find the complete set in books of this kind. Therefore they have been compiled here. In all formulas we assume that the ring is thin, i.e. that the di­ mensions of its cross section are small compared with the radius. The axis of the ring is supposed to pass through the centroids of all cross sections. One principal axis of these sections lies in the plane of the ring. Besides those explained in the figures, the following notations have been used: A = area of cross section, 11 = moment of inertia for the centroidal axis in the plane of the ring, I 2 = moment of inertia for the centroida.l axis normal to the plane of the ring, J T = torsional rigidity factor of the section. Where moments and angular displacements have been represented by arrows, the corkscrew rule applies for the interpretation. 1. Radial Load (Fig. A.l) The load per unit length of the axis is assumed to be p = Pn cosnO. Stress resultants: Pna2 N =- ~·a cos nO M =- - --cos nO. n·- 1 ' 2 n 2 - 1 Displacements: 508 APPENDIX These formulas are not valid for n = 0 and n = 1. For n = 0 (axisym­ metric load) they must be replaced by the well-known formulas pa" N=pa, V= 0, W=EA.
    [Show full text]
  • Geometric Properties of Central Catadioptric Line Images
    Geometric Properties of Central Catadioptric Line Images Joao˜ P. 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/f g 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 [5–9]. The general approach of combining mirrors with conventional imaging systems is referred to as catadioptric image formation [3].
    [Show full text]