Gaussian curvature

In the theory of surfaces in three-dimensional Euclidean space ( ), a field of differential geometry , Gaussian curvature (the Gaussian measure of curvature ), named after the mathematician Carl Friedrich Gauß , is the most important term for curvature besides mean curvature . ${\displaystyle \mathbb {R} ^{3}}$

definition

Let a regular area in and a point of this area be given. The Gaussian curvature of the surface at this point is the product of the two main curvatures and . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle K}$ ${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$

${\displaystyle K\,=\,k_{1}\cdot k_{2}={\frac {1}{r_{1}}}\cdot {\frac {1}{r_{2}}}}$

Where and are the two main radii of curvature. ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$

Examples

• In the case of a sphere (surface) with a radius , the Gaussian curvature is given by . ${\displaystyle r}$ ${\displaystyle K=1/r^{2}}$

• At any point on the curved surface of a right circular cylinder , the Gaussian curvature is zero.

• At any point on the curved surface of a right circular cone , the Gaussian curvature is zero.

calculation

• Are , , and , , the coefficients of the first and second fundamental form , then the following formula applies: ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle L}$ ${\displaystyle M}$ ${\displaystyle N}$

${\displaystyle K={\frac {LN-M^{2}}{EG-F^{2}}}}$

• If the area under consideration is the graph of a function over the parameter range , i.e. for all , then applies to the Gaussian curvature: ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle X(u,v)=(u,v,f(u,v))}$ ${\displaystyle (u,v)\in U}$

${\displaystyle K={\frac {f_{uu}f_{vv}-f_{uv}^{2}}{{(1+f_{u}^{2}+f_{v}^{2})}^{2}}}}$

Here and denote the first and , and the second partial derivatives of . ${\displaystyle f_{u}}$ ${\displaystyle f_{v}}$ ${\displaystyle f_{uu}}$ ${\displaystyle f_{uv}}$ ${\displaystyle f_{vv}}$ ${\displaystyle f}$

• If the area is given as a set of zeros of a function with a regular value , then the Gaussian curvature is calculated from the formula ${\displaystyle f^{-1}(0)}$ ${\displaystyle f\colon \mathbb {R} ^{3}\to \mathbb {R} }$ ${\displaystyle 0\in \mathbb {R} }$

${\displaystyle K={\frac {{\nabla f}^{T}\cdot \operatorname {adj} (H_{f})\cdot \nabla f}{|\nabla f|^{4}}}.}$

Here, the amount of the gradient and the cofactor of the Hessian matrix of . ${\displaystyle |\nabla f|}$ ${\displaystyle \operatorname {adj} (H_{f})}$ ${\displaystyle f}$

properties

sign

In elliptical points , the Gaussian curvature is positive ( ), in hyperbolic points negative ( ) and in parabolic or flat points it disappears. ${\displaystyle K>0}$ ${\displaystyle K<0}$

Examples:

• In the case of a bicycle tube (= torus), the points on the rim are hyperbolic and the points on the outside are elliptical. The two dividing lines of these two areas are two circles whose points are parabolic.
• An ellipsoid has only elliptical points, a paraboloid (= saddle surface) has only hyperbolic points.

Property of internal geometry

The Gaussian curvature only depends on the internal geometry of the given surface (see CF Gauss's Theorema egregium ). This sentence is a corollary from Brioschi's formula:

${\displaystyle K={\frac {1}{(EG-F^{2})^{2}}}\left({\begin{vmatrix}-{\frac {1}{2}}E_{vv}+F_{uv}-{\frac {1}{2}}G_{uu}&{\frac {1}{2}}E_{u}&F_{u}-{\frac {1}{2}}E_{v}\\F_{v}-{\frac {1}{2}}G_{u}&E&F\\{\frac {1}{2}}G_{v}&F&G\end{vmatrix}}-{\begin{vmatrix}0&{\frac {1}{2}}E_{v}&{\frac {1}{2}}G_{u}\\{\frac {1}{2}}E_{v}&E&F\\{\frac {1}{2}}G_{u}&F&G\end{vmatrix}}\right)}$

Where , and are the coefficients of the first fundamental form. The terms , etc. stand for the first and second partial derivatives according to the parameters and , with which the given area is parameterized. This equation is, among other things, one of the necessary integration conditions of the Gauß-Weingarten equations . ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle E_{u}}$ ${\displaystyle F_{uv}}$ ${\displaystyle u}$ ${\displaystyle v}$

Another formula for calculating the Gaussian curvature is:

${\displaystyle K=-{\frac {1}{2{\sqrt {EG-F^{2}}}}}\left(\left({\frac {E_{v}-F_{u}}{\sqrt {EG-F^{2}}}}\right)_{v}+\left({\frac {G_{u}-F_{v}}{\sqrt {EG-F^{2}}}}\right)_{u}\right)-{\frac {1}{4\left(EG-F^{2}\right)^{2}}}{\begin{vmatrix}E&E_{u}&E_{v}\\F&F_{u}&F_{v}\\G&G_{u}&G_{v}\end{vmatrix}}}$

In the case of orthogonal parameterization ( ) this formula is reduced to ${\displaystyle F=0}$

${\displaystyle K=-{\frac {1}{2{\sqrt {EG}}}}\left(\left({\frac {E_{v}}{\sqrt {EG}}}\right)_{v}+\left({\frac {G_{u}}{\sqrt {EG}}}\right)_{u}\right)}$

If the area is parameterized isothermally, i. i.e., it applies and , then writes itself ${\displaystyle 0<E=G}$ ${\displaystyle F=0}$

${\displaystyle K=-{\frac {1}{2E}}\Delta \log E}$

with the Laplace operator

${\displaystyle \Delta ={\frac {\partial ^{2}}{\partial u^{2}}}+{\frac {\partial ^{2}}{\partial v^{2}}}}$.

Total curvature

The inside angle sum of a surface triangle on a negatively curved surface is less than 180 °.

The surface integral

${\displaystyle \iint _{T}K\,dA}$

the Gaussian curvature over a subset of a surface is called its total curvature . In the case of polygons whose edges are geodesic , there is a relationship between the total curvature and the sum of the interior angles. For example, the sum of the interior angles of a geodetic triangle applies : ${\displaystyle K}$ ${\displaystyle T}$ ${\displaystyle \alpha +\beta +\gamma }$

${\displaystyle \alpha +\beta +\gamma =\pi +\iint _{T}K\,dA.}$

The total curvature of a geodetic triangle thus corresponds to the deviation of the internal angle sum from : The internal angle sum of a triangle located on a positively curved surface exceeds , on a negatively curved surface the internal angle sum is below . If the Gaussian curvature is zero, the sum of the interior angles is exactly as in the flat case . ${\displaystyle \pi }$ ${\displaystyle \pi }$ ${\displaystyle \pi }$ ${\displaystyle \pi }$

A generalization of this fact is the Gauss-Bonnet theorem , which describes a connection between the Gaussian curvature of a surface and the geodetic curvature of the associated boundary curve.

literature

• Manfredo Perdigão do Carmo: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Upper Saddle River NJ 1976, ISBN 0-13-212589-7 .

Individual evidence

1. ↑ Michael Spivak: A comprehensive introduction to differential geometry . 3. Edition. Volume 3. Publish or Perish, Houston, Texas 1999, ISBN 0-914098-72-1 , Chapter 3. A compendium of surfaces.