RULED SURFACE MEDIA FACADES Using Ruled Surface Principles For
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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. -
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). -
1 on the Design and Invariants of a Ruled Surface Ferhat Taş Ph.D
On the Design and Invariants of a Ruled Surface Ferhat Taş Ph.D., Research Assistant. Department of Mathematics, Faculty of Science, Istanbul University, Vezneciler, 34134, Istanbul,Turkey. [email protected] ABSTRACT This paper deals with a kind of design of a ruled surface. It combines concepts from the fields of computer aided geometric design and kinematics. A dual unit spherical Bézier- like curve on the dual unit sphere (DUS) is obtained with respect the control points by a new method. So, with the aid of Study [1] transference principle, a dual unit spherical Bézier-like curve corresponds to a ruled surface. Furthermore, closed ruled surfaces are determined via control points and integral invariants of these surfaces are investigated. The results are illustrated by examples. Key Words: Kinematics, Bézier curves, E. Study’s map, Spherical interpolation. Classification: 53A17-53A25 1 Introduction Many scientists like mechanical engineers, computer scientists and mathematicians are interested in ruled surfaces because the surfaces are widely using in mechanics, robotics and some other industrial areas. Some of them have investigated its differential geometric properties and the others, its representation mathematically for the design of these surfaces. Study [1] used dual numbers and dual vectors in his research on the geometry of lines and kinematics in the 3-dimensional space. He showed that there 1 exists a one-to-one correspondence between the position vectors of DUS and the directed lines of space 3. So, a one-parameter motion of a point on DUS corresponds to a ruled surface in 3-dimensionalℝ real space. Hoschek [2] found integral invariants for characterizing the closed ruled surfaces. -
OBJ (Application/Pdf)
THE PROJECTIVE DIFFERENTIAL GEOMETRY OF RULED SURFACES A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY WILLIAM ALBERT JONES, JR. DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA AUGUST 1949 t-lii f£l ACKNOWLEDGMENTS A special expression of appreciation is due Mr. C. B. Dansby, my advisor, whose wise tolerant counsel and broad vision have been of inestimable value in making this thesis a success. The Author ii TABLE OP CONTENTS Chapter Page I. INTRODUCTION 1 1. Historical Sketch 1 2. The General Aim of this Study. 1 3. Methods of Approach... 1 II. FUNDAMENTAL CONCEPTS PRECEDING THE STUDY OP RULED SURFACES 3 1. A Linear Space of n-dimens ions 3 2. A Ruled Surface Defined 3 3. Elements of the Theory of Analytic Surfaces 3 4. Developable Surfaces. 13 III. FOUNDATIONS FOR THE THEORY OF RULED SURFACES IN Sn# 18 1. The Parametric Vector Equation of a Ruled Surface 16 2. Osculating Linear Spaces of a Ruled Surface 22 IV. RULED SURFACES IN ORDINARY SPACE, S3 25 1. The Differential Equations of a Ruled Surface 25 2. The Transformation of the Dependent Variables. 26 3. The Transformation of the Parameter 37 V. CONCLUSIONS 47 BIBLIOGRAPHY 51 üi LIST OF FIGURES Figure Page 1. A Proper Analytic Surface 5 2. The Locus of the Tangent Lines to Any Curve at a Point Px of a Surface 10 3. The Developable Surface 14 4. The Non-developable Ruled Surface 19 5. The Transformed Ruled Surface 27a CHAPTER I INTRODUCTION 1. -
On the Computation of Foldings
On the Computation of Foldings The process of determining the development (or net) of a polyhedron or of a developable surface is called unfolding and has a unique result, apart from the placement of different components in the plane. The reverse process called folding is much more complex. In the case of polyhedra it H. Stachel leads to a system of algebraic equations. A given development can Professor emeritus Institute of Discrete Mathematics correspond to several or even to infinitely many incongruent polyhedra. and Geometry The same holds also for smooth surfaces. In the paper two examples of Vienna University of Technology such foldings are presented. Austria In both cases the spatial realisations bound solids, for which mathe- matical models are required. In the first example, the cylinders with curved creases are given. In this case the involved curves can be exactly described. In the second example, even the ruling of the involved developable surface is unknown. Here, the obtained model is only an approximation. Keywords: folding, curved folding, developable surfaces, revolute surfaces of constant curvature. 1. UNFOLDING AND FOLDING surface Φ is unique, apart from the placement of the components in the plane. The unfolding induces an In Descriptive Geometry there are standard procedures isometry Φ → Φ 0 : each curve c on Φ has the same available for the construction of the development ( net length as its planar counterpart c ⊂ Φ . Hence, the or unfolding ) of polyhedra or piecewise linear surfaces, 0 0 i.e., polyhedral structures. The same holds for development shows in the plane the interior metric of developable smooth surfaces. -
On the Motion of the Oloid Toy
On the motion of the Oloid toy On the motion of the Oloid toy Alexander S. Kuleshov Mont Hubbard Dale L. Peterson Gilbert Gede [email protected] Abstract Analysis and simulation are performed for the commercially available toy known as the Oloid. While rolling on the fixed horizontal plane the Oloid moves very swinging but smooth: it never falls over its edges. The trajectories of points of contact of the Oloid with the supporting plane are found analytically. 1 Introduction Let us consider the motion of the Oloid on a fixed horizontal plane. The Oloid is a developable surface comprises of two circles of radius R whose planes of symmetry make a right angle between each other with the distance between the centers of the circles equals to their radius R. The resulting convex hull is called Oloid. The Oloid have been constructed for the first time by Paul Schatz [2,3]. The geometric properties of the surface of the Oloid have been discussed in the paper [1]. The Oloid is also used for technical applications. Special mixing-machines are constructed using such bodies [4]. In our paper we make the complete kinematical analysis of motion of this object on the horizontal plane. Further we briefly describe basic facts from Kinematics and Differential Geometry which we will use in our investigation. The Frenet - Serret formulas. Consider a particle which moves along a continuous differentiable curve in three - dimensional Euclidean Space 3. We can introduce the R following coordinate system: the origin of this system is in the moving particle, τ is the unit vector tangent to the curve, pointing in the direction of motion, ν is the derivative of τ with respect to the arc-length parameter of the curve, divided by its length and β is the cross product of τ and ν: β = [τ ν]. -
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. -
The Polycons: the Sphericon (Or Tetracon) Has Found Its Family
The polycons: the sphericon (or tetracon) has found its family David Hirscha and Katherine A. Seatonb a Nachalat Binyamin Arts and Crafts Fair, Tel Aviv, Israel; b Department of Mathematics and Statistics, La Trobe University VIC 3086, Australia ARTICLE HISTORY Compiled December 23, 2019 ABSTRACT This paper introduces a new family of solids, which we call polycons, which generalise the sphericon in a natural way. The static properties of the polycons are derived, and their rolling behaviour is described and compared to that of other developable rollers such as the oloid and particular polysphericons. The paper concludes with a discussion of the polycons as stationary and kinetic works of art. KEYWORDS sphericon; polycons; tetracon; ruled surface; developable roller 1. Introduction In 1980 inventor David Hirsch, one of the authors of this paper, patented `a device for generating a meander motion' [9], describing the object that is now known as the sphericon. This discovery was independent of that of woodturner Colin Roberts [22], which came to public attention through the writings of Stewart [28], P¨oppe [21] and Phillips [19] almost twenty years later. The object was named for how it rolls | overall in a line (like a sphere), but with turns about its vertices and developing its whole surface (like a cone). It was realised both by members of the woodturning [17, 26] and mathematical [16, 20] communities that the sphericon could be generalised to a series of objects, called sometimes polysphericons or, when precision is required and as will be elucidated in Section 4, the (N; k)-icons. These objects are for the most part constructed from frusta of a number of cones of differing apex angle and height. -
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 . -
(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. -
The Extended Oloid and Its Inscribed Quadrics
The extended oloid and its inscribed quadrics Uwe B¨asel and Hans Dirnb¨ock Abstract The oloid is the convex hull of two circles with equal radius in perpen- dicular planes so that the center of each circle lies on the other circle. It is part of a developable surface which we call extended oloid. We determine the tangential system of all inscribed quadrics Qλ of the extended oloid O where λ is the system parameter. From this result we conclude parameter equations of the touching curve Cλ between O and Qλ, the edge of regression R of O, and the asymptotes of R. Properties of the curves Cλ are investigated, including the case that λ ! ±∞. The self-polar tetrahedron of the tangential system Qλ is obtained. The common generating lines of O and any ruled surface Qλ are determined. Furthermore, we derive the curves which are the images of Cλ and R when O is developed onto the plane. Mathematics Subject Classification: 51N05, 53A05 Keywords: oloid, extended oloid, developable, tangential system of quadrics, touching curve, edge of regression, self-polar tetrahedron, ruled surface 1 Introduction The oloid was discovered by Paul Schatz in 1929. It is the convex hull of two circles with equal radius r in perpendicular planes so that the center of each circle lies on the other circle. The oloid has the remarkable properties that it develops its entire surface while rolling, and its surface area is equal to 4πr2. The surface of the oloid is part of a developable surface. [2], [8] In the following this developable surface is called extended oloid. -
Surfaces in 3-Space
Differential Geometry of Some SURFACES IN 3-SPACE Nicholas Wheeler December 2015 Introduction. Recent correspondence with Ahmed Sebbar concerning the theory of unimodular 3 3 circulant matrices1 × x y z det z x y = x3 + y3 + z3 3xyz = 1 y z x − brought to my attention a surface Σ in R3 which, I was informed, is encountered in work of H. Jonas (1915, 1921) and, because of its form when plotted, is known as “Jonas’ hexenhut” (witch’s hat). I was led by Google from “hexenhut” to a monograph B¨acklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, by C. Rogers & W. K. Schief (2002). These are subjects in which I have had longstanding interest, but which I have not thought about for many years. I am inspired by those authors’ splendid book to revisit this subject area. In part one I assemble the tools that play essential roles in the theory of surfaces in R3, and in part two use those tools to develop the properties of some specific surfaces—particularly the pseudosphere, because it was the cradle in which was born the sine-Gordon equation, which a century later became central to the physical theory of solitons.2 part one Concepts & Tools Essential to the Theory of Surfaces in 3-Space Fundamental forms. Relative to a Cartesian frame in R3, surfaces Σ can be described implicitly f(x, y, z) = 0 but for the purposes of differential geometry must be described parametrically x(u, v) r(u, v) = y(u, v) z(u, v) 1 See “Simplest generalization of Pell’s Problem,” (September, 2015).