Английская Википедия:Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature Шаблон:Mvar of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, Шаблон:Math and Шаблон:Math, at the given point: <math display="block"> K = \kappa_1 \kappa_2.</math> The Gaussian radius of curvature is the reciprocal of Шаблон:Mvar. For example, a sphere of radius Шаблон:Mvar has Gaussian curvature Шаблон:Math everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the Theorema egregium.
Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.
Informal definition
At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector 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 “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these Шаблон:Math, Шаблон:Math. The Gaussian curvature is the product of the two principal curvatures Шаблон:Math.
The sign of the Gaussian curvature can be used to characterise the surface.
- If both principal curvatures are of the same sign: Шаблон:Math, 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: Шаблон:Math, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point. At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the asymptotic curves for that point.
- If one of the principal curvatures is zero: Шаблон:Math, 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 the geometry of the surface is spherical geometry. Spheres and patches of spheres have this geometry, but there exist other examples as well, such as the lemon / American football.
When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.
Relation to principal curvatures
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 from that point. We represent the surface by the implicit function theorem as the graph of a function, Шаблон:Mvar, of two variables, in such a way that the point Шаблон:Mvar is a critical point, that is, the gradient of Шаблон:Mvar vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at Шаблон:Mvar is the determinant of the Hessian matrix of Шаблон:Mvar (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2×2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between a cup/cap versus a saddle point.
Alternative definitions
It is also given by <math display="block">K = \frac{\bigl\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\bigr\rangle}{\det g},</math> where Шаблон:Math is the covariant derivative and Шаблон:Mvar is the metric tensor.
At a point Шаблон:Math on a regular surface in Шаблон:Math, the Gaussian curvature is also given by <math display="block">K(\mathbf{p}) = \det S(\mathbf{p}),</math> where Шаблон:Mvar is the shape operator.
A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates.
Total curvature
The surface integral of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a geodesic triangle equals the deviation of the sum of its angles from Шаблон:Pi. The sum of the angles of a triangle on a surface of positive curvature will exceed Шаблон:Pi, while the sum of the angles of a triangle on a surface of negative curvature will be less than Шаблон:Pi. On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely Шаблон:Pi radians. <math display="block">\sum_{i=1}^3 \theta_i = \pi + \iint_T K \,dA.</math>
A more general result is the Gauss–Bonnet theorem.
Important theorems
Theorema egregium
Шаблон:Main Gauss's Theorema egregium (Latin: "remarkable theorem") states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. In fact, it can be found given the full knowledge of the first fundamental form and expressed via the first fundamental form and its partial derivatives of first and second order. Equivalently, the determinant of the second fundamental form of a surface in Шаблон:Math can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface Шаблон:Mvar in Шаблон:Math certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the intrinsic metric of the surface without any further reference to the ambient space: it is an intrinsic invariant. In particular, the Gaussian curvature is invariant under isometric deformations of the surface.
In contemporary differential geometry, a "surface", viewed abstractly, is a two-dimensional differentiable manifold. To connect this point of view with the classical theory of surfaces, such an abstract surface is embedded into Шаблон:Math and endowed with the Riemannian metric given by the first fundamental form. Suppose that the image of the embedding is a surface Шаблон:Mvar in Шаблон:Math. A local isometry is a diffeomorphism Шаблон:Math between open regions of Шаблон:Math whose restriction to Шаблон:Math is an isometry onto its image. Theorema egregium is then stated as follows:
For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat).[1]Шаблон:Page needed On the other hand, since a sphere of radius Шаблон:Mvar has constant positive curvature Шаблон:Math and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally. Thus any planar representation of even a small part of a sphere must distort the distances. Therefore, no cartographic projection is perfect.
Gauss–Bonnet theorem
Шаблон:Main The Gauss–Bonnet theorem relates the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties.
- <math>\int_M K\,dA+\int_{\partial M}k_g\,ds=2\pi\chi(M), \, </math>
Surfaces of constant curvature
- Minding's theorem (1839) states that all surfaces with the same constant curvature Шаблон:Mvar are locally isometric. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called developable surfaces. Minding also raised the question of whether a closed surface with constant positive curvature is necessarily rigid.
- Liebmann's theorem (1900) answered Minding's question. The only regular (of class Шаблон:Math) closed surfaces in Шаблон:Math with constant positive Gaussian curvature are spheres.[2] If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof uses Hilbert's lemma that non-umbilical points of extreme principal curvature have non-positive Gaussian curvature.[3]
- Hilbert's theorem (1901) states that there exists no complete analytic (class Шаблон:Math) regular surface in Шаблон:Math of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class Шаблон:Math immersed in Шаблон:Math, but breaks down for Шаблон:Math-surfaces. The pseudosphere has constant negative Gaussian curvature except at its boundary circle, where the gaussian curvature is not defined.
There are other surfaces which have constant positive Gaussian curvature. Manfredo do Carmo considers surfaces of revolution <math>(\phi(v) \cos(u), \phi(v) \sin(u), \psi(v))</math> where <math>\phi(v) = C \cos v</math>, and <math display="inline"> \psi(v) = \int_0^v \sqrt{1-C^2 \sin^2 v'}\ dv'</math> (an incomplete Elliptic integral of the second kind). These surfaces all have constant Gaussian curvature of 1, but, for <math>C\ne 1</math> either have a boundary or a singular point. do Carmo also gives three different examples of surface with constant negative Gaussian curvature, one of which is pseudosphere.[4]
There are many other possible bounded surfaces with constant Gaussian curvature. Whilst the sphere is rigid and can not be bent using an isometry, if a small region removed, or even a cut along a small segment, then the resulting surface can be bent. Such bending preserves Gaussian curvature so any such bending of a sphere with a region removed will also have constant Gaussian curvature.[5]
Alternative formulas
- Gaussian curvature of a surface in Шаблон:Math can be expressed as the ratio of the determinants of the second and first fundamental forms Шаблон:Math and Шаблон:Math: <math display="block">K = \frac{\det(\mathrm{I\!I})}{\det(\mathrm I)} = \frac{LN-M^2}{EG-F^2}.</math>
- The Шаблон:Vanchor (after Francesco Brioschi) gives Gaussian curvature solely in terms of the first fundamental form: <math display="block"> K =\frac{\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}}{\left(EG - F^2\right)^2} </math>
- For an orthogonal parametrization (Шаблон:Math), Gaussian curvature is: <math display="block">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).</math>
- For a surface described as graph of a function Шаблон:Math, Gaussian curvature is:[6] <math display="block">K = \frac{F_{xx}\cdot F_{yy}- F_{xy}^2}{\left(1+F_x^2+ F_y^2\right)^2}</math>
- For an implicitly defined surface, Шаблон:Math, the Gaussian curvature can be expressed in terms of the gradient Шаблон:Math and Hessian matrix Шаблон:Math:[7][8] <math display="block">
K = -\frac{
\begin{vmatrix}
H(F) & \nabla F^{\mathsf T} \\ \nabla F & 0 \end{vmatrix} }{ |\nabla F|^4 } =-\frac{ \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 } </math>
- For a surface with metric conformal to the Euclidean one, so Шаблон:Math and Шаблон:Math, the Gauss curvature is given by (Шаблон:Math being the usual Laplace operator): <math display="block"> K = -\frac{1}{2e^\sigma}\Delta \sigma.</math>
- Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:[9] <math display="block"> K = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3}</math>
- Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane:[9] <math display="block">K = \lim_{r\to 0^+}12\frac{\pi r^2-A(r)}{\pi r^4 } </math>
- Gaussian curvature may be expressed with the Christoffel symbols:[10] <math display="block">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)</math>
See also
- Earth's Gaussian radius of curvature
- Sectional curvature
- Mean curvature
- Gauss map
- Riemann curvature tensor
- Principal curvature
References
Books
External links
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