DOCSLIB.ORG

Involute-Evolute Curve Couples in the Euclidean 4-Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Involute-Evolute Curve Couples in the Euclidean 4-Space Int. J. Open Problems Compt. Math., Vol. 2, No. 2, June 2009 Involute-Evolute Curve Couples in the Euclidean 4-Space Emin Özyılmaz1 and Süha Yılmaz2 1Ege University, Faculty of Science, Dept. of Math., Bornova-Izmir, Turkey. 2Dokuz Eylül University, Buca Educational Faculty, Buca-Izmir, Turkey. E-mail: [email protected], [email protected] Abstract In this work, for regular involute-evolute curve couples, it is proven that evolute’s Frenet apparatus can be formed by involute’s apparatus in four dimensional Euclidean space when involute curve has constant Frenet curvatures. By this way, another orthonormal frame of the same space is obtained via the method expressed in [5]. Moreover, it is observed that evolute curve cannot be an inclined curve. In an analogous way, we use method of [6]. Keywords: Classical Differential Geometry, Euclidean space, Involute- Evolute curve couples, Inclined curves. 1 Introduction The idea of a string involute is due to C. Huygens (1658), who is also known for his work in optics. He discovered involutes while trying to build a more accurate clock (see [1]). The involute of a given curve is a well-known concept in Euclidean-3 space E 3 (see [3]). It is well-known that, if a curve differentiable in an open interval, at each point, a set of mutually orthogonal unit vectors can be constructed. And these vectors are called Frenet frame or moving frame vectors. The rates of these frame vectors along the curve define curvatures of the curves. The set, whose elements are frame vectors and curvatures of a curve, is called Frenet apparatus of the curves. In recent years, the theory of degenerate submanifolds has been treated by researchers and some classical differential geometry topics have been extended to 169 Involute-Evolute Curve Couples Lorentz manifolds. For instance, in [6], the authors extended and studied space- like involute-evolute curves in Minkowski space-time. In the presented paper, in an analogous way as in [6], we calculate Frenet apparatus of the evolute curve by apparatus of involute W-curve. We use the method expressed in [5]. Thereafter, we prove that evolute of an involute cannot be an inclined curve in four dimensional Euclidean space. 2 Preliminaries To meet the requirements in the next sections, here, the basic elements of the theory of curves in the space E 4 are briefly presented (A more complete elementary treatment can be found in [3]). G Let α : I ⊂ R → E 4 be an arbitrary curve in the Euclidean space E 4 . G Recall that the curve α is said to be of unit speed (or parametrized by arclength G G function s ) if α ′,α ′ =1, where .,. is the standard scalar (inner) product of E 4 given by G G a,b = a1b1 + a2b2 + a3b3 + a4b4 (1) G G 4 for each a = (a1 , a2 , a3 , a4 ) , b = (b1 ,b2 ,b3 ,b4 )∈ E . In particular, the norm of a G G G G vector a ∈ E 4 is given by a = a, a . G G G G Denote by {T (s), N(s), B(s), E(s)} the moving Frenet frame along the unit G speed curve α . Then the Frenet formulas are given by (see [2]) G G ⎡T ′ ⎤ ⎡ 0 κ 0 0⎤⎡T ⎤ ⎢ G ⎥ ⎢ G ⎥ ′ ⎢ ⎥ ⎢N ⎥ ⎢−κ 0 τ 0⎥⎢N⎥ ⎢ G ⎥ = ⎢ G ⎥ . (2) B′ ⎢ 0 −τ 0 σ ⎥ B ⎢ G ⎥ ⎢ ⎥⎢ G ⎥ ⎢ ⎥ 0 0 −σ 0 ⎢ ⎥ ⎣E′⎦ ⎣ ⎦⎣E ⎦ G G G G Here T,N, B andE are called, respectively, the tangent, the normal, the binormal and the trinormal vector fields of the curves. And the functions κ(s),τ (s) and σ (s) are called, respectively, the first, the second and the third curvature of the G curveα . And, recall that a regular curve is called a W-curve if it has constant G Frenet curvatures [4]. Let α : I ⊂ R → E 4 be a regular curve. If tangent vector G G G field T of α forms a constant angle with unit vector U , this curve is called an G G G inclined curve in E 4 . Let φ and ξ be unit speed regular curves in E 4 φ is an E min Özyılmaz and Süha Yılmaz 170 G G G G G involute of ξ if φ lies on the tangent line to ξ at ξ (s ) and the tangents to ξ , G G G 0 G G ξ (s ) and φ are perpendicular for each s . φ is an evolute of ξ if ξ is an 0 G 0 G G G involute of φ . And this curve couple defined by φ = ξ + µ T. In the same space, in [5], author presented a vector product and a method to determine Frenet apparatus of the regular curves as follows. G G Definition 2.1 Let a = (a , a , a , a ), b = (b ,b ,b ,b ) and 1 2 3 4 G1 2 3 4 G 4 G G c = (c1 , c2 , c3 , c4 ) be vectors in E .The vector product of a, b and c is defined by the determinant G G G G e1 e2 e3 e4 G G G a a a a a ∧ b ∧ c = 1 2 3 4 , b1 b2 b3 b4 c1 c2 c3 c4 where G G G G G G G G e ∧ e ∧ e = e , e ∧ e ∧ e = e , G G 1G G2 G 3 G 4 G2 G3 G 4 G 1 G G e3 ∧ e4 ∧ e1 = e2 ,,e4 ∧ e1 ∧ e2 = e3 e3 ∧ e2 ∧ e1 = e4 . G G Theorem 2.2 Let α = α(t) be an arbitrary curve in E4 . Frenet apparatus G of the α can be calculated by the following equations. G G α ′ T = G , (3) α ′ G 2 G G G G G α ′ α ′′ − α ′,α ′′ α ′ N = , (4) G 2 G G G G α ′ α ′′ − α ′,α ′′ α ′ G G G G B =η E ∧T ∧ N, (5) G G G G T ∧ N ∧α ′′′ E =η G G G , (6) T ∧ N ∧α ′′′ 171 Involute-Evolute Curve Couples G 2 G G G G α ′ α ′′ − α ′,α ′′ α ′ κ = , (7) G 4 α ′ G G G G T ∧ N ∧α ′′′ α ′ τ = , (8) G 2 G G G G α ′ α ′′ − α ′,α ′′ α ′ G G α (IV ) , E σ = G G G G (9) T ∧ N ∧ α ′′′ α ′ G G G G where η is taken ±1 to make 1 determinant of [T, N, B, E]. matrix. 3 Main Results G G G Theorem 3.1 Let φ be involute of ξ and ξ be a W-curve in E 4 . The Frenet G G G G G G apparatus of φ ({Tφ , Nφ , Bφ , Eφ ,κ φ ,τ φ ,σ φ }) can be formed by apparatus of ξ G G G G ({T, N, B, E,κ,τ ,σ }). Proof From definition of involute-evolute curve couples we know G G G φ = ξ + µ T. (10) Differentiating both sides with respect to s , we obtain G dφ dsφ G dµ G G = T + T + µκN. (11) ds ds ds φ G G Definition of such kind curves yields [3], Tφ ⊥ T. Therefore, we have dµ + 1 = 0. (12) ds Hence µ = c − s. Then, we write G G G φ = ξ + (c − s)T (13) and G ds G T φ = (c − s)κN. (14) φ ds Taking the norm of both sides (here . denotes derivative according to s ) G G Tφ = N (15) and G φ = (c − s)κ. (16) E min Özyılmaz and Süha Yılmaz 172 Considering the presented method, we calculate differentiations of (14) four times. We write, respectively, G G G G φ = −κ 2 (c − s)T −κN +κτ(c − s)B. (17) G G G G φ = {}− 2κ 2T − κ(c − s)[κ 2 +τ 2 ]N − 2κτB + κτσ(c − s)E . (18) ....G G G G G φ = {}κ 2 (c − s)[][]κ 2 +τ 2 T − 2κ κ 2 +τ 2 N −κτ(c − s) [κ 2 +τ 2 +σ 2 ]B − 2κτσE . (19) Considering (4), we form G 2 G G G G G G φ φ − φ ,φ φ == κ(c − s)[− κT +τB]. (20) Since, we have the principal normal and theG firstG curvature of the curve G − κT + τB Nφ = (21) κ 2 + τ 2 and κ 2 + τ 2 κ φ = . (22) [](c − s)κ 3 G G G Using vector product of Tφ ∧ Nφ ∧ φ , we get G G G G T N B E G G G 0 1 0 0 T ∧ N ∧ φ = κ τ φ φ − 0 0 (23) κ 2 +τ 2 κ 2 +τ 2 φ1 φ2 φ3 φ4 κτ G G G = []τσ (c − s)T + κσ(c − s)B + 3κE . κ 2 +τ 2 We obtain trinormal vector G G G G τσ (c − s)T +κσ(c − s)B + 3κE Eφ =η . (24) [][]τσ (c − s) 2 + κσ (c − s) 2 + 9κ 2 By this way, we easily have the second curvature []τσ (c − s) 2 + [κσ (c − s)]2 + 9κ 2 τ = . (25) φ κ 2 +τ 2 G Considering (9) and (24), we have the third curvature of the curve φ as 173 Involute-Evolute Curve Couples −κ 2τσ [σ 2 (c − s) 2 − 3] σ φ = . (26) [][]τσ (c − s) 2 + κσ (c − s) 2 + 9κ 2 G G G Finally, the vector product of Eφ ∧ Tφ ∧ Nφ follows that G G G G η 2 2 2 Bφ = {−3κτ T −3κ B +σ (c − s)(κ +τ )E}, (27) A.
Load more

Recommended publications
  • Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3-Space E3
    Fundamentals of Contemporary Mathematical Sciences (2020) 1(2) 63 { 70 Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3-space E3 Abdullah Yıldırım 1,∗ Feryat Kaya 2 1 Harran University, Faculty of Arts and Sciences, Department of Mathematics S¸anlıurfa, T¨urkiye 2 S¸ehit Abdulkadir O˘guzAnatolian Imam Hatip High School S¸anlıurfa, T¨urkiye, [email protected] Received: 29 February 2020 Accepted: 29 June 2020 Abstract: In this study, evolute-involute curves are researched. Characterization of evolute-involute curves lying on the surface are examined according to Darboux frame and some curves are obtained. Keywords: Curve, surface, geodesic, curvature, frame. 1. Introduction The interest of special curves has increased recently. Some of these are associated curves. They are curves where one of the Frenet vectors at opposite points is linearly dependent to the other curve. One of the best examples of these curves is the evolute-involute partner curves. An involute thought known to have been used in his optical work came up in 1658 by C. Huygens. C. Huygens discovered involute curves while trying to make more accurate measurement studies [5]. Many researches have been conducted about evolute-involute partner curves. Some of them conducted recently are Bilici and C¸alı¸skan [4], Ozyılmaz¨ and Yılmaz [9], As and Sarıo˘glugil[2]. Bekta¸sand Y¨uceconsider the notion of the involute-evolute curves lying on the surfaces for a special situation. They determine the special involute-evolute partner D−curves in E3: By using the Darboux frame of the curves they obtain the necessary and sufficient conditions between κg , ∗ − ∗ ∗ τg; κn and κn for a curve to be the special involute partner D curve.
    [Show full text]
  • Computer-Aided Design and Kinematic Simulation of Huygens's
    applied sciences Article Computer-Aided Design and Kinematic Simulation of Huygens’s Pendulum Clock Gloria Del Río-Cidoncha 1, José Ignacio Rojas-Sola 2,* and Francisco Javier González-Cabanes 3 1 Department of Engineering Graphics, University of Seville, 41092 Seville, Spain; [email protected] 2 Department of Engineering Graphics, Design, and Projects, University of Jaen, 23071 Jaen, Spain 3 University of Seville, 41092 Seville, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-953-212452 Received: 25 November 2019; Accepted: 9 January 2020; Published: 10 January 2020 Abstract: This article presents both the three-dimensional modelling of the isochronous pendulum clock and the simulation of its movement, as designed by the Dutch physicist, mathematician, and astronomer Christiaan Huygens, and published in 1673. This invention was chosen for this research not only due to the major technological advance that it represented as the first reliable meter of time, but also for its historical interest, since this timepiece embodied the theory of pendular movement enunciated by Huygens, which remains in force today. This 3D modelling is based on the information provided in the only plan of assembly found as an illustration in the book Horologium Oscillatorium, whereby each of its pieces has been sized and modelled, its final assembly has been carried out, and its operation has been correctly verified by means of CATIA V5 software. Likewise, the kinematic simulation of the pendulum has been carried out, following the approximation of the string by a simple chain of seven links as a composite pendulum. The results have demonstrated the exactitude of the clock.
    [Show full text]
  • Variation Analysis of Involute Spline Tooth Contact
    Brigham Young University BYU ScholarsArchive Theses and Dissertations 2006-02-22 Variation Analysis of Involute Spline Tooth Contact Brian J. De Caires Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mechanical Engineering Commons BYU ScholarsArchive Citation De Caires, Brian J., "Variation Analysis of Involute Spline Tooth Contact" (2006). Theses and Dissertations. 375. https://scholarsarchive.byu.edu/etd/375 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. VARIATION ANALYSIS OF INVOLUTE SPLINE TOOTH CONTACT By Brian J. K. DeCaires A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Mechanical Engineering Brigham Young University April 2006 Copyright © 2006 Brian J. K. DeCaires All Rights Reserved BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a thesis submitted by Brian J. K. DeCaires This thesis has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. ___________________________ ______________________________________ Date Kenneth W. Chase, Chair ___________________________ ______________________________________ Date Carl D. Sorensen ___________________________
    [Show full text]
  • Interactive Involute Gear Analysis and Tooth Profile Generation Using Working Model 2D
    AC 2008-1325: INTERACTIVE INVOLUTE GEAR ANALYSIS AND TOOTH PROFILE GENERATION USING WORKING MODEL 2D Petru-Aurelian Simionescu, University of Alabama at Birmingham Petru-Aurelian Simionescu is currently an Assistant Professor of Mechanical Engineering at The University of Alabama at Birmingham. His teaching and research interests are in the areas of Dynamics, Vibrations, Optimal design of mechanical systems, Mechanisms and Robotics, CAD and Computer Graphics. Page 13.781.1 Page © American Society for Engineering Education, 2008 Interactive Involute Gear Analysis and Tooth Profile Generation using Working Model 2D Abstract Working Model 2D (WM 2D) is a powerful, easy to use planar multibody software that has been adopted by many instructors teaching Statics, Dynamics, Mechanisms, Machine Design, as well as by practicing engineers. Its programming and import-export capabilities facilitate simulating the motion of complex shape bodies subject to constraints. In this paper a number of WM 2D applications will be described that allow students to understand the basics properties of involute- gears and how they are manufactured. Other applications allow students to study the kinematics of planetary gears trains, which is known to be less intuitive than that of fix-axle transmissions. Introduction There are numerous reports on the use of Working Model 2D in teaching Mechanical Engineering disciplines, including Statics, Dynamics, Mechanisms, Vibrations, Controls and Machine Design1-9. Working Model 2D (WM 2D), currently available form Design Simulation Technologies10, is a planar multibody software, capable of performing kinematic and dynamic simulation of interconnected bodies subject to a variety of constraints. The versatility of the software is given by its geometry and data import/export capabilities, and scripting through formula and WM Basic language system.
    [Show full text]
  • The Cycloid Scott Morrison
    The cycloid Scott Morrison “The time has come”, the old man said, “to talk of many things: Of tangents, cusps and evolutes, of curves and rolling rings, and why the cycloid’s tautochrone, and pendulums on strings.” October 1997 1 Everyone is well aware of the fact that pendulums are used to keep time in old clocks, and most would be aware that this is because even as the pendu- lum loses energy, and winds down, it still keeps time fairly well. It should be clear from the outset that a pendulum is basically an object moving back and forth tracing out a circle; hence, we can ignore the string or shaft, or whatever, that supports the bob, and only consider the circular motion of the bob, driven by gravity. It’s important to notice now that the angle the tangent to the circle makes with the horizontal is the same as the angle the line from the bob to the centre makes with the vertical. The force on the bob at any moment is propor- tional to the sine of the angle at which the bob is currently moving. The net force is also directed perpendicular to the string, that is, in the instantaneous direction of motion. Because this force only changes the angle of the bob, and not the radius of the movement (a pendulum bob is always the same distance from its fixed point), we can write: θθ&& ∝sin Now, if θ is always small, which means the pendulum isn’t moving much, then sinθθ≈. This is very useful, as it lets us claim: θθ&& ∝ which tells us we have simple harmonic motion going on.
    [Show full text]
  • A Tale of the Cycloid in Four Acts
    A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries.
    [Show full text]
  • Mathematics, the Language of Watchmaking
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE 90LEARNINGprovidedL by Infoscience E- École polytechniqueA fédérale de LausanneRNINGLEARNINGLEARNING Mathematics, the language of watchmaking Morley’s theorem (1898) states that if you trisect the angles of any triangle and extend the trisecting lines until they meet, the small triangle formed in the centre will always be equilateral. Ilan Vardi 1 I am often asked to explain mathematics; is it just about numbers and equations? The best answer that I’ve found is that mathematics uses numbers and equations like a language. However what distinguishes it from other subjects of thought – philosophy, for example – is that in maths com- plete understanding is sought, mostly by discover- ing the order in things. That is why we cannot have real maths without formal proofs and why mathe- maticians study very simple forms to make pro- found discoveries. One good example is the triangle, the simplest geometric shape that has been studied since antiquity. Nevertheless the foliot world had to wait 2,000 years for Morley’s theorem, one of the few mathematical results that can be expressed in a diagram. Horology is of interest to a mathematician because pallet verge it enables a complete understanding of how a watch or clock works. His job is to impose a sequence, just as a conductor controls an orches- tra or a computer’s real-time clock controls data regulating processing. Comprehension of a watch can be weight compared to a violin where science can only con- firm the preferences of its maker.
    [Show full text]
  • Parametric Equations and Polar Coordinates
    10 PARAMETRIC EQUATIONS AND POLAR COORDINATES Parametric equations and polar coordinates enable us to describe a great variety of new curves—some practical, some beautiful, some fanciful, some strange. So far we have described plane curves by giving y as a function of x ͓y ෇ f ͑x͔͒ or x as a function of y ͓x ෇ t͑y͔͒ or by giving a relation between x and y that defines y implicitly as a function of x ͓ f ͑x, y͒ ෇ 0͔. In this chapter we discuss two new methods for describing curves. Some curves, such as the cycloid, are best handled when both x and y are given in terms of a third variable t called a parameter ͓x ෇ f ͑t͒, y ෇ t͑t͔͒ . Other curves, such as the cardioid, have their most convenient description when we use a new coordinate system, called the polar coordinate system. 620 10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS y Imagine that a particle moves along the curve C shown in Figure 1. It is impossible to C describe C by an equation of the form y ෇ f ͑x͒ because C fails the Vertical Line Test. But (x, y)={f(t), g(t)} the x- and y-coordinates of the particle are functions of time and so we can write x ෇ f ͑t͒ and y ෇ t͑t͒. Such a pair of equations is often a convenient way of describing a curve and gives rise to the following definition. Suppose that x and y are both given as functions of a third variable t (called a param- 0 x eter ) by the equations x ෇ f ͑t͒ y ෇ t͑t͒ FIGURE 1 (called parametric equations ).
    [Show full text]
  • Cycloid Article(Final04)
    The Helen of Geometry John Martin The seventeenth century is one of the most exciting periods in the history of mathematics. The first half of the century saw the invention of analytic geometry and the discovery of new methods for finding tangents, areas, and volumes. These results set the stage for the development of the calculus during the second half. One curve played a central role in this drama and was used by nearly every mathematician of the time as an example for demonstrating new techniques. That curve was the cycloid. The cycloid is the curve traced out by a point on the circumference of a circle, called the generating circle, which rolls along a straight line without slipping (see Figure 1). It has been called it the “Helen of Geometry,” not just because of its many beautiful properties but also for the conflicts it engendered. Figure 1. The cycloid. This article recounts the history of the cycloid, showing how it inspired a generation of great mathematicians to create some outstanding mathematics. This is also a story of how pride, pettiness, and jealousy led to bitter disagreements among those men. Early history Since the wheel was invented around 3000 B.C., it seems that the cycloid might have been discovered at an early date. There is no evidence that this was the case. The earliest mention of a curve generated by a -1-(Final) point on a moving circle appears in 1501, when Charles de Bouvelles [7] used such a curve in his mechanical solution to the problem of squaring the circle.
    [Show full text]
  • Simulating High Flux Isotope Reactor Core Thermal-Hydraulics Via Interdimensional Model Coupling
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 5-2014 Simulating High Flux Isotope Reactor Core Thermal-Hydraulics via Interdimensional Model Coupling Adam Ross Travis University of Tennessee - Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Mechanical Engineering Commons Recommended Citation Travis, Adam Ross, "Simulating High Flux Isotope Reactor Core Thermal-Hydraulics via Interdimensional Model Coupling. " Master's Thesis, University of Tennessee, 2014. https://trace.tennessee.edu/utk_gradthes/2759 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Adam Ross Travis entitled "Simulating High Flux Isotope Reactor Core Thermal-Hydraulics via Interdimensional Model Coupling." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mechanical Engineering. Kivanc Ekici, Major Professor We have read this thesis and recommend its acceptance: Jay Frankel, Rao Arimilli Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Simulating High Flux Isotope Reactor Core Thermal-Hydraulics via Interdimensional Model Coupling A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville Adam Ross Travis May 2014 Copyright © 2014 by Adam Ross Travis All rights reserved.
    [Show full text]
  • Differential Geometry of Curves and Surfaces 1
    DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 1. Curves in the Plane 1.1. Points, Vectors, and Their Coordinates. Points and vectors are fundamental objects in Geometry. The notion of point is intuitive and clear to everyone. The notion of vector is a bit more delicate. In fact, rather than saying what a vector is, we prefer to say what a vector has, namely: direction, sense, and length (or magnitude). It can be represented by an arrow, and the main idea is that two arrows represent the same vector if they have the same direction, sense, and length. An arrow representing a vector has a tail and a tip. From the (rough) definition above, we deduce that in order to represent (if you want, to draw) a given vector as an arrow, it is necessary and sufficient to prescribe its tail. a c b b a c a a b P Figure 1. We see four copies of the vector a, three of the vector b, and two of the vector c. We also see a point P . An important instrument in handling points, vectors, and (consequently) many other geometric objects is the Cartesian coordinate system in the plane. This consists of a point O, called the origin, and two perpendicular lines going through O, called coordinate axes. Each line has a positive direction, indicated by an arrow (see Figure 2). We denote by R the a P y a y x O O x Figure 2. The point P has coordinates x, y. The vector a has also coordinates x, y.
    [Show full text]
  • Some Philowphical Prehistory of General Relativity
    - ----HOWARD STEI N ----- Some Philowphical Prehistory of General Relativity As history, my remarks will form rather a medley. If they can claim any sort of unity (apart from a general concern with space and time), it derives from two or three philosophi<lal motifs: the notion- metaphysical, if you will-of structure in the world, or vera causa; 1 the epistemological princi­ ple of the primacy ofexperience , as touchstone of both the content and the admissibility of knowledge claims; and a somewhat delicate issue of scien­ tific method that arises from the confrontation of that notion and that principle. The historical figures to be touched on are Leibniz, Huygens, Newton; glancingly, Kant; Mach , Helmholtz, and Riemann. The heroes of my story are Newton and Riemann, who seem to me to have expressed (although laconically) the clearest and the deepest views of the matters concerned. The story has no villains; but certain attributions often made to the credit of Leibniz and of Mach will come under criticism. It is well known that Leibniz denied, in some sense, to space the status of a vera ctwsa. Jn what precise sense he intended the denial is perhaps less well known; indeed, as I shall soon explain, I myself consider that sense in some respects difficult if not impossible to determine. The fact !hat Leibniz characterizes space as not "real" but "ideal," or as an "entity of reason" or abstraction, by itself decides nothing; for he also tells us that lhc structure thus abstracted is the structure--or, as he puts it, the "order"- of the situations of coexistent things; furthermore, of the situa­ liuns of all actual or 11ossible coexistent things; or, again (in the fourth lctlcr lo Clarke).
    [Show full text]