All Topics  
Theorema Egregium

 

 
   Email Print
   Bookmark   Link
 

 

Theorema Egregium
 
 
  Topics Theorema Egregium Topic Home

Gauss's Theorema Egregium (Latin: "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss
Carl Friedrich Gauss

Johann Carl Friedrich Gauss. was a Germans mathematician and scientist who contributed significantly to many fields, including number theory, statistics, mathematical analysis, Differential geometry and topology, geodesy, electrostatics, astronomy and optics....
 that concerns the curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 of surfaces. Informally, the theorem says that the Gaussian curvature
Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
 of a surface can be determined entirely by measuring angles and distances on the surface itself, without further reference to the particular way in which the surface is situated in the ambient 3-dimensional Euclidean space.






Discussion
Ask a question about 'Theorema Egregium'
Start a new discussion about 'Theorema Egregium'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Gauss's Theorema Egregium (Latin: "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss
Carl Friedrich Gauss

Johann Carl Friedrich Gauss. was a Germans mathematician and scientist who contributed significantly to many fields, including number theory, statistics, mathematical analysis, Differential geometry and topology, geodesy, electrostatics, astronomy and optics....
 that concerns the curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 of surfaces. Informally, the theorem says that the Gaussian curvature
Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
 of a surface can be determined entirely by measuring angles and distances on the surface itself, without further reference to the particular way in which the surface is situated in the ambient 3-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant
Invariant (mathematics)

In mathematics, an invariant is something that does not change under a set of Transformation s. The property of being an invariant is invariance....
 of a surface.

Gauss presented the theorem in this way (translated from Latin):

Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.


The theorem is "remarkable" because the definition of Gaussian curvature makes direct use of the position of the surface in space. So it is quite surprising that the end result does not depend on the embedding.

In modern mathematical language, the theorem may be stated as follows:

The Gaussian curvature
Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
 of a surface is invariant under local isometry.


Elementary applications


Helicatenoid
A sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 of radius R has constant Gaussian curvature which is equal to R−2. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances: mathematically speaking, a sphere and a plane are not isometric
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
, even locally. This fact is of enormous significance for cartography
Cartography

File:Mediterranean chart fourteenth century2.jpgCartography is the study and practice of making Geography Map. Combining science, aesthetics, and technique, cartography builds on the premise that we can model reality in ways that communicate spatial information effectively....
: it implies that no perfect map of Earth can be created, even for a portion of the Earth's surface. Thus every cartographic projection
Map projection

A map projection is any method of representing the surface of a sphere or other shape on a Plane . Map projections are necessary for creating maps....
 necessarily distorts at least some distances.

The catenoid
Catenoid

A catenoid is a three-dimensional shape made by rotating a catenary curve around the axis. Not counting the plane, it is the first minimal surface to be discovered....
 and the helicoid
Helicoid

The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776....
 are two very different-looking surfaces. Nevertheless, each of them can be continuously bent into the other: they are locally isometric. It follows from Theorema Egregium that the Gaussian curvature at the two points of the catenoid and helicoid corresponding to each other under this bending is the same.

See also

  • Second fundamental form
    Second fundamental form

    In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a differential geometry of surfaces in the three dimensional Euclidean space, usually denoted by II....
  • Gaussian curvature
    Gaussian curvature

    In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
  • Differential geometry of surfaces
    Differential geometry of surfaces

    In mathematics, the differential geometry of surfaces deals with smooth manifold surfaces with various additional structures, most often, a Riemannian metric....