Gaussian Curvature

Gaussian curvature, sometimes also called total curvature (Kreyszig 1991, p. 131), is an intrinsic property of a space independent of the coordinate system used to describe it. The Gaussian curvature of a regular surface in R^3 at a point p is formally defined as


where S is the shape operator and det denotes the determinant.

If x:U->R^3 is a regular patch, then the Gaussian curvature is given by


where E, F, and G are coefficients of the first fundamental form and e, f, and g are coefficients of the second fundamental form (Gray 1997, p. 377). The Gaussian curvature can be given entirely in terms of the first fundamental form


and the metric discriminant




where Gamma_(ij)^k are Christoffel symbols of the first kind. Equivalently,

 K=1/(g^2)|E F (partialF)/(partialv)-1/2(partialG)/(partialu); F G 1/2(partialG)/(partialv); 1/2(partialE)/(partialu) k_(23) k_(33)|-1/(g^2)|E F 1/2(partialE)/(partialv); F G 1/2(partialG)/(partialu); 1/2(partialE)/(partialv) 1/2(partialG)/(partialu) 0|,



Writing this out,


The Gaussian curvature is also given by


(Gray 1997, p. 380), as well as


where epsilon^(ij) is the permutation symbol, N^^ is the unit normal vector and T^^ is the unit tangent vector. The Gaussian curvature is also given by


where R is the scalar curvature, kappa_1 and kappa_2 the principal curvatures, and R_1 and R_2 the principal radii of curvature. For a Monge patch with z=h(u,v),


The Gaussian curvature of a surface defined implicitly by F(x,y,z)=0 is given by


(Trott 2004, pp. 1285-1286).

The Gaussian curvature K and mean curvature H satisfy


with equality only at umbilic points, since


If p is a point on a regular surface M subset R^3 and v_(p) and w_(p) are tangent vectors to M at p, then the Gaussian curvature of M at p is related to the shape operator S by


Let Z be a nonvanishing vector field on M which is everywhere perpendicular to M, and let V and W be vector fields tangent to M such that VxW=Z, then


(Gray 1997, p. 410).

For a sphere, the Gaussian curvature is K=1/a^2. For Euclidean space, the Gaussian curvature is K=0. For Gauss-Bolyai-Lobachevsky space, the Gaussian curvature is K=-1/a^2. A developable surface is a regular surface and special class of minimal surface on which Gaussian curvature vanishes everywhere.

A point p on a regular surface M in R^3 is classified based on the sign of K(p) as given in the following table (Gray 1997, p. 375), where S is the shape operator.

A surface on which the Gaussian curvature K is everywhere positive is called synclastic, while a surface on which K is everywhere negative is called anticlastic. Surfaces with constant Gaussian curvature include the cone, cylinder, Kuen surface, plane, pseudosphere, and sphere. Of these, the cone and cylinder are the only developable surfaces of revolution.

See also

Anticlastic, Brioschi Formula, Developable Surface, Elliptic Point, Hyperbolic Point, Integral Curvature, Mean Curvature, Metric Tensor, Minimal Surface, Parabolic Point, Planar Point, Scalar Curvature, Synclastic, Total Curvature, Umbilic Point Explore this topic in the MathWorld classroom

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Geometry Center. "Gaussian Curvature.", A. "The Gaussian and Mean Curvatures" and "Surfaces of Constant Gaussian Curvature." §16.5 and Ch. 21 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 373-380 and 481-500, 1997.Kreyszig, E. Differential Geometry. New York: Dover, p. 131, 1991.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004.

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Gaussian Curvature

Cite this as:

Weisstein, Eric W. "Gaussian Curvature." From MathWorld--A Wolfram Web Resource.

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