Geometrical and Graphical Representations Analysis of Lissajous Figures in Rotor Dynamic System
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Lissajous Curve - Wikipedia, the Free Encyclopedia
Lissajous curve - Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Lissajous_curve Lissajous curve From Wikipedia, the free encyclopedia In mathematics, a Lissajous curve /ˈlɪsəʒuː/, also known as Lissajous figure or Bowditch curve /ˈbaʊdɪtʃ/, is the graph of a system of parametric equations which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857. The appearance of the figure is highly sensitive to the ratio a/b. For a ratio of 1, the figure is an ellipse, with special cases including circles (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (a/b = 2, δ = π/2). Other ratios produce more complicated curves, which are closed only if a/b is rational. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures. Lissajous figures where a = 1, b = N (N is a natural number) and are Chebyshev polynomials of the first kind of degree N. This property is exploited to produce a set of points, called Padua points, at which a function may be sampled in order to compute either a bivariate interpolation or quadrature of the function over the domain [-1,1]×[-1,1]. Lissajous figure on an oscilloscope, Contents displaying a 1:3 relationship between the frequencies of the vertical and horizontal 1 Examples sinusoidal inputs, respectively. 2 Generation 2.1 Practical application 3 Application for the case of a = b 4 Popular culture 5 See also 6 Notes 7 External links 7.1 Interactive demos Examples The animation below shows the curve adaptation with continuously increasing fraction from 0 to 1 in steps of 0.01. -
Lissajous Figures and Chebyshev Polynomials Julio Castineira˜ Merino
Lissajous Figures and Chebyshev Polynomials Julio Castineira˜ Merino Julio Castineira˜ Merino ([email protected]) graduated with a degree in mathematics from the Universidad Complutense in Madrid, Spain. His academic interests include geometry, numerical analysis and computer science. He plays golf in his free time and enjoys music and movies, as well as spending time with family and friends. To my wife Julia and my children Isabel, Lucia, and Rodrigo. The aim of this paper is to obtain implicit equations for Lissajous figures in the form f (x, y) = 0 using some elementary properties of Chebyshev polynomials. A Lissajous figure is the trajectory of a moving point whose rectangular coordi- nates are simple harmonic motions. The equation of a simple harmonic motion is x = a cos(ωt + ϕ) where t is the time and the constants a, ω,andϕ are the amplitude, the frequency, and the phase respectively. Parametric equations for Lissajous figures thus are x = a cos(ω1t + ϕ1) y = b cos(ω2t + ϕ2). The constants a and b determine the size of the curve while its shape depends on the ratio of the frequencies. If the frequencies are equal, the curve is either an ellipse or a line segment, the latter occurring if the difference of the phases is a multiple of π. This property can be used to study an unknown signal. If we apply the unknown signal to the vertical axis of an oscilloscope and then vary the horizontal frequency, when the oscilloscope shows an ellipse the signal’s frequency has been determined. Lissajous figures were actually discovered by the American astronomer and mathe- matician Nathaniel Bowditch in 1815 when he was studying the movement of a com- pound pendulum. -
Rational Curves
Chapter 22 Rational Curves 597 598 CHAPTER 22. RATIONAL CURVES 22.1 Rational Curves and Multiprojective Maps In this chapter, rational curves are investigated. After a quick review of the traditional parametric definition in terms of homogeneous polynomials, we explore the possibility of defining rational curves in terms of polar forms. Because the polynomials involved are homogeneous, the polar forms are actually multilinear, and rational curves are defined in terms of multiprojective maps. Then, using the contruction at the end of the previous chapter, we define rational curves in BR-form. This allows us to handle rational curves in terms of control points in the hat space obtained from . We present two versions of the de Casteljau algorithm for rational curves and show howE subdivision canE be easily extended from the polynomial case. We also present a very simple method for! approximating closed rational curves. This method only uses two control polygons obtained from the original control polygon. We conclude this chapter with a quick tour in a gallery of rational curves. We begin by defining rational polynomial curves in a traditional manner, and then, we show how this definition can be advantageously recast in terms of symmetric multiprojective maps. Keep in mind that rational polynomial curves really live in the projective completion of the affine space . This is the reason why homogeneous polynomials are involved. We will assume thatE the homogenization AE of the affine line A is identified with the direct sum R R1. Then, every element of A"is of the form (t, z) R2. ⊕ ∈ ! A rational curve of degree m in an affine space of dimension n is specified by n fractions, say E ! F1(t) F2(t) Fn(t) x1(t)= ,x2(t)= ,...,xn(t)= , Fn+1(t) Fn+1(t) Fn+1(t) where F1(X),...,Fn+1(X) are polynomials of degree at most m. -
Rheology of Complex Fluid Films for Biological and Mechanical Adhesive Locomotion by Randy H
Rheology of Complex Fluid Films for Biological and Mechanical Adhesive Locomotion by Randy H. Ewoldt B.S., Mechanical Engineering Iowa State University (2004) Submitted to the Department of Mechanical Engineering in partial ful¯llment of the requirements for the degree of Master of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2006 °c Massachusetts Institute of Technology 2006. All rights reserved. Author.............................................................. Department of Mechanical Engineering May 12, 2006 Certi¯ed by. Anette Hosoi Assistant Professor, Mechanical Engineering Thesis Supervisor Certi¯ed by. Gareth H. McKinley Professor, Mechanical Engineering Thesis Supervisor Accepted by . Lallit Anand Chairman, Department Committee on Graduate Students 2 Rheology of Complex Fluid Films for Biological and Mechanical Adhesive Locomotion by Randy H. Ewoldt Submitted to the Department of Mechanical Engineering on May 12, 2006, in partial ful¯llment of the requirements for the degree of Master of Science in Mechanical Engineering Abstract Many gastropods, such as snails and slugs, crawl using adhesive locomotion, a tech- nique that allows the organisms to climb walls and walk across ceilings. These animals stick to the crawling surface by excreting a thin layer of biopolymer mucin gel, known as pedal mucus, and their acrobatic ability is due in large part to the rheological prop- erties of this slime. The primary application of the present research is to enable a mechanical crawler to climb walls and walk across ceilings using adhesive locomotion. A properly selected slime simulant will enable a mechanical crawler to optimally perform while climbing in the horizontal, inclined, and inverted positions. To this end, the rheology of gastropod pedal mucus is examined in greater detail than any previously published work. -
9.1 Plane Curves and Parametric Equations
P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO GTBL001-09 GTBL001-Smith-v16.cls November 16, 2005 11:41 716 CHAPTER 9 .. Parametric Equations and Polar Coordinates 9-2 9.1 PLANE CURVES AND PARAMETRIC EQUATIONS We often find it convenient to describe the location of a point (x, y)inthe plane in terms of a parameter. For instance, in tracking the movement of a satellite, we would naturally want to give its location in terms of time. In this way, we not only know the path it follows, but we also know when it passes through each point. Given any pair of functions x(t) and y(t) defined on the same domain D, the equations x = x(t), y = y(t) are called parametric equations. Notice that for each choice of t, the parametric equations specify a point (x, y) = (x(t), y(t)) in the xy-plane. The collection of all such points is called the graph of the parametric equations. In the case where x(t) and y(t) are continuous functions and D is an interval of the real line, the graph is a curve in the xy-plane, referred to as a plane curve. The choice of the letter t to denote the independent variable (called the parameter) should make you think of time, which is often what the parameter represents. For instance, we might represent the position (x(t), y(t)) of a moving object as a function of the time t.Infact, you might recall that in section 5.5, we used a pair of equations of this type to describe two-dimensional projectile motion. -
9.5 Parametric Equations 925
P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 925 Section 9.5 Parametric Equations 925 Group Exercise Preview Exercises 69. Many public and private organizations and schools provide Exercises 70–72 will help you prepare for the material covered educational materials and information for the blind and in the next section. In each exercise, graph the equation in a visually impaired. Using your library, resources on the World rectangular coordinate system. Wide Web, or local organizations, investigate how your group 70. y2 = 4 x + 1 or college could make a contribution to enhance the study of 1 2 mathematics for the blind and visually impaired. In relation 1 2 71. y = x + 1, x Ú 0 to conic sections, group members should discuss how to 2 create graphs in tactile, or touchable, form that show blind 2 students the visual structure of the conics, including x2 y 72. + = 1 asymptotes, intercepts, end behavior, and rotations. 25 4 Section 9.5 Parametric Equations Objectives ᕡ hat a baseball game! You got to see the great Use point plotting to graph WDerek Jeter of the New York Yankees blast plane curves described by a powerful homer. In less than eight seconds, parametric equations. the parabolic path of his home run took the ᕢ Eliminate the parameter. ball a horizontal distance of over 1000 feet. Is ᕣ Find parametric equations there a way to model this path that gives for functions. both the ball’s location and the time that it is ᕤ Understand the advantages of in each of its positions? In this section, we parametric representations. -
Arxiv:2006.14066V2 [Math.AG] 10 Aug 2020 10
EXPRESSIVE CURVES SERGEY FOMIN AND EUGENII SHUSTIN Abstract. We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve C is expressive if (a) each irreducible component of C can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of C in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of C in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a){(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more. Contents Introduction 2 1. Plane curves and their singularities 6 2. Intersections of polar curves at infinity 8 3. L1-regular curves 12 4. Polynomial and trigonometric curves 18 5. Expressive curves and polynomials 24 6. Divides 30 7. L1-regular expressive curves 32 8. More expressivity criteria 41 9. Bending, doubling, and unfolding 43 arXiv:2006.14066v2 [math.AG] 10 Aug 2020 10. Arrangements of lines, parabolas, and circles 49 11. Shifts and dilations 52 12. Arrangements of polynomial curves 59 13. -
2Dcurves in .Pdf Format (1882 Kb) Curve Literature Last Update: 2003−06−14 Higher Last Updated: Lennard−Jones 2002−03−25 Potential
2d curves A collection of 631 named two−dimensional curves algebraic curves are a polynomial in x other curves and y kid curve line (1st degree) 3d curve conic (2nd) cubic (3rd) derived curves quartic (4th) sextic (6th) barycentric octic (8th) caustic other (otherth) cissoid conchoid transcendental curves curvature discrete cyclic exponential derivative fractal envelope gamma & related hyperbolism isochronous inverse power isoptic power exponential parallel spiral pedal trigonometric pursuit reflecting wall roulette strophoid tractrix Some programs to draw your own For isoptic cubics: curves: • Isoptikon (866 kB) from the • GraphEq (2.2 MB) from University of Crete Pedagoguery Software • GraphSight (696 kB) from Cradlefields ($19 to register) 2dcurves in .pdf format (1882 kB) curve literature last update: 2003−06−14 higher last updated: Lennard−Jones 2002−03−25 potential Atoms of an inert gas are in equilibrium with each other by means of an attracting van der Waals force and a repelling forces as result of overlapping electron orbits. These two forces together give the so−called Lennard−Jones potential, which is in fact a curve of thirteenth degree. backgrounds main last updated: 2002−11−16 history I collected curves when I was a young boy. Then, the papers rested in a box for decades. But when I found them, I picked the collection up again, some years spending much work on it, some years less. questions I have been thinking a long time about two questions: 1. what is the unity of curve? Stated differently as: when is a curve different from another one? 2. which equation belongs to a curve? 1. -
Math Girls Talk About Trigonometry
MATH GIRLS TALK ABOUT TRIGONOMETRY Originally published as S¯ugakuG¯aruNo Himitsu N¯otoMarui Sankaku Kans¯u Copyright © 2014 Hiroshi Yuki Softbank Creative Corp., Tokyo English translation © 2014 by Tony Gonzalez Edited by Joseph Reeder and M.D. Hendon Cover design by Kasia Bytnerowicz All rights reserved. No portion of this book in excess of fair use considerations may be reproduced or transmitted in any form or by any means without written permission from the copyright holders. Published 2014 by Bento Books, Inc. Austin, Texas 78732 bentobooks.com ISBN 978-1-939326-26-3 (hardcover) ISBN 978-1-939326-25-6 (trade paperback) Library of Congress Control Number: 2014958329 Printed in the United States of America First edition, December 2014 Math Girls Talk About Trigonometry Contents Prologue1 1 Round Triangles 3 1.1 Starting the Journey 3 1.2 Right Triangles 4 1.3 Naming Angles 5 1.4 Naming Vertices and Sides 7 1.5 The Sine Function 9 1.6 How to Remember Sine 14 1.7 The Cosine Function 15 1.8 Removing Conditions 18 1.9 Sine Curves 26 Appendix: Greek Letters 37 Appendix: Values of trigonometric functions for common angles 38 Problems for Chapter 1 41 2 Back and Forth 43 2.1 In My Room 43 2.2 Drawing Graphs 50 2.3 Further Ahead 59 2.4 Slowing Things Down 60 2.5 Doubling Down 63 2.6 More Graphs 73 Appendix: Lissajous curve graphing paper 80 Problems for Chapter 2 81 3 Around and Around 83 3.1 In the Library 83 3.2 Vectors 89 3.3 Multiplying Vectors by a Real Number 93 3.4 Adding Vectors 95 3.5 Rotations 96 3.6 Graphically Rotating a Point