3 Curvature and the Notion of Space
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The Gaussian Curvature 13/09/2007 Renzo Mattioli
The Gaussian Curvature 13/09/2007 Renzo Mattioli The Gaussian Curvature Map Renzo Mattioli (Optikon 2000, Roma*) *Author is a full time employee of Optikon 2000 Spa (Int.: C3) Historical background: Theorema Egregium by Carl Friderich Gauss Refractive On Line 2007 1 The Gaussian Curvature 13/09/2007 Renzo Mattioli Historical background: C.F. Gauss (1777-1855) was professor of mathematics in Göttingen (D). Has developed theories adopted in – Analysis – Number theory – Differential Geometry – Statistic (Guaussian distribution) – Physics (magnetism, electrostatics, optics, Astronomy) "Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss.“ Kronecker, L. Gaussian Curvature of a surface (Definition) Gaussian curvature is the product of steepest and flattest local curvatures, at each point. Negative GC Zero GC Positive GC Refractive On Line 2007 2 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gauss’ Theorema Egregium (1828) Gaussian curvature (C1 x C2) of a flexible surface is invariant. (non-elastic isometric distortion) C1=0 C1=0 C2>0 C2=0 C1 x C2 = 0 C1 x C2 = 0 Gauss’ Theorema Egregium Gaussian curvature is an intrinsic property of surfaces C1 C2 C1 C2 Curvature Gaussian Curvature Gaussian C2 C2 C1 x C2 C1 x C2 C1 C1 Refractive On Line 2007 3 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gauss’ Theorema Egregium • “A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection… preserves angles but fails to preserve area.” (Wikipedia) The Gaussian Map in Corneal Topography Refractive On Line 2007 4 The Gaussian Curvature 13/09/2007 Renzo Mattioli Proposed by B.Barsky et al. -
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. -
Manifolds and Riemannian Geometry Steinmetz Symposium
Manifolds and Riemannian Geometry Steinmetz Symposium Daniel Resnick, Christina Tonnesen-Friedman (Advisor) 14 May 2021 2 What is Differential Geometry? Figure: Surfaces of different curvature. Differential geometry studies surfaces and how to distinguish between them| and other such objects unique to each type of surface. Daniel Resnick (Union College) Manifolds and Riemannian Geometry 14 May 2021 2 / 16 3 Defining a Manifold We begin by defining a manifold. Definition A topological space M is locally Euclidean of dimension n if every point p 2 M has a neighborhood U such that there is a homeomorphism φ from U to an open subset of Rn. We call the pair (U; φ : U ! Rn) a chart. Definition A topological space M is called a topological manifold if the space is Hausdorff, second countable, and locally Euclidean. Daniel Resnick (Union College) Manifolds and Riemannian Geometry 14 May 2021 3 / 16 4 The Riemann Metric Geometry deals with lengths, angles, and areas, and \measurements" of some kind, and we will encapsulate all of these into one object. Definition The tangent space Tp(M) is the vector space of every tangent vector to a point p 2 M. Definition (Important!) A Riemann metric on a manifold to each point p in M of an inner product h ; ip on the tangent space TpM; moreover, the assignment p 7! h ; i is required to be C 1 in the following sense: if X and Y are C 1 vector fields on M, then 1 p 7! hXp; Ypip is a C function on M.A Riemann manifold is a pair (M; h ; i) consisting of the manifold M together with the Riemann metric h ; i on M. -
Euclid's Error: Non-Euclidean Geometries Present in Nature And
International Journal for Cross-Disciplinary Subjects in Education (IJCDSE), Volume 1, Issue 4, December 2010 Euclid’s Error: Non-Euclidean Geometries Present in Nature and Art, Absent in Non-Higher and Higher Education Cristina Alexandra Sousa Universidade Portucalense Infante D. Henrique, Portugal Abstract This analysis begins with an historical view of surfaces, we are faced with the impossibility of Geometry. One presents the evolution of Geometry solving problems through the same geometry. (commonly known as Euclidean Geometry) since its Unlike what happens with the initial four beginning until Euclid’s Postulates. Next, new postulates of Euclid, the Fifth Postulate, the famous geometric worlds beyond the Fifth Postulate are Parallel Postulate, revealed a lack intuitive appeal, presented, discovered by the forerunners of the Non- and several were the mathematicians who, Euclidean Geometries, as a result of the flaw that throughout history, tried to show it. Many retreated many mathematicians encountered when they before the findings that this would be untrue, some attempted to prove Euclid’s Fifth Postulate (the had the courage and determination to make such a Parallel Postulate). Unlike what happens with the falsehood, thus opening new doors to Geometry. initial four Postulates of Euclid, the Fifth Postulate One puts up, then, two questions. Where can be revealed a lack of intuitive appeal, and several were found the clear concepts of such Geometries? And the mathematicians who, throughout history, tried to how important is the knowledge and study of show it. Geometries, beyond the Euclidean, to a better understanding of the world around us? The study, After this brief perspective, a reflection is made now developed, seeks to answer these questions. -
Carl Friedrich Gauss
GENERAL I ARTICLE Carl Friedrich Gauss D Surya Ramana D Surya Ramana is a Carl Friedrich Gauss, whose mathematical work in the 19th junior research fellow century earned him the title of Prince among Mathematicians ,1 at the Institute of was born into the family of Gebhard Dietrich Gauss, a bricklayer Mathematical Sciences, at the German town of Brunswick, on 30 April 1777. Chennai. He works in the area of analytic number theory. Gauss was a child prodigy whose calculating prowess showed itself fairly early in his life. Stories of his childhood are full of anecdotes involving his abilities that first amazed his mother, then his teachers at school and much more recently his biographers to the present day. For instance, when he was barely ten years old he surprised his teachers by calculating the sum of the first hundred natural numbers using the fact that they So varied was the mathe f<lrmed an arithmetical progression. matical work of Gauss that this article is limited in scope by While his talents impressed his mother Dorothea, they caused the author's own special a great deal of anxiety to his father. Apparently afraid of losing interests. a hand that would otherwise ply some useful trade, he very reluctantly permitted his son to enroll in a gymnasium (high school) at Brunswick. Gauss studied here from 1788-1791. Perhaps conscious of the fact that his father would not sponsor his studies anymore, Gauss began to look for a patron who would finance his education beyond the gymnasium. Encouraged by his teachers and his mother, Gauss appeared in the court of Duke Carl Wilhelm Ferdinand of Brunswick to demonstrate his computational abilities. -
Sebastian's Space and Forms
GENERAL ARTICLE Sebastian’s Space and Forms Michele eMMer A strong message that mathematics revealed in the late nineteenth and non-Euclidean hyperbolic geometry) “imaginary geometry,” the early twentieth centuries is that geometry and space can be the because it was in such strong contrast with common sense. realm of freedom and imagination, abstraction and rigor. An example For some years non-Euclidean geometry remained marginal of this message lies in the infi nite variety of forms of mathematical AbStrAct inspiration that the Mexican sculptor Sebastian invented, rediscovered to the fi eld, a sort of unusual and curious genre, until it was and used throughout all his artistic activity. incorporated into and became an integral part of mathema- tics through the general ideas of G.F.B. Riemann (1826–1866). Riemann described a global vision of geometry as the study of intellectUAl ScAndAl varieties of any dimension in any kind of space. According to At the beginning of the twentieth century, Robert Musil re- Riemann, geometry had to deal not necessarily with points or fl ected on the role of mathematics in his short essay “Th e space in the traditional sense but with sets of ordered n-ples. Mathematical Man,” writing that mathematics is an ideal in- Th e erlangen Program of Felix Klein (1849–1925) descri- tellectual apparatus whose task is to anticipate every possible bed geometry as the study of the properties of fi gures that case. Only when one looks not toward its possible utility, were invariant with respect to a particular group of transfor- but within mathematics itself one sees the real face of this mations. -
Moon-Earth-Sun: the Oldest Three-Body Problem
Moon-Earth-Sun: The oldest three-body problem Martin C. Gutzwiller IBM Research Center, Yorktown Heights, New York 10598 The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler’s laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today’s many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem. -
Alexis Claude Clairaut
Alexis Claude Clairaut Born: 7 May 1713 in Paris, France Died: 17 May 1765 in Paris, France Alexis Clairaut's father, Jean-Baptiste Clairaut, taught mathematics in Paris and showed his quality by being elected to the Berlin Academy. Alexis's mother, Catherine Petit, had twenty children although only Alexis survived to adulthood. Jean-Baptiste Clairaut educated his son at home and set unbelievably high standards. Alexis used Euclid's Elements while learning to read and by the age of nine he had mastered the excellent mathematics textbook of Guisnée Application de l'algèbre à la géométrie which provided a good introduction to the differential and integral calculus as well as analytical geometry. In the following year, Clairaut went on to study de L'Hôpital's books, in particular his famous text Analyse des infiniment petits pour l'intelligence des lignes courbes. Few people have read their first paper to an academy at the age of 13, but this was the incredible achievement of Clairaut's in 1726 when he read his paper Quatre problèmes sur de nouvelles courbes to the Paris Academy. Although we have already noted that Clairaut was the only one of twenty children of his parents to reach adulthood, he did have a younger brother who, at the age of 14, read a mathematics paper to the Academy in 1730. This younger brother died in 1732 at the age of 16. Clairaut began to undertake research on double curvature curves which he completed in 1729. As a result of this work he was proposed for membership of the Paris Academy on 4 September 1729 but the king did not confirm his election until 1731. -
Basics of the Differential Geometry of Surfaces
Chapter 20 Basics of the Differential Geometry of Surfaces 20.1 Introduction The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We simply want to introduce the concepts needed to understand the notion of Gaussian curvature, mean curvature, principal curvatures, and geodesic lines. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate differential geometry course offered in the fall of 1994. Most of the topics covered in this course have been included, except a presentation of the global Gauss–Bonnet–Hopf theorem, some material on special coordinate systems, and Hilbert’s theorem on surfaces of constant negative curvature. What is a surface? A precise answer cannot really be given without introducing the concept of a manifold. An informal answer is to say that a surface is a set of points in R3 such that for every point p on the surface there is a small (perhaps very small) neighborhood U of p that is continuously deformable into a little flat open disk. Thus, a surface should really have some topology. Also,locally,unlessthe point p is “singular,” the surface looks like a plane. Properties of surfaces can be classified into local properties and global prop- erties.Intheolderliterature,thestudyoflocalpropertieswascalled geometry in the small,andthestudyofglobalpropertieswascalledgeometry in the large.Lo- cal properties are the properties that hold in a small neighborhood of a point on a surface. Curvature is a local property. Local properties canbestudiedmoreconve- niently by assuming that the surface is parametrized locally. -
Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time. -
From Abraham De Moivre to Johann Carl Friedrich Gauss
International Journal of Engineering Science Invention (IJESI) ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org ||Volume 7 Issue 6 Ver V || June 2018 || PP 28-34 A Brief Historical Overview Of the Gaussian Curve: From Abraham De Moivre to Johann Carl Friedrich Gauss Edel Alexandre Silva Pontes1 1Department of Mathematics, Federal Institute of Alagoas, Brazil Abstract : If there were only one law of probability to be known, this would be the Gaussian distribution. Faced with this uneasiness, this article intends to discuss about this distribution associated with its graph called the Gaussian curve. Due to the scarcity of texts in the area and the great demand of students and researchers for more information about this distribution, this article aimed to present a material on the history of the Gaussian curve and its relations. In the eighteenth and nineteenth centuries, there were several mathematicians who developed research on the curve, including Abraham de Moivre, Pierre Simon Laplace, Adrien-Marie Legendre, Francis Galton and Johann Carl Friedrich Gauss. Some researchers refer to the Gaussian curve as the "curve of nature itself" because of its versatility and inherent nature in almost everything we find. Virtually all probability distributions were somehow part or originated from the Gaussian distribution. We believe that the work described, the study of the Gaussian curve, its history and applications, is a valuable contribution to the students and researchers of the different areas of science, due to the lack of more detailed research on the subject. Keywords - History of Mathematics, Distribution of Probabilities, Gaussian Curve. ----------------------------------------------------------------------------------------------------------------------------- --------- Date of Submission: 09-06-2018 Date of acceptance: 25-06-2018 ----------------------------------------------------------------------------------------------------------------------------- ---------- I.