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# Topic: Theorema Egregium

###### In the News (Sat 25 May 13)

 [No title]   (Site not responding. Last check: 2007-11-03) From: jrl16@po.cwru.edu (Yossi Lonke) Subject: Gauss's Wonderful Theorem Date: 4 Jun 1999 08:25:44 -0400 Newsgroups: sci.math Keywords: Theorema egregium Hi All, A famous theorem due to Gauss, states that the Gaussian curvature of a surface remains invariant under surface-isometries. In particular, it is impossible to lay a plane triangle on a sphere. Gauss was so pleased with his theorem, that he named it "Theorema Egregium", which translates from Latin to : The wonderful theorem. www.math.niu.edu /~rusin/known-math/99/egregium   (188 words)

 Theorema Egregium - Wikipedia, the free encyclopedia The Theorema Egregium ('Remarkable Theorem') is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, that is, it does not depend on how the surface might be embedded in (3-dimensional) space. A somewhat whimsical application of the Theorema Egregium is seen in a common pizza-eating strategy: A slice of pizza can be seen as a surface with constant Gaussian curvature 0. en.wikipedia.org /wiki/Theorema_Egregium   (367 words)

 Gauss's Theorema Egregium -- from Wolfram MathWorld Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-space may be understood intrinsically to that surface. Gaussian curvature of the surface without ever venturing into full three-dimensional space; they can observe the curvature of the surface they live in without even knowing about the three-dimensional space in which they are embedded. Gray, A. "Gauss's Theorema Egregium." §22.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. mathworld.wolfram.com /GausssTheoremaEgregium.html   (212 words)

 Lecture Notes on Differential Geometry Intrinsic metric and isometries of surfaces, Gauss's Theorema Egregium, Brioschi's formula for Gaussian curvature. Gauss's formulas, Christoffel symbols, Gauss and Codazzi-Mainardi equations, Riemann curvature tensor, and a second proof of Gauss's Theorema Egregium. Self adjointness of the shape operator, Riemann curvature tensor of surfaces, Gauss and Codazzi Mainardi equations, and Theorema Egregium revisited. www.math.gatech.edu /~ghomi/LectureNotes   (357 words)

 Diff Geom Sprg 2000 worksheets While the focus of this class is on theory, there are nonetheless many places where it is appropriate to use a computer algebra system for visualization, for experimentation and quickly calculating some nontrivial examples. egregium.mws: A formal proof (by involved, but straightforward, calculation) of Gauss' Theorema Egregium. The following three are older worksheets from a class that explored the use of MAPLE to calculate and visualize objects and their properties. math.la.asu.edu /~kawski/classes/mat494s00/worksheets.html   (451 words)

 Math 442 - Differential Geometry - Description   (Site not responding. Last check: 2007-11-03) Other geometric ideas, such as geodesics on surfaces, will be introduced. Once we understand how to express Gaussian curvature in terms of the connection forms we are led to Gauss's Theorema Egregium, and the realization that an abstract (Riemannian) surface has a geometry of its own. We will see that there are many interesting surfaces which do not arise as surfaces in 3-space at all. www.math.uic.edu /~culler/math442/description.html   (271 words)

 GAUSS, Karl Friedrich, Disquisitiones generales circa superficies curvas.   (Site not responding. Last check: 2007-11-03) This work virtually created a new field of mathematical investigation, and led directly to the work of Riemann and the mathematical foundations of the general theory of relativity. Gauss here develops the equations of curved surfaces, the principle of the invariance under isometries of total curvature (the theorema egregium), with the derivative of conformal mapping (Gaussian mapping), and the theorems of angles of geodesic triangles and the sum of angles in small geodesic triangles (the Gauss-Bonnet theorem). Implicit in this work is a non-Euclidean geometry which Bolyai, Lobachevskii, and Riemann formally developed, and it was this paper which inspired Riemann's classic Ueber die Hypothesen, welche der Geometrie zu Grunde liegen, 1867. www.polybiblio.com /watbooks/2551.html   (393 words)

 diff geo images page These topics were explored in exercises in the course.Click on the picture to see the animated view. This graphic shows the first frame of an animated view of the Theorema Egregium of Gauss, that the (Gauss) curvature K of a surface is invariant under isometry. The green/yellow surface shown is a portion of the round sphere of radius 1. www.math.uiowa.edu /~wseaman/DGImage53100.htm   (3122 words)

 Read This: The Mathematics of Soap Films This chapter is particularly well-written, with great care taken not to lose students merely through terminology. A simple example of this is the first sentence in the proof of Theorem 2.5.3, Gauss' famous "Theorema Egregium". Oprea states "Because mixed partial derivatives are equal no matter the order of differentiation,...." instead of just saying that mixed partials commute. www.maa.org /reviews/maplesoap.html   (1455 words)

 MATH 320: Course Description Computing curvature and torsion for curves that are NOT parametrized by arclength (solution of Problem 12abc on p. Proof of THEOREMA EGREGIUM of Gauss: ps and pdf Due dates will be announced in class and on the course web page. www.math.mcgill.ca /~jakobson/courses/math320.html   (626 words)

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