A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, Regions, and Themes The Mercator projection, shown here, preserves angles but fails to preserve area. The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569 In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane

Gauss's Theorema Egregium (Latin: "Remarkable Theorem"), is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry Informally, the theorem says that the Gaussian curvature 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 situtated in the ambient 3-dimensional Euclidean space. In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures In Mathematics, the differential geometry of surfaces deals with smooth Surfaces with various additional structures most often a Riemannian metric Thus the Gaussian curvature is an intrinsic invariant of a surface. The term intrinsic denotes a characteristic or property of some thing or action which is essential and specific to that thing or action and which is wholly independent In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance.

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 of a surface is invariant under local isometry. In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures For the Mechanical engineering and Architecture usage see Isometric projection.

Elementary applications

Animation showing the deformation of a helicoid into a catenoid. Generated with Mac OS X Grapher. Grapher is a Software program bundled with Mac OS X v104 and 10

A sphere of radius R has constant Gaussian curvature which is equal to R⁻². "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe 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, even locally. For the Mechanical engineering and Architecture usage see Isometric projection. This fact is of enormous significance for cartography: it implies that no perfect map of Earth can be created, even for a portion of the Earth's surface. Thus every cartographic projection necessarily distorts at least some distances. A map projection is any method of representing the Surface of a sphere or other shape on a plane. ^[1]

Catenoid and the helicoid are two very different-looking surfaces. A catenoid is a three- Dimensional Shape made by rotating a Catenary Curve around the x axis The helicoid, after the plane and the Catenoid, is the third Minimal surface to be known Nevertheless, each of them can be continuosly 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.

Notes

1. ^ Geodetical applications were one of the primary motivations for Gauss's "investigations of the curved surfaces". Geodesy (dʒiːˈɒdɪsi also called geodetics, a branch of Earth sciences, is the scientific discipline that deals

See also

• Second fundamental form
• Gaussian curvature

References

• Karl Friedrich Gauss, General Investigations of Curved Surfaces of 1827 and 1825, (1902) The Princeton University Library. In Differential geometry, the second fundamental form is a Quadratic form on the Tangent plane of a smooth surface in the three dimensional In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures (A translation of Gauss's original paper. )