Developable Surfaces in Euclidean Space
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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. -
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. -
An Introduction to Topology the Classification Theorem for Surfaces by E
An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5. -
Parametric Modelling of Architectural Developables Roel Van De Straat Scientific Research Mentor: Dr
MSc thesis: Computation & Performance parametric modelling of architectural developables Roel van de Straat scientific research mentor: dr. ir. R.M.F. Stouffs design research mentor: ir. F. Heinzelmann third mentor: ir. J.L. Coenders Computation & Performance parametric modelling of architectural developables MSc thesis: Computation & Performance parametric modelling of architectural developables Roel van de Straat 1041266 Delft, April 2011 Delft University of Technology Faculty of Architecture Computation & Performance parametric modelling of architectural developables preface The idea of deriving analytical and structural information from geometrical complex design with relative simple design tools was one that was at the base of defining the research question during the early phases of the graduation period, starting in September of 2009. Ultimately, the research focussed an approach actually reversely to this initial idea by concentrating on using analytical and structural logic to inform the design process with the aid of digital design tools. Generally, defining architectural characteristics with an analytical approach is of increasing interest and importance with the emergence of more complex shapes in the building industry. This also means that embedding structural, manufacturing and construction aspects early on in the design process is of interest. This interest largely relates to notions of surface rationalisation and a design approach with which initial design sketches can be transferred to rationalised designs which focus on a strong integration with manufacturability and constructability. In order to exemplify this, the design of the Chesa Futura in Sankt Moritz, Switzerland by Foster and Partners is discussed. From the initial design sketch, there were many possible approaches for surfacing techniques defining the seemingly freeform design. -
Section 2.6 Cylindrical and Spherical Coordinates
Section 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in the plane can be uniquely described by its distance to the origin r = dist (P, O) and the angle µ, 0 µ < 2¼ : · Y P(x,y) r θ O X We call (r, µ) the polar coordinate of P. Suppose that P has Cartesian (stan- dard rectangular) coordinate (x, y) .Then the relation between two coordinate systems is displayed through the following conversion formula: x = r cos µ Polar Coord. to Cartesian Coord.: y = r sin µ ½ r = x2 + y2 Cartesian Coord. to Polar Coord.: y tan µ = ( p x 0 µ < ¼ if y > 0, 2¼ µ < ¼ if y 0. · · · Note that function tan µ has period ¼, and the principal value for inverse tangent function is ¼ y ¼ < arctan < . ¡ 2 x 2 1 So the angle should be determined by y arctan , if x > 0 xy 8 arctan + ¼, if x < 0 µ = > ¼ x > > , if x = 0, y > 0 < 2 ¼ , if x = 0, y < 0 > ¡ 2 > > Example 6.1. Fin:>d (a) Cartesian Coord. of P whose Polar Coord. is ¼ 2, , and (b) Polar Coord. of Q whose Cartesian Coord. is ( 1, 1) . 3 ¡ ¡ ³ So´l. (a) ¼ x = 2 cos = 1, 3 ¼ y = 2 sin = p3. 3 (b) r = p1 + 1 = p2 1 ¼ ¼ 5¼ tan µ = ¡ = 1 = µ = or µ = + ¼ = . 1 ) 4 4 4 ¡ 5¼ Since ( 1, 1) is in the third quadrant, we choose µ = so ¡ ¡ 4 5¼ p2, is Polar Coord. -
Area, Volume and Surface Area
The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 YEARS 810 Area, Volume and Surface Area (Measurement and Geometry: Module 11) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 810 {4} A guide for teachers AREA, VOLUME AND SURFACE AREA ASSUMED KNOWLEDGE • Knowledge of the areas of rectangles, triangles, circles and composite figures. • The definitions of a parallelogram and a rhombus. • Familiarity with the basic properties of parallel lines. • Familiarity with the volume of a rectangular prism. • Basic knowledge of congruence and similarity. • Since some formulas will be involved, the students will need some experience with substitution and also with the distributive law. -
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. -
Developability of Triangle Meshes
Developability of Triangle Meshes ODED STEIN and EITAN GRINSPUN, Columbia University KEENAN CRANE, Carnegie Mellon University Fig. 1. By encouraging discrete developability, a given mesh evolves toward a shape comprised of flattenable pieces separated by highly regular seam curves. Developable surfaces are those that can be made by smoothly bending flat pieces—most industrial applications still rely on manual interaction pieces without stretching or shearing. We introduce a definition of devel- and designer expertise [Chang 2015]. opability for triangle meshes which exactly captures two key properties of The goal of this paper is to develop mathematical and computa- smooth developable surfaces, namely flattenability and presence of straight tional foundations for developability in the simplicial setting, and ruling lines. This definition provides a starting point for algorithms inde- show how this perspective inspires new approaches to developable velopable surface modeling—we consider a variational approach that drives design. Our starting point is a new definition of developability for a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient descent on an energy with support in the triangle meshes (Section 3). This definition is motivated by the be- vertex star, without the need to explicitly cluster patches or identify seams. havior of developable surfaces that are twice differentiable rather We briefly explore applications to developable design and manufacturing. than those that are merely continuous: instead of just asking for flattenability, we also seek a definition that naturally leads towell- CCS Concepts: • Mathematics of computing → Discretization; • Com- ruling lines puting methodologies → Mesh geometry models; defined . Moreover, unlike existing notions of discrete developability, it applies to general triangulated surfaces, with no Additional Key Words and Phrases: developable surface modeling, discrete special conditions on combinatorics. -
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. -
Analytic Geometry
STATISTIC ANALYTIC GEOMETRY SESSION 3 STATISTIC SESSION 3 Session 3 Analytic Geometry Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres Point, Line, Plane and Solid A Point has no dimensions, only position A Line is one-dimensional A Plane is two dimensional (2D) A Solid is three-dimensional (3D) Plane Geometry Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper). 2D Shapes Activity: Sorting Shapes Triangles Right Angled Triangles Interactive Triangles Quadrilaterals (Rhombus, Parallelogram, etc) Rectangle, Rhombus, Square, Parallelogram, Trapezoid and Kite Interactive Quadrilaterals Shapes Freeplay Perimeter Area Area of Plane Shapes Area Calculation Tool Area of Polygon by Drawing Activity: Garden Area General Drawing Tool Polygons A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons. Here are some more: Pentagon Pentagra m Hexagon Properties of Regular Polygons Diagonals of Polygons Interactive Polygons The Circle Circle Pi Circle Sector and Segment Circle Area by Sectors Annulus Activity: Dropping a Coin onto a Grid Circle Theorems (Advanced Topic) Symbols There are many special symbols used in Geometry. Here is a short reference for you: -
Lecture 20 Dr. KH Ko Prof. NM Patrikalakis
13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 20 Dr. K. H. Ko Prof. N. M. Patrikalakis Copyrightc 2003Massa chusettsInstitut eo fT echnology ≤ Contents 20 Advanced topics in differential geometry 2 20.1 Geodesics ........................................... 2 20.1.1 Motivation ...................................... 2 20.1.2 Definition ....................................... 2 20.1.3 Governing equations ................................. 3 20.1.4 Two-point boundary value problem ......................... 5 20.1.5 Example ........................................ 8 20.2 Developable surface ...................................... 10 20.2.1 Motivation ...................................... 10 20.2.2 Definition ....................................... 10 20.2.3 Developable surface in terms of B´eziersurface ................... 12 20.2.4 Development of developable surface (flattening) .................. 13 20.3 Umbilics ............................................ 15 20.3.1 Motivation ...................................... 15 20.3.2 Definition ....................................... 15 20.3.3 Computation of umbilical points .......................... 15 20.3.4 Classification ..................................... 16 20.3.5 Characteristic lines .................................. 18 20.4 Parabolic, ridge and sub-parabolic points ......................... 21 20.4.1 Motivation ...................................... 21 20.4.2 Focal surfaces ..................................... 21 20.4.3 Parabolic points ................................... 22 -
Title CLAD HELICES and DEVELOPABLE SURFACES( Fulltext )
CLAD HELICES AND DEVELOPABLE SURFACES( Title fulltext ) Author(s) TAKAHASHI,Takeshi; TAKEUCHI,Nobuko Citation 東京学芸大学紀要. 自然科学系, 66: 1-9 Issue Date 2014-09-30 URL http://hdl.handle.net/2309/136938 Publisher 東京学芸大学学術情報委員会 Rights Bulletin of Tokyo Gakugei University, Division of Natural Sciences, 66: pp.1~ 9 ,2014 CLAD HELICES AND DEVELOPABLE SURFACES Takeshi TAKAHASHI* and Nobuko TAKEUCHI** Department of Mathematics (Received for Publication; May 23, 2014) TAKAHASHI, T and TAKEUCHI, N.: Clad Helices and Developable Surfaces. Bull. Tokyo Gakugei Univ. Div. Nat. Sci., 66: 1-9 (2014) ISSN 1880-4330 Abstract We define new special curves in Euclidean 3-space which are generalizations of the notion of helices. Then we find a geometric invariant of a space curve which is related to the singularities of the special developable surface of the original curve. Keywords: cylindrical helices, slant helices, clad helices, g-clad helices, developable surfaces, singularities Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Koganei-shi, Tokyo 184-8501, Japan 1. Introduction In this paper we define the notion of clad helices and g-clad helices which are generalizations of the notion of helices. Then we can find them as geodesics on the tangent developable(cf., §3). In §2 we describe basic notions and properties of space curves. We review the classification of singularities of the Darboux developable of a space curve in §4. We introduce the notion of the principal normal Darboux developable of a space curve. Then we find a geometric invariant of a clad helix which is related to the singularities of the principal normal Darboux developable of the original curve.