Curvature, Natural Frames, and Acceleration for Plane and Space Curves
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The Osculating Circumference Problem
THE OSCULATING CIRCUMFERENCE PROBLEM School I.S.I.S.S. M. Casagrande Pieve di Soligo, Treviso Italy Students Silvia Micheletto (I) Azzurra Soa Pizzolotto (IV) Luca Barisan (II) Simona Sota (IV) Gaia Barella (IV) Paolo Barisan (V) Marco Casagrande (IV) Anna De Biasi (V) Chiara De Rosso (IV) Silvia Giovani (V) Maddalena Favaro (IV) Klara Metaliu (V) Teachers Fabio Breda Francesco Maria Cardano Davide Palma Francesco Zampieri Researcher Alberto Zanardo, University of Padova Italy Year 2019/2020 The osculating circumference problem ISISS M. Casagrande, Pieve di Soligo, Treviso Italy Abstract The aim of the article is to study the osculating circumference i.e. the circumference which best approximates the graph of a curve at one of its points. We will dene this circumference and we will describe several methods to nd it. Finally we will introduce the notions of round points and crossing points, points of the curve where the circumference has interesting properties. ??? Contents Introduction 3 1 The osculating circumference 5 1.1 Particular cases: the formulas cannot be applied . 9 1.2 Curvature . 14 2 Round points 15 3 Beyond round points 22 3.1 Crossing points . 23 4 Osculating circumference of a conic section 25 4.1 The rst method . 25 4.2 The second method . 26 References 31 2 The osculating circumference problem ISISS M. Casagrande, Pieve di Soligo, Treviso Italy Introduction Our research was born from an analytic geometry problem studied during the third year of high school. The problem. Let P : y = x2 be a parabola and A and B two points of the curve symmetrical about the axis of the parabola. -
Plotting Graphs of Parametric Equations
Parametric Curve Plotter This Java Applet plots the graph of various primary curves described by parametric equations, and the graphs of some of the auxiliary curves associ- ated with the primary curves, such as the evolute, involute, parallel, pedal, and reciprocal curves. It has been tested on Netscape 3.0, 4.04, and 4.05, Internet Explorer 4.04, and on the Hot Java browser. It has not yet been run through the official Java compilers from Sun, but this is imminent. It seems to work best on Netscape 4.04. There are problems with Netscape 4.05 and it should be avoided until fixed. The applet was developed on a 200 MHz Pentium MMX system at 1024×768 resolution. (Cost: $4000 in March, 1997. Replacement cost: $695 in June, 1998, if you can find one this slow on sale!) For various reasons, it runs extremely slowly on Macintoshes, and slowly on Sun Sparc stations. The curves should move as quickly as Roadrunner cartoons: for a benchmark, check out the circular trigonometric diagrams on my Java page. This applet was mainly done as an exercise in Java programming leading up to more serious interactive animations in three-dimensional graphics, so no attempt was made to be comprehensive. Those who wish to see much more comprehensive sets of curves should look at: http : ==www − groups:dcs:st − and:ac:uk= ∼ history=Java= http : ==www:best:com= ∼ xah=SpecialP laneCurve dir=specialP laneCurves:html http : ==www:astro:virginia:edu= ∼ eww6n=math=math0:html Disclaimer: Like all computer graphics systems used to illustrate mathematical con- cepts, it cannot be error-free. -
Complex Analysis
8 Complex Representations of Functions “He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical.” Henry David Thoreau (1817-1862) 8.1 Complex Representations of Waves We have seen that we can determine the frequency content of a function f (t) defined on an interval [0, T] by looking for the Fourier coefficients in the Fourier series expansion ¥ a0 2pnt 2pnt f (t) = + ∑ an cos + bn sin . 2 n=1 T T The coefficients take forms like 2 Z T 2pnt an = f (t) cos dt. T 0 T However, trigonometric functions can be written in a complex exponen- tial form. Using Euler’s formula, which was obtained using the Maclaurin expansion of ex in Example A.36, eiq = cos q + i sin q, the complex conjugate is found by replacing i with −i to obtain e−iq = cos q − i sin q. Adding these expressions, we have 2 cos q = eiq + e−iq. Subtracting the exponentials leads to an expression for the sine function. Thus, we have the important result that sines and cosines can be written as complex exponentials: 286 partial differential equations eiq + e−iq cos q = , 2 eiq − e−iq sin q = .( 8.1) 2i So, we can write 2pnt 1 2pint − 2pint cos = (e T + e T ). -
4 Complex Analysis
4 Complex Analysis “He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical.” Henry David Thoreau (1817 - 1862) We have seen that we can seek the frequency content of a signal f (t) defined on an interval [0, T] by looking for the the Fourier coefficients in the Fourier series expansion In this chapter we introduce complex numbers and complex functions. We a ¥ 2pnt 2pnt will later see that the rich structure of f (t) = 0 + a cos + b sin . 2 ∑ n T n T complex functions will lead to a deeper n=1 understanding of analysis, interesting techniques for computing integrals, and The coefficients can be written as integrals such as a natural way to express analog and dis- crete signals. 2 Z T 2pnt an = f (t) cos dt. T 0 T However, we have also seen that, using Euler’s Formula, trigonometric func- tions can be written in a complex exponential form, 2pnt e2pint/T + e−2pint/T cos = . T 2 We can use these ideas to rewrite the trigonometric Fourier series as a sum over complex exponentials in the form ¥ 2pint/T f (t) = ∑ cne , n=−¥ where the Fourier coefficients now take the form Z T −2pint/T cn = f (t)e dt. -
Gottfried Wilhelm Leibnitz (Or Leibniz) Was Born at Leipzig on June 21 (O.S.), 1646, and Died in Hanover on November 14, 1716. H
Gottfried Wilhelm Leibnitz (or Leibniz) was born at Leipzig on June 21 (O.S.), 1646, and died in Hanover on November 14, 1716. His father died before he was six, and the teaching at the school to which he was then sent was inefficient, but his industry triumphed over all difficulties; by the time he was twelve he had taught himself to read Latin easily, and had begun Greek; and before he was twenty he had mastered the ordinary text-books on mathematics, philosophy, theology and law. Refused the degree of doctor of laws at Leipzig by those who were jealous of his youth and learning, he moved to Nuremberg. An essay which there wrote on the study of law was dedicated to the Elector of Mainz, and led to his appointment by the elector on a commission for the revision of some statutes, from which he was subsequently promoted to the diplomatic service. In the latter capacity he supported (unsuccessfully) the claims of the German candidate for the crown of Poland. The violent seizure of various small places in Alsace in 1670 excited universal alarm in Germany as to the designs of Louis XIV.; and Leibnitz drew up a scheme by which it was proposed to offer German co-operation, if France liked to take Egypt, and use the possessions of that country as a basis for attack against Holland in Asia, provided France would agree to leave Germany undisturbed. This bears a curious resemblance to the similar plan by which Napoleon I. proposed to attack England. In 1672 Leibnitz went to Paris on the invitation of the French government to explain the details of the scheme, but nothing came of it. -
Geometry in the Age of Enlightenment
Geometry in the Age of Enlightenment Raymond O. Wells, Jr. ∗ July 2, 2015 Contents 1 Introduction 1 2 Algebraic Geometry 3 2.1 Algebraic Curves of Degree Two: Descartes and Fermat . 5 2.2 Algebraic Curves of Degree Three: Newton and Euler . 11 3 Differential Geometry 13 3.1 Curvature of curves in the plane . 17 3.2 Curvature of curves in space . 26 3.3 Curvature of a surface in space: Euler in 1767 . 28 4 Conclusion 30 1 Introduction The Age of Enlightenment is a term that refers to a time of dramatic changes in western society in the arts, in science, in political thinking, and, in particular, in philosophical discourse. It is generally recognized as being the period from the mid 17th century to the latter part of the 18th century. It was a successor to the renaissance and reformation periods and was followed by what is termed the romanticism of the 19th century. In his book A History of Western Philosophy [25] Bertrand Russell (1872{1970) gives a very lucid description of this time arXiv:1507.00060v1 [math.HO] 30 Jun 2015 period in intellectual history, especially in Book III, Chapter VI{Chapter XVII. He singles out Ren´eDescartes as being the founder of the era of new philosophy in 1637 and continues to describe other philosophers who also made significant contributions to mathematics as well, such as Newton and Leibniz. This time of intellectual fervor included literature (e.g. Voltaire), music and the world of visual arts as well. One of the most significant developments was perhaps in the political world: here the absolutism of the church and of the monarchies ∗Jacobs University Bremen; University of Colorado at Boulder; [email protected] 1 were questioned by the political philosophers of this era, ushering in the Glo- rious Revolution in England (1689), the American Revolution (1776), and the bloody French Revolution (1789). -
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. -
Osculating Curves and Surfaces*
OSCULATING CURVES AND SURFACES* BY PHILIP FRANKLIN 1. Introduction The limit of a sequence of curves, each of which intersects a fixed curve in « +1 distinct points, which close down to a given point in the limit, is a curve having contact of the «th order with the fixed curve at the given point. If we substitute surface for curve in the above statement, the limiting surface need not osculate the fixed surface, no matter what value » has, unless the points satisfy certain conditions. We shall obtain such conditions on the points, our conditions being necessary and sufficient for contact of the first order, and sufficient for contact of higher order. By taking as the sequence of curves parabolas of the «th order, we obtain theorems on the expression of the »th derivative of a function at a point as a single limit. For the surfaces, we take paraboloids of high order, and obtain such expressions for the partial derivatives. In this case, our earlier restriction, or some other, on the way in which the points close down to the limiting point is necessary. In connection with our discussion of osculating curves, we obtain some interesting theorems on osculating conies, or curves of any given type, which follow from a generalization of Rolle's theorem on the derivative to a theorem on the vanishing of certain differential operators. 2. Osculating curves Our first results concerning curves are more or less well known, and are presented chiefly for the sake of completeness and to orient the later results, t We begin with a fixed curve c, whose equation, in some neighbor- hood of the point x —a, \x —a\<Q, may be written y=f(x). -
The Frenet–Serret Formulas∗
The Frenet–Serret formulas∗ Attila M´at´e Brooklyn College of the City University of New York January 19, 2017 Contents Contents 1 1 The Frenet–Serret frame of a space curve 1 2 The Frenet–Serret formulas 3 3 The first three derivatives of r 3 4 Examples and discussion 4 4.1 Thecurvatureofacircle ........................... ........ 4 4.2 Thecurvatureandthetorsionofahelix . ........... 5 1 The Frenet–Serret frame of a space curve We will consider smooth curves given by a parametric equation in a three-dimensional space. That is, writing bold-face letters of vectors in three dimension, a curve is described as r = F(t), where F′ is continuous in some interval I; here the prime indicates derivative. The length of such a curve between parameter values t I and t I can be described as 0 ∈ 1 ∈ t1 t1 ′ dr (1) σ(t1)= F (t) dt = dt Z | | Z dt t0 t0 where, for a vector u we denote by u its length; here we assume t is fixed and t is variable, so | | 0 1 we only indicated the dependence of the arc length on t1. Clearly, σ is an increasing continuous function, so it has an inverse σ−1; it is customary to write s = σ(t). The equation − def (2) r = F(σ 1(s)) s J = σ(t): t I ∈ { ∈ } is called the re-parametrization of the curve r = F(t) (t I) with respect to arc length. It is clear that J is an interval. To simplify the description, we will∈ always assume that r = F(t) and s = σ(t), ∗Written for the course Mathematics 2201 (Multivariable Calculus) at Brooklyn College of CUNY. -
Osculating Curves: Around the Tait-Kneser Theorem
Osculating curves: around the Tait-Kneser Theorem E. Ghys S. Tabachnikov V. Timorin 1 Tait and Kneser The notion of osculating circle (or circle of curvature) of a smooth plane curve is familiar to every student of calculus and elementary differential geometry: this is the circle that approximates the curve at a point better than all other circles. One may say that the osculating circle passes through three infinitesimally close points on the curve. More specifically, pick three points on the curve and draw a circle through these points. As the points tend to each other, there is a limiting position of the circle: this is the osculating circle. Its radius is the radius of curvature of the curve, and the reciprocal of the radius is the curvature of the curve. If both the curve and the osculating circle are represented locally as graphs of smooth functions then not only the values of these functions but also their first and second derivatives coincide at the point of contact. Ask your mathematical friend to sketch an arc of a curve and a few osculating circles. Chances are, you will see something like Figure 1. Figure 1: Osculating circles? arXiv:1207.5662v1 [math.DG] 24 Jul 2012 This is wrong! The following theorem was discovered by Peter Guthrie Tait in the end of the 19th century [9] and rediscovered by Adolf Kneser early in the 20th century [4]. Theorem 1 The osculating circles of an arc with monotonic positive curvature are pairwise disjoint and nested. Tait's paper is so short that we quote it almost verbatim (omitting some old-fashioned terms): 1 When the curvature of a plane curve continuously increases or diminishes (as in the case with logarithmic spiral for instance) no two of the circles of curvature can intersect each other. -
Arxiv:1911.12155V13 [Math.CA] 1 Jul 2020 2.7 Double Integration
On logarithmic integrals, harmonic sums and variations Ming Hao Zhao Abstract Based on various non-MZV approaches we evaluate certain logarithmic integrals and har- monic sums. More specifically, 85 LIs, 89 ESs, 263 PLIs, 28 non-alt QESs, 39 GESs, 26 BESs, 14 IBESs with weight ≤ 5, 193 QLIs, 172 QPLIs, 83 QESs, 7 QBESs, 3 IQBESs with weight ≤ 4. With help of MZV theory we evaluate other 10 QESs with weight ≤ 4. High weight hypergeometric series, nonhomogeneous integrals and other results are also established. Contents 1 Preliminaries 5 1.1 Special functions . .5 1.2 Definition of subjects . .6 1.2.1 LI and PLI . .6 1.2.2 QLI and QPLI . .7 1.2.3 ES and QES . .8 1.2.4 Fibonacci basis . .9 1.3 Main results . 11 2 LI 11 2.1 Brute force . 11 2.2 General formulas . 12 2.3 Integration by parts . 14 2.4 Fractional transformation . 14 2.5 Beta derivatives . 15 2.6 Contour integration . 16 arXiv:1911.12155v13 [math.CA] 1 Jul 2020 2.7 Double integration . 16 2.8 Hypergeometric identities . 16 2.9 Solution . 17 1 3 ES 17 3.1 General formulas . 17 3.2 Symmetric relations . 18 3.3 Partial fractions . 18 3.4 Contour integration . 18 3.5 Solution . 18 3.5.1 Preparation . 18 3.5.2 Confrontation . 24 4 PLI 27 4.1 Brute force . 27 4.2 General formulas . 27 4.3 Integration by parts . 28 4.4 Fractional transformation . 28 4.5 Complementary integrals . 28 4.6 Polylog identities . 28 4.7 Series expansion . -
The Surface Area of a Scalene Cone As Solved by Varignon, Leibniz, and Euler
Euleriana Volume 1 Issue 1 Article 3 2021 The Surface Area of a Scalene Cone as Solved by Varignon, Leibniz, and Euler Daniel J. Curtin Northern Kentucky University, [email protected] Follow this and additional works at: https://scholarlycommons.pacific.edu/euleriana Part of the Other Mathematics Commons Recommended Citation Curtin, Daniel J. (2021) "The Surface Area of a Scalene Cone as Solved by Varignon, Leibniz, and Euler," Euleriana: 1(1), p. 10, Article 3. Available at: https://scholarlycommons.pacific.edu/euleriana/vol1/iss1/3 This Translation & Commentary is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Euleriana by an authorized editor of Scholarly Commons. For more information, please contact [email protected]. Curtin: The Surface Area of a Scalene Cone as Solved by Varignon, Leibniz The Scalene Cone: Euler, Varignon, and Leibniz Daniel J. Curtin, Northern Kentucky University (Emeritus) 148 Brentwood Place, Fort Thomas, KY 41075 [email protected] Abstract The work translated alongside this article is Euler's contribution to the problem of the scalene cone, in which he discusses the earlier work of Varignon and Leibniz. He improves Varignon's solution and corrects Leib- niz's more ambitious solution. Here I will discuss the problem, provide extensive notes on Euler's paper, and lay out its relation to the other two. For ease of reference translations of the papers of Varignon and Leibniz are appended, with the Latin originals. Euler's version in Latin is available at https://scholarlycommons.pacific.edu/euler-works/133 1 The Problem 1.1 The History of the Problem Finding the surface area of a right cone, a cone with circular base and vertex directly over the center of the circle, was known from antiquity, certainly by Archimedes.