Theorema Egregium
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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. 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. Thus the Gaussian curvature is an intrinsic invariant 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 of a surface is invariant under local isometry.
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[edit] Elementary applications
A sphere 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, even locally. 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.[1]
Catenoid and the helicoid are two very different-looking surfaces. 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.
[edit] Notes
- ^ Geodetical applications were one of the primary motivations for Gauss's "investigations of the curved surfaces".
[edit] See also
[edit] References
- Karl Friedrich Gauss, General Investigations of Curved Surfaces of 1827 and 1825, (1902) The Princeton University Library. (A translation of Gauss's original paper.)
