Ngô Quốc Anh

December 10, 2009

R-G: Curvature

Filed under: Riemannian geometry — Ngô Quốc Anh @ 13:17

In this topic we discuss Gauss’ work to explain the concept of curvature.

The curvature of a curve in a plane is determined by how fast its unit normal vector n (or the tangent vector for that matter) changes as we move along the curve. A measure of curvature is the ratio of the small change |dn| in the unit normal vector to the distance ds moved by the point on the curve.

A straight line has zero curvature because the unit normals are all parallel and do not change. A circle of radius R has curvature \frac{1}{R} because for the distance ds that the point P moves, the unit normal vector changes by an angle \frac{ds}{R} so |dn| = \frac{ds}{R}.

Gauss defined the curvature of a surface analogously.

  • Curvature of plane curves

For a plane curve C, the mathematical definition of curvature uses a parametric representation of C with respect to the arc length parametrization. It can be computed given any regular parametrization by a more complicated formula given below.

Curvature: Let \gamma (s) be a regular parametric curve, where s is the arc length, or natural parameter. This determines the unit tangent vector T, the unit normal vector N(s), the curvature \kappa (s), the oriented or signed curvature k(s), and the radius of curvature R(s) at each point:

\displaystyle T(s)=\gamma'(s),\quad T'(s)=\kappa (s)N(s),\quad \kappa (s) = \|\gamma''(s)\| = \left|k(s)\right|, \quad R(s)=\frac{1}{\kappa(s)}.

Local expressions. For a plane curve given parametrically as c(t) = (x(t),y(t)), the curvature is

\displaystyle\kappa = \frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}},

and the signed curvature k is

\displaystyle k = \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}.

For the less general case of a plane curve given explicitly as y = f(x) the curvature is

\displaystyle\kappa = \frac{|y''|}{(1+y'^2)^{3/2}}.

Slightly abusing notation, the signed curvature may also be written in this way as

\displaystyle k=\frac{y''}{(1+y'^2)^{3/2}}

with the understanding that the curve is traversed in the direction of increasing x.

  • Curvature of space curves

For a parametrically defined space curve as c(t) = (x(t),y(t),z(t)), its curvature is:

\displaystyle F[x,y,z]=\frac{\sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}.

Given a function r(t) with values in R^3, the curvature at a given value of t is

\displaystyle \kappa = \frac{|\dot{r} \times \ddot{r}|}{|\dot{r}|^3}

where \dot{r} and \ddot{r} correspond to the first and second derivatives of r(t), respectively, and \times is the cross (vector) product. (Note that this formula is the vector notation of F[x,y,z] above.)

  • Curves on surfaces

When a one dimensional curve lies on a two dimensional surface embedded in three dimensions R^3, further measures of curvature are available, which take the surface’s unit-normal vector, u into account. These are the normal curvature, geodesic curvature and geodesic torsion.

Any non-singular curve on a smooth surface will have its tangent vector T lying in the tangent plane of the surface orthogonal to the normal vector. The normal curvature, k_n, is the curvature of the curve projected onto the plane containing the curve’s tangent T and the surface normal u; the geodesic curvature, k_g, is the curvature of the curve projected onto the surface’s tangent plane; and the geodesic torsion (or relative torsion), \tau_r, measures the rate of change of the surface normal around the curve’s tangent.

Let the curve be a unit speed curve and let t=u \times T so that T, u, t form an orthonormal basis: the Darboux frame. The above quantities are related by

\displaystyle\begin{pmatrix} T'\\ t'\\ u' \end{pmatrix} = \begin{pmatrix} 0&\kappa_g&k_n\\ -\kappa_g&0&\tau_r\\ -\kappa_n&-\tau_r&0 \end{pmatrix} \begin{pmatrix} T\\ t\\ u \end{pmatrix}.

Principal curvature: All curves with the same tangent vector will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing T and u.

Taking all possible tangent vectors then the maximum and minimum values of the normal curvature at a point are called the principal curvatures, \kappa_1 and \kappa_2, and the directions of the corresponding tangent vectors are called principal directions.

  • Curvature of surfaces (two dimensions)

Gaussian curvature: In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. Here we adopt the convention that a curvature is taken to be positive if the curve turns in the same direction as the surface’s chosen normal, otherwise negative.

The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures. It has the dimension of 1/length2 and is positive for spheres, negative for one-sheet hyperboloids and zero for planes. It determines whether a surface is locally convex (when it is positive) or locally saddle (when it is negative).

Symbolically, the Gaussian curvature K is defined as

\displaystyle K= \kappa_1 \kappa_2

where \kappa_1 and \kappa_2 are the principal curvatures. It is also given by

\displaystyle K= \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)e_1, e_2\rangle}{\det g},

where \nabla_i = \nabla_{e_i} is the covariant derivative and g is the metric tensor. At a point p on a regular surface in \mathbb R^3, the Gaussian curvature is also given by

\displaystyle K(p) = \det(S(p)),

where S is the shape operator.

The above definition of Gaussian curvature is extrinsic in that it uses the surface’s embedding in \mathbb R^3 normal vectors, external planes etc. Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; the cylinder has extrinsic curvature, but no intrinsic curvature.

Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss‘ celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. She runs around P while the thread is completely stretched and measures the length C(r) of one complete trip around P. If the surface were flat, she would find C(r)=2\pi r. On curved surfaces, the formula for C(r) will be different, and the Gaussian curvature K at the point P can be computed by the Bertrand–Diquet–Puiseux theorem as

\displaystyle K = \mathop {\lim }\limits_{r \to 0} (2\pi r - {\text{C}}(r))\frac{3}{{\pi {r^3}}}.

The integral of the Gaussian curvature over the whole surface is closely related to the surface’s Euler characteristic; see the Gauss-Bonnet theorem. The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the (angular) defect; the analog for the Gauss-Bonnet theorem is Descartes’ theorem on total angular defect. Because curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.

Mean curvature: The mean curvature is equal to the sum of the principal curvatures, \kappa_1+\kappa_2, over 2, that is

\displaystyle H=\frac{\kappa_1+\kappa_2}{2}.

It has the dimension of 1/length. Mean curvature is closely related to the first variation of surface area, in particular a minimal surface such as a soap film, has mean curvature zero and a soap bubble has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.

  • Curvature of surfaces (three dimensions)

By extension of the former argument, a space of three or more dimensions can be intrinsically curved; the full mathematical description is described at curvature of Riemannian manifolds (Riemann curvature tensor, Sectional curvature, Ricci curvature, Scalar curvature, Einstein curvature tensor, Weyl curvature tensor).

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