Gaussian curvature is a numerical quantity associated with an area of a surface that describes the intrinsic geometric property of that area. It is different from the curvature of a curve, for that is an extrinsic geometric property defining how it is bent in a plane or space. The Gaussian curvature remains the same no matter how a surface is bent as long as it is not distorted. It is defined as the product of the principal curvatures, the maximum and minimum values of normal curvature at a point on a surface. Since it is the product of two curvatures, Gaussian curvature has the units of curvature squared. If the Gaussian curvature is a positive value then the surface is locally either a peak (both the maximum and minimum values are positive, hence the product is positive) or a valley (both the maximum and minimum values are negative, hence the product is positive). A negative value indicates that the surface has saddle points locally, that is either the maximum or minimum value is negative and the other is positive. A value of zero is indicative of a surface that is flat in at least one direction. Surfaces such as a plane and a cylinder have zero Gaussian curvature.
Gaussian curvature can be thought of in another way. The Gaussian curvature of a surface determines when one surface can or cannot be bent into another. The total Gaussian curvature of a region on a surface can be determined by opening up the surface and measuring the angle by which it is opened up. For instance, if one cuts a small portion of a sphere out and cuts open the portion so that the material is free to flatten out one can measure this angle. The angle between the tangents to the curve at the two sides of the cut is the total Gaussian curvature. This numerical quantity associated to an area of a surface is very closely related to angle defect. In fact, the total Gaussian curvature of a region of a polyhedron containing only one vertex is the angle defect at that vertex. To think of Gaussian curvature in this way it is useful to think of trying to coerce a flat sheet of paper into a sphere. It cannot be done unless some of the paper is removed or distorted in some way. The Gaussian curvature of a sphere is not equal to that of a flat sheet of paper and therefore one cannot shape a piece of paper into a sphere. On the other hand it is possible to fold a flat sheet of paper into a cylinder. Both the cylinder and flat sheet of paper have Gaussian curvatures of zero and so can be inter-converted from one to the other.
The Gaussian curvature of a function describing a two-dimensional object vanishes wherever the function is locally one-dimensional. So for a plane the Gaussian curvature is zero as it is for a cylinder. The formal form for the Gaussian curvature of a regular surface in three-dimensional space at any point p is G(p) = det(S(p)), Where S is the shape operator and det signifies the determinant. The shape operator, or Weingertan map or second fundamental tensor as it is sometimes called, is an extrinsic curvature. Gaussian curvature is also given by G(p) = 1/R1R2 = K1K2, Where R1R2 are the principal radii of curvature and K1K2 are the principal curvatures. While a surface on which the Gaussian curvature is always positive is called synclastic a surface on which the Gaussian curvature is everywhere negative is called anticlastic. Surfaces with constant Gaussian curvature include the sphere, cone, cylinder, and plane. In areas of the surface which bend highly the curvature is high, whereas areas where bending is slight the curvature is low.
Gaussian curvature is used to explore and describe curved surfaces and spaces. While doing geodetic survey work for the governments of Hanover and Denmark in 1821 a German scientist, Carl Friedrich Gauss, developed the concept of Gaussian curvature. He used a Gauss map to define curvature. A Gauss map is a function describing the distance to a sphere from an orientable surface in Euclidean space. It associates its oriented normal vector to every point on the surface. As well as playing a fundamental role in Einstein's theory of gravitation, smooth surface points can be catalogued by their Gaussian and toric curvature. Since Gaussian curvature has units of curvature squared many corneal topography advocates are tempted to redefine Gaussian curvature as its square root. The square root of Gaussian curvature is known as geometric mean curvature or root Gaussian curvature in these situations although it is of no real use in corneal topography. Gaussian curvature is especially useful in designing pressing process for items such as car bodies. Shapes with nonzero Gaussian curvature will require that some material is stretched or deformed in the pressing process. To do this effectively there has to be a balance in the stretching and amount of material and such stress a material can tolerate.
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