DOCSLIB.ORG

Superellipse (Lamé Curve)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Superellipse (Lamé Curve) 6 Superellipse (Lamé curve) 6.1 Equations of a superellipse A superellipse (horizontally long) is expressed as follows. Implicit Equation x n y n + = 10<b a , n > 0 (1.1) a b Explicit Equation 1 n x n y = b 1 - 0<b a , n > 0 (1.1') a When a =3, b=2 , the superellipses for n = 9.9 , 2, 1 and 2/3 are drawn as follows. Fig1 These are called an ellipse when n= 2, are called a diamond when n = 1, and are called an asteroid when n = 2/3. These are known well. parametric representation of an ellipse In order to ask for the area and the arc length of a super-ellipse, it is necessary to calculus the equations. However, it is difficult for (1.1) and (1.1'). Then, in order to make these easy, parametric representation of (1.1) is often used. As far as the 1st quadrant, (1.1) is as follows . xn yn 0xa + = 1 a n b n 0yb Let us compare this with the following trigonometric equation. sin 2 + cos 2 = 1 Then, we obtain xn yn = sin 2 , = cos 2 a n b n i.e. - 1 - 2 2 x = a sin n ,y = bcos n The variable and the domain are changed as follows. x: 0a : 0/2 By this representation, the area and the arc length of a superellipse can be calculated clockwise from the y-axis. cf. If we represent this as usual as follows, 2 2 x = a cos n ,y = bsin n the area and the arc length of a superellipse can be calculated counterclockwise from the x-axis. However, from this representation, we cannot obtain the Legendre forms of elliptic integrals - 2 - 6.2 Area of a superellipse Formula 6.2.1 When n,a,b ()b a are positive numbers respectively and ()z is the gamma function, the area S of the ellipse of degree n is given by the following expression. 1 2 1+ n S = 4ab 2 1+ n Proof In order to obtain the area of a superellipse, we integrate with the following equation in the 1st quadrant, and should just multiply it by 4. 1 n xn y = b 1- 0<b a , n > 0 a n That is, 1 n a xn S = 4b 1- n dx 0 a However, it is known that this integration cannot be expressed with elementary functions. Then, we use the parametric representation mentioned in the previous section. That is, 2 2 x = a sin n ,y = bcos n The variable and the domain are changed as follows. x: 0a : 0/2 And dx is 2 2a -1 dx = sin n cos d n Then, 2 2 S a 2 2a -1 = ydx = bcos n ( sin n cos)d 4 0 0 n 2 2 2 2 n -1 n +1 = ab sin cos d n 0 According to "岩波 数学公式Ⅰ" p 243 , 2 1 +1 +1 sin cos d = , , > -1 0 2 2 2 Using this, 2 2 2 2 -1+1 +1+1 2 -1 +1 1 n n 1 1 1 sin n cos n d = , = , +1 0 2 2 2 2 n n - 3 - 1 1 1 2 +1 1+ n n n n = = 1 1 2 2 2 + +1 1+ n n n Substituting this for the avove, 1 2 1 2 1+ 1+ S 2 n n n = ab = ab 4 n 2 2 2 1+ 1+ n n Quadrupling both sides, we obtain the desired expression. Note Especially when a =b =1 , 1 2 2 = 4 1+ / 1+ n n n n n n is the area of the unit circle ||x +||y =1 n > 0 , and should be called constant of the circle of degree n . That is, the area of an ellipse of degree n is given by abn , where a is the major axis and b is the minor axis. In fact, when n =2 , 1 2 1 1 2 1+ 2 2 2 1 2 = 4 = 4 = 4 = 2 2 1! 2 1+ 2 2 2 This is the area of the circle x + y = 1 , and the area of an ellipse of degree2 is given by ab . Example: Area of an ellipse of degree 2.3, a =3, b =2 As the result of calculating this area in numerical integral and the formula, both were completely consistent. - 4 - 6.3 A part of area of a superellipse In this section, we calculate the area of the light-blue portion of the following figure. Formula 6.3.1 When n,a,b ()b a are positive numbers respectively, the area s()x from 0 to x of the ellipse of degree n in the 1st quadrant is given by the following expression. r nr+1 1/n ()-1 x (1.1) s()x = bΣ nr r=0 r a nr+1 Proof The area s()x from 0 to x of the superellipse in the 1st quadrant is given by the following integral. 1 n x xn s()x = b 1- n dx 0< x a 0 a This integral is not expressed with elementary functions except for a special case. So, we try the termwise integration. Applying generalized binomial theorem to this integrand, 1 n nr n x r 1/n x 1- n = Σ()-1 a r=0 r a Integrating the right side with respect to x from 0 to x term by term, 1 n nr x n x x r 1/n x 1- n dx = Σ()-1 dx 0 a r=0 r 0 a r nr+1 1/n ()-1 x = Σ nr r=0 r a nr+1 Multiplying this by b , we obtain (1.1) . Example: Area of the ellipse of degree 2.3, a =3, b =2 on x=02 - 5 - We calculate the area of the light-blue portion of the above figure. As the result of calculating by numerical integral and (1.1) , both were completely consistent. By-products Especially integrating this to a , 1 n n r nr+1 r a x 1/n ()-1 a 1/n ()-1 b 1- n dx = bΣ nr = abΣ 0 a r=0 r a nr+1 r=0 r nr+1 Then, the area S of the superellipse is r 1/n ()-1 S = 4abΣ (1.2) r=0 r nr+1 Therefore also, 2 r 1 2 1/n ()-1 1+ / 1+ = Σ (1.3) n n r=0 r nr+1 - 6 - 6.4 Arc length of a superellipse 6.4.1 Arc length of an oblong superellipse In this sub-section, we calculate the length bP of the following oblong superellipse. 1 n xn y = b 1- 0<ba , n > 0 (1.0) a n Formula 6.4.1 n 1 - b n-1 n Let n,a,b ()b a are positive numbers, x be a number s.t. 0< x a 1 + a and ()z s be Pochhammer's symbol. Then, the arc length l()x from 0 to x of (1.0) is given by the followings. 2()n-1 r -1+s r n r ns 1/2 n b 2 x2(-1 ) + +1 (1.1) l()x = ΣΣ nr ns r=0 s=0 r 2()n -1 r a 2 + 2()n -1 r+ns+1 -1 n 2r n r ns 1/2 2()n -1 r 1 b x2(-1 ) + +1 (1.1') = ΣΣ 2nr+ns r=0 s=0 r n s s! a 2()n -1 r+ns+1 2()r-s n r s ns r 1/2 2()n -1 ()r-s 1 b x2()-1 ()- + +1 (1.1") = ΣΣ 2n()r-s +ns r=0 s=0 r-s n s s! a 2()n -1 ()r-s +ns+1 Proof Length l()x of the curve y = f()x x on a plane is given by the following formula. x dy 2 l()x = 1+ dx x dx Differentiating both sides of (1.0) with respect to x, 1 n-1 n -1 - n dy b xn nxn-1 b xn-1 xn = 1- - = - 1- dx n a n a n a a n-1a n n-1 n b xn/a n = - a 1-xn/a n - 7 - Then, 2()n-1 1 n 2 x b 2 xn/a n (1) l()x = 1+ 2 n n dx 0 a 1-x /a This integral is not expressed with elementary functions except for a special case. So, we expand this integrand to series. n 1 2(n-1 ) b - b 2 x n/a n 0< x a 1 + n-1 n 0 < n 1 a a 2 1-x n/a n According to genralized binomial thorem, 2()n-1 1 2()n-1 r 2 2 n n n 2r n n n b x /a 1/2 b x /a 1+ 2 n n = Σ n n a 1-x /a r=0 r a 1-x /a Then, 2()n-1 r 2r n n n 1/2 b x x /a (2) l()x = Σ n n dx r=0 r a 0 1-x /a Since this integral is also a non-elementary function, we try the termwise integration. 1 s+0 = 1 + x1 + x2 + = Σ xs 1-x s=0 0 Differentiating both sides of this with respect to x and dividing it by factorial one by one, 1 1 s+1 0 1 2 3 s 2 = 1x + 2x + 3x + 4x + = Σ x ()1-x 1! s=0 1 1 1 s+2 0 1 2 3 s 3 = 1x + 3x + 6x + 10x + = Σ x ()1-x 2! s=0 2 1 s+m-1 s m = Σ x ()1-x s=0 m-1 m Multiplying both sides by x , m x s+m-1 m+s m = Σ x ()1-x s=0 m-1 n n Replacing x with x /a , n n m n()m+s x /a s+m-1 x n n = Σ 1-x /a s=0 m-1 a Integrating both sides with respect to x from 0 to x, n n m n()m+s x x /a s+m-1 x x n n dx = Σ dx 0 1-x /a s=0 m-1 0 a i.e.
Load more

Recommended publications
  • Polar Coordinates, Arc Length and the Lemniscate Curve
    Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Calculus Mathematics via Primary Historical Sources (TRIUMPHS) Summer 2018 Gaussian Guesswork: Polar Coordinates, Arc Length and the Lemniscate Curve Janet Heine Barnett Colorado State University-Pueblo, [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_calculus Click here to let us know how access to this document benefits ou.y Recommended Citation Barnett, Janet Heine, "Gaussian Guesswork: Polar Coordinates, Arc Length and the Lemniscate Curve" (2018). Calculus. 3. https://digitalcommons.ursinus.edu/triumphs_calculus/3 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Calculus by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. Gaussian Guesswork: Polar Coordinates, Arc Length and the Lemniscate Curve Janet Heine Barnett∗ October 26, 2020 Just prior to his 19th birthday, the mathematical genius Carl Freidrich Gauss (1777{1855) began a \mathematical diary" in which he recorded his mathematical discoveries for nearly 20 years. Among these discoveries is the existence of a beautiful relationship between three particular numbers: • the ratio of the circumference of a circle to its diameter, or π; Z 1 dx • a specific value of a certain (elliptic1) integral, which Gauss denoted2 by $ = 2 p ; and 4 0 1 − x p p • a number called \the arithmetic-geometric mean" of 1 and 2, which he denoted as M( 2; 1). Like many of his discoveries, Gauss uncovered this particular relationship through a combination of the use of analogy and the examination of computational data, a practice that historian Adrian Rice called \Gaussian Guesswork" in his Math Horizons article subtitled \Why 1:19814023473559220744 ::: is such a beautiful number" [Rice, November 2009].
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Application of G. Lame's and J. Gielis' Formulas for Description of Shells
    MATEC Web of Conferences 239, 06007 (2018) https://doi.org/10.1051/matecconf /201823906007 TransSiberia 2018 Application of G. Lame’s and J. Gielis’ formulas for description of shells superplastic forming Aleksandr Anishchenko1, Volodymyr Kukhar1, Viktor Artiukh2,* and Olga Arkhipova3 1Pryazovskyi State Technical University, Universytetska, 7, Mariupol, 87555, Ukraine 2Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya, 29, St. Petersburg, 195251, Russia 3Tyumen Industrial University, Volodarskogo str., 38, Tyumen, 625000, Russia Abstract. The paper shows that the contour of a sheet blank at all stages of superplastic forming can be described by application of the universal equations known as a "superformula" and "superellipse". The work provides information on the ranges of values of the coefficients entering these equations and shows that their magnitudes make it possible to additionally determine the level of superplastic properties of the workpiece metal, the shape of the shell being manufactured, the stage of superplastic forming, and the presence of additional operations to regulate the flow of the deformable metal. The paper has the results of approximation by the proposed equations of shell contours that manufactured by superplastic forming by different methods. 1 Introduction Superplastic forming (SPF) of casings of hemisphere type, spherical shells, boxes, box sections and the like of sheet blanks is a relatively new technological process of plastic working of metals, that sprang up owing to discovery of the phenomenon of superplasticity in metals and alloys. SPF was first copied in 1964 from the process of gas forming of thermoplastics [1, 2]. Some of the materials developed for superplastic forming are known [2]: Aluminium-Magnesium (Al-Mg) alloys, Bismuth-Tin (Bi-Sn), Aluminium-Lithium (Al-Li), Zinc-Aluminium (Zn-Al) alloys, URANUS 45N (UR 45N) two phase stainless steel, Nickel (Ni) alloy INCONEL-718, Titanium (Ti-6Al-N) and Pb-Sn alloys.
    [Show full text]
  • Download Author Version (PDF)
    Soft Matter Accepted Manuscript This is an Accepted Manuscript, which has been through the Royal Society of Chemistry peer review process and has been accepted for publication. Accepted Manuscripts are published online shortly after acceptance, before technical editing, formatting and proof reading. Using this free service, authors can make their results available to the community, in citable form, before we publish the edited article. We will replace this Accepted Manuscript with the edited and formatted Advance Article as soon as it is available. You can find more information about Accepted Manuscripts in the Information for Authors. Please note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal’s standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains. www.rsc.org/softmatter Page 1 of 12 Soft Matter Shape-controlled orientation and assembly of colloids with sharp edges in nematic liquid crystals Daniel A. Beller,∗a Mohamed A. Gharbi,a,b and Iris B. Liub Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX First published on the web Xth XXXXXXXXXX 200X DOI: 10.1039/b000000x The assembly of colloids in nematic liquid crystals via topological defects has been extensively studied for spherical particles, and investigations of other colloid shapes have revealed a wide array of new assembly behaviors. We show, using Landau-de Gennes numerical modeling, that nematic defect configurations and colloidal assembly can be strongly influenced by fine details of colloid shape, in particular the presence of sharp edges.
    [Show full text]
  • NOTES Reconsidering Leibniz's Analytic Solution of the Catenary
    NOTES Reconsidering Leibniz's Analytic Solution of the Catenary Problem: The Letter to Rudolph von Bodenhausen of August 1691 Mike Raugh January 23, 2017 Leibniz published his solution of the catenary problem as a classical ruler-and-compass con- struction in the June 1691 issue of Acta Eruditorum, without comment about the analysis used to derive it.1 However, in a private letter to Rudolph Christian von Bodenhausen later in the same year he explained his analysis.2 Here I take up Leibniz's argument at a crucial point in the letter to show that a simple observation leads easily and much more quickly to the solution than the path followed by Leibniz. The argument up to the crucial point affords a showcase in the techniques of Leibniz's calculus, so I take advantage of the opportunity to discuss it in the Appendix. Leibniz begins by deriving a differential equation for the catenary, which in our modern orientation of an x − y coordinate system would be written as, dy n Z p = (n = dx2 + dy2); (1) dx a where (x; z) represents cartesian coordinates for a point on the catenary, n is the arc length from that point to the lowest point, the fraction on the left is a ratio of differentials, and a is a constant representing unity used throughout the derivation to maintain homogeneity.3 The equation characterizes the catenary, but to solve it n must be eliminated. 1Leibniz, Gottfried Wilhelm, \De linea in quam flexile se pondere curvat" in Die Mathematischen Zeitschriftenartikel, Chap 15, pp 115{124, (German translation and comments by Hess und Babin), Georg Olms Verlag, 2011.
    [Show full text]
  • The Arc Length of a Random Lemniscate
    The arc length of a random lemniscate Erik Lundberg, Florida Atlantic University joint work with Koushik Ramachandran [email protected] SEAM 2017 Random lemniscates Topic of my talk last year at SEAM: Random rational lemniscates on the sphere. Topic of my talk today: Random polynomial lemniscates in the plane. Polynomial lemniscates: three guises I Complex Analysis: pre-image of unit circle under polynomial map jp(z)j = 1 I Potential Theory: level set of logarithmic potential log jp(z)j = 0 I Algebraic Geometry: special real-algebraic curve p(x + iy)p(x + iy) − 1 = 0 Polynomial lemniscates: three guises I Complex Analysis: pre-image of unit circle under polynomial map jp(z)j = 1 I Potential Theory: level set of logarithmic potential log jp(z)j = 0 I Algebraic Geometry: special real-algebraic curve p(x + iy)p(x + iy) − 1 = 0 Polynomial lemniscates: three guises I Complex Analysis: pre-image of unit circle under polynomial map jp(z)j = 1 I Potential Theory: level set of logarithmic potential log jp(z)j = 0 I Algebraic Geometry: special real-algebraic curve p(x + iy)p(x + iy) − 1 = 0 Prevalence of lemniscates I Approximation theory: Hilbert's lemniscate theorem I Conformal mapping: Bell representation of n-connected domains I Holomorphic dynamics: Mandelbrot and Julia lemniscates I Numerical analysis: Arnoldi lemniscates I 2-D shapes: proper lemniscates and conformal welding I Harmonic mapping: critical sets of complex harmonic polynomials I Gravitational lensing: critical sets of lensing maps Prevalence of lemniscates I Approximation theory:
    [Show full text]
  • Shapes of Coil Ends in Racetrack Layout for Superconducting Magnets Attilio Milanese / TE-MSC Department
    CERN TE-Note-2010-04 [email protected] TE-MSC EDMS no : 1057916 Shapes of coil ends in racetrack layout for superconducting magnets Attilio Milanese / TE-MSC department Keywords: racetrack coils, coil winding, least energy curve, smooth curve, superconducting magnets Summary Racetrack coils have received considerable attention for Nb3Sn magnets, both built using the React-and-Wind and Wind-and-React techniques. The geometry usually consists of a series of straight parts connected with circular arcs. Therefore, at the interface between these sections, a finite change in curvature is imposed on the cable. Alternative transition curves are analyzed here, with a particular focus on the total strain energy and the minimum / maximum radii of curvature. The study is presented in dimensionless form and the various alternatives are detailed in mathematical terms, so to be used for drafting or simulations. Extensions for the design of flared ends are also briefly discussed. This study is within the framework of the EuCARD WP7-HFM project. In particular, the proposed curve can be used for the end design of the high field model magnet (Task 7.3). 1. Introduction The racetrack configuration has been recently employed for several Nb3Sn coils wound with Rutherford cables: see, for example, the common coils dipoles designed at BNL [1], FNL [2] and LBNL [3], the HD1 [4] and CERN SMC [5]. Moreover, LBNL designed, built and tested HD2 [6], a dipole with racetrack ends that flare out to clear a circular aperture. The geometry of the coil is shown in Fig. 1, together with a picture taken during coil winding [7].
    [Show full text]
  • Discrete and Smooth Curves Differential Geometry for Computer Science (Spring 2013), Stanford University Due Monday, April 15, in Class
    Homework 1: Discrete and Smooth Curves Differential Geometry for Computer Science (Spring 2013), Stanford University Due Monday, April 15, in class This is the first homework assignment for CS 468. We will have assignments approximately every two weeks. Check the course website for assignment materials and the late policy. Although you may discuss problems with your peers in CS 468, your homework is expected to be your own work. Problem 1 (20 points). Consider the parametrized curve g(s) := a cos(s/L), a sin(s/L), bs/L for s 2 R, where the numbers a, b, L 2 R satisfy a2 + b2 = L2. (a) Show that the parameter s is the arc-length and find an expression for the length of g for s 2 [0, L]. (b) Determine the geodesic curvature vector, geodesic curvature, torsion, normal, and binormal of g. (c) Determine the osculating plane of g. (d) Show that the lines containing the normal N(s) and passing through g(s) meet the z-axis under a constant angle equal to p/2. (e) Show that the tangent lines to g make a constant angle with the z-axis. Problem 2 (10 points). Let g : I ! R3 be a curve parametrized by arc length. Show that the torsion of g is given by ... hg˙ (s) × g¨(s), g(s)i ( ) = − t s 2 jkg(s)j where × is the vector cross product. Problem 3 (20 points). Let g : I ! R3 be a curve and let g˜ : I ! R3 be the reparametrization of g defined by g˜ := g ◦ f where f : I ! I is a diffeomorphism.
    [Show full text]
  • Arc Length and Curvature
    Arc Length and Curvature For a ‘smooth’ parametric curve: xftygt= (), ==() and zht() for atb≤≤ OR for a ‘smooth’ vector function: rijk()tftgtht= ()++ () () for atb≤≤ the arc length of the curve for the t values atb≤ ≤ is defined as the integral b 222 ('())('())('())f tgthtdt++ ∫ a b For vector functions, the integrand can be simplified this way: r'(tdt ) ∫a Ex) Find the arc length along the curve rjk()tt= (cos())33+ (sin()) t for 0/2≤ t ≤π . Unit Tangent and Unit Normal Vectors r'(t ) Unit Tangent Vector Æ T()t = r'(t ) T'(t ) Unit Normal Vector Æ N()t = (can be a lengthy calculation) T'(t ) The unit normal vector is designed to be perpendicular to the unit tangent vector AND it points ‘into the turn’ along a vector curve’s path. 2 Ex) For the vector function rij()tt=++− (2 3) (5 t ), find T()t and N()t . (this blank page was NOT a mistake ... we’ll need it) Ex) Graph the vector function rij()tt= (2++− 3) (5 t2 ) and its unit tangent and unit normal vectors at t = 1. Curvature Curvature, denoted by the Greek letter ‘ κ ’ (kappa) measures the rate at which the ‘tilt’ of the unit tangent vector is changing with respect to the arc length. The derivation of the curvature formula is a rather lengthy one, so I will leave it out. Curvature for a vector function r()t rr'(tt )× ''( ) Æ κ= r'(t ) 3 Curvature for a scalar function yfx= () fx''( ) Æ κ= (This numerator represents absolute value, not vector magnitude.) (1+ (fx '( ))23/2 ) YOU WILL BE GIVEN THESE FORMULAS ON THE TEST!! NO NEED TO MEMORIZE!! One visualization of curvature comes from using osculating circles.
    [Show full text]
  • Capturing Human Activity Spaces: New Geometries
    R. K. Rai, M. Balmer, M. Rieser, V. S. Vaze, S. Schönfelder, K. W. Axhausen Capturing human activity spaces: New geometries Journal article | Accepted manuscript (Postprint) This version is available at https://doi.org/10.14279/depositonce-10352 Rai, R. K., Balmer, M., Rieser, M., Vaze, V. S., Schönfelder, S., & Axhausen, K. W. (2007). Capturing Human Activity Spaces. Transportation Research Record: Journal of the Transportation Research Board, 2021(1), 70–80. https://doi.org/10.3141/2021-09 © 2007 SAGE Publications Terms of Use Copyright applies. A non-exclusive, non-transferable and limited right to use is granted. This document is intended solely for personal, non-commercial use. R.K. Rai, M. Balmer, M. Rieser V.S. Vaze, S. Schönfelder, K.W. Axhausen 1 Capturing human activity spaces: New geometries 2006-11-15 R.K. Rai IIT Guwahati Guwahati 781039, Assam, India Phone: +91-986-420-23-53 Email: [email protected] Michael Balmer Institute for Transport Planning and Systems (IVT), ETH Zurich 8093 Zurich, Switzerland Phone: +41-44-633-27-80 Fax: +41-44-633-10-57 Email: [email protected] Marcel Rieser VSP, TU-Berlin 10587 Berlin, Germany Phone: +49-30-314-25-258 Fax: +49-30-314-26-269 Email: [email protected] V.S. Vaze IIT Bombay Powai, 400057 Mumbai, India Email: [email protected] Stefan Schönfelder TRAFICO Verkehrsplanung Fillgradergasse 6/2, 1060 Vienna, Austria Phone: +43-1-586-4181 Fax: +43-1-586-41-8110 Email: [email protected] Kay W. Axhausen Institute for Transport Planning and Systems (IVT), ETH Zurich 8093 Zurich, Switzerland Phone: +41-44-633-39-43 Fax: +41-44-633-10-57 Email: [email protected] Words: 6234 Figures/Tables: 6/1 Total: 7984 R.K.
    [Show full text]
  • 11.1 Circumference and Arc Length
    11.1 Circumference and Arc Length EEssentialssential QQuestionuestion How can you fi nd the length of a circular arc? Finding the Length of a Circular Arc Work with a partner. Find the length of each red circular arc. a. entire circle b. one-fourth of a circle y y 5 5 C 3 3 1 A 1 A B −5 −3 −1 1 3 5 x −5 −3 −1 1 3 5 x −3 −3 −5 −5 c. one-third of a circle d. fi ve-eighths of a circle y y C 4 4 2 2 A B A B −4 −2 2 4 x −4 −2 2 4 x − − 2 C 2 −4 −4 Using Arc Length Work with a partner. The rider is attempting to stop with the front tire of the motorcycle in the painted rectangular box for a skills test. The front tire makes exactly one-half additional revolution before stopping. LOOKING FOR The diameter of the tire is 25 inches. Is the REGULARITY front tire still in contact with the IN REPEATED painted box? Explain. REASONING To be profi cient in math, you need to notice if 3 ft calculations are repeated and look both for general methods and for shortcuts. CCommunicateommunicate YourYour AnswerAnswer 3. How can you fi nd the length of a circular arc? 4. A motorcycle tire has a diameter of 24 inches. Approximately how many inches does the motorcycle travel when its front tire makes three-fourths of a revolution? Section 11.1 Circumference and Arc Length 593 hhs_geo_pe_1101.indds_geo_pe_1101.indd 593593 11/19/15/19/15 33:10:10 PMPM 11.1 Lesson WWhathat YYouou WWillill LLearnearn Use the formula for circumference.
    [Show full text]
  • Notes Sec 8.1 Arc Length
    Section 8.1 MATH 1152Q Arc Length Notes Arc Length Suppose that a curve C is defined by the equation yfx= ( ), where f is continuous and axb££. We obtain a polygonal approximation to C by dividing the interval [ab, ] into n subintervals with endpoints x0 , x1 , , xn and equal width Dx . If yfxii= ( ), then the point Pi (xyii, ) lies on C and the polygon with vertices P0 , P1 , , Pn is an approximation to C . The length L of C is approximately the length of this polygon and the approximation gets better as we let n increase. Therefore, we define the length L of the curve C with equation yfx= ( ), axb££, as the limit of the lengths of these inscribed polygons (provided the limit exists). n L = lim PPii-1 n®¥ å i=1 The definition of arc length given above is not very convenient for computational purposes, but we can derive an integral formula for L in the case where f has a continuous derivative. Such a function f is called smooth because a small change in x produces a small change in fx'( ). If we let Dyyyiii=--1, then 22 PPii-1 =(xxii---11) +( yy ii-) 22 =(Dxy) +(Di ) éù2 2 æöDyi =(Dx) êú1+ç÷ ëûêúèøDx 2 æöDyi =+1 ç÷ ×Dx èøDx By applying the Mean Value Theorem to f on the interval [xxii-1, ], we find that there is a * number xi between xi-1 and xi such that * fx( ii) - fx( --11) = f'( x iii)( x- x) Page 1 of 3 Thus we have 2 æöDyi PPii-1 =+1 ç÷ ×Dx èøDx 2 =+1'éùfx* ×D x ëû( i ) Therefore, n L = lim PPii-1 n®¥ å i=1 n 2 * =+lim 1éùfx ' i ×D x n®¥ å ëû( ) i=1 b 2 =+1'éùfx( ) dx òa ëû 2 The integral exists because the integrand 1'+ ëûéùfx( ) is continuous.
    [Show full text]