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Gaussian curvature

 

Gaussian curvature

From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere).

In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point. It is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in any space. It is named after Carl Friedrich Gauss, and is the content of his Theorema egregium.

Symbolically, the Gaussian curvature Κ is defined as

\Kappa = \kappa_1 \kappa_2,

where κ1 and κ2 are the principal curvatures.

Contents

  • Informal definition 1
  • Relation to geometries 2
  • Further informal discussion 3
  • Alternative definitions 4
  • Total curvature 5
  • Important theorems 6
    • Theorema egregium 6.1
    • Gauss–Bonnet theorem 6.2
  • Surfaces of constant curvature 7
  • Alternative formulas 8
  • See also 9
  • References 10
  • External links 11

Informal definition

Saddle surface with normal planes in directions of principal curvatures

At any point on a surface we can find a normal vector which is at right angles to the surface; planes containing the normal are called normal planes. The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these κ1, κ2. The Gaussian curvature is the product of the two principal curvatures Κ = κ1 κ2.

The sign of the Gaussian curvature can be used to characterise the surface.

  • If both principal curvatures are the same sign: κ1κ2 > 0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
  • If the principal curvatures have different signs: κ1κ2 < 0, then the Gaussian curvature is negative and the surface is said to have a hyperbolic point. At such points the surface will be saddle shaped. For two directions the sectional curvatures will be zero giving the asymptotic directions.
  • If one of the principal curvatures is zero: κ1κ2 = 0, the Gaussian curvature is zero and the surface is said to have a parabolic point.

Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line.

Relation to geometries

When a surface has a constant zero Gaussian curvature then it is a developable surface and the geometry of the surface is Euclidean geometry.

When a surface has a constant positive Gaussian curvature then it is a sphere and the geometry of the surface is spherical geometry.

When a surface has a constant negative Gaussian curvature then if this curvature then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.

Further informal discussion

In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point. We represent the surface by the implicit function theorem as the graph of a function, f, of two variables, in such a way that the point p is a critical point, i.e., the gradient of f vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2-by-2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between cup/cap versus saddle point.

Alternative definitions

It is also given by

\Kappa = \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\rangle}{\det g},

where \nabla_i = \nabla_{(EG-F^2)^2}

  • For an orthogonal parametrization (i.e., F = 0), Gaussian curvature is:
K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v}\frac{E_v}{\sqrt{EG}}\right).
  • For a surface described as graph of a function z = F(x, y), Gaussian curvature is:
K = \frac{F_{xx}\cdot F_{yy}- F_{xy}^2}{(1+F_x^2+ F_y^2)^2}
  • For a surface F(x,y,z) = 0, the Gaussian curvature can be expressed in terms of the gradient \nabla F and Hessian matrix H(F):[5][6]
K=-\frac{ \det \begin{vmatrix} H(F) & \nabla F^{\mathsf T} \\ \nabla F & 0 \end{vmatrix} }{ |\nabla F|^4 } =-\frac{ \det\begin{vmatrix} F_{xx} & F_{xy} & F_{xz} & F_x \\ F_{xy} & F_{yy} & F_{yz} & F_y \\ F_{xz} & F_{yz} & F_{zz} & F_z \\ F_{x} & F_{y} & F_{z} & 0 \\ \end{vmatrix} }{ |\nabla F|^4 }
  • For a surface with metric conformal to the Euclidean one, so F = 0 and E = G = eσ, the Gauss curvature is given by (Δ being the usual Laplace operator):
K = -\frac{1}{2e^\sigma}\Delta \sigma,
  • Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:[7]
K = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3}
  • Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane:[7]
K = \lim_{r\to 0^+}12\frac{\pi r^2-A(r)}{\pi r^4 }
K = -\frac{1}{E} \left( \frac{\partial}{\partial u}\Gamma_{12}^2 - \frac{\partial}{\partial v}\Gamma_{11}^2 + \Gamma_{12}^1\Gamma_{11}^2 - \Gamma_{11}^1\Gamma_{12}^2 + \Gamma_{12}^2\Gamma_{12}^2 - \Gamma_{11}^2\Gamma_{22}^2\right)

See also

References

  1. ^ Porteous, I. R., Geometric Differentiation. Cambridge University Press, 1994. ISBN 0-521-39063-X
  2. ^ Kühnel, Wolfgang (2006). Differential Geometry: Curves - Surfaces - Manifolds. American Mathematical Society.  
  3. ^ Gray, Mary (1997), "28.4 Hilbert's Lemma and Liebmann's Theorem", Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.), CRC Press, pp. 652–654,  .
  4. ^ . Springer Online Reference Works.Hilbert theorem
  5. ^ Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces". Computer Aided Geometric Design 22 (7): 632.  
  6. ^ Spivak, M (1975). A Comprehensive Introduction to Differential Geometry 3. Publish or Perish, Boston. 
  7. ^ a b Bertrand–Diquet–Puiseux theorem
  8. ^ Struik, Dirk (1988). Lectures on Classical Differential Geometry. Courier Dover Publications.  

External links

  • Hazewinkel, Michiel, ed. (2001), "Gaussian curvature",  
  • Curvature in two spacelike dimensions
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