Geodesic curvature

The geodesic curvature is a term used in classical differential geometry and called at a curve on a surface that portion of the curvature of this curve, which can be measured in the area. Clearly, it is the curvature of the projected tangent in the graph.

The geodesic curvature is a dependent of the surface property of the curve. It belongs to the inner geometry of the surface, that is, it can be determined in space without knowledge of the curvature of the surface. Curves with geodesic curvature 0 are called geodesics. Forming the shortest distance between two points in the area.

Definition

In three-dimensional space () is a surface with the unit normal vector and a parameterized by arc length differentiable curve were on. Then say

The geodesic curvature of respect.

Related to the normal curvature

The ( spatial) curvature vector can be divided according to the derivation of equations of Burali - Forti in two parts: a portion that is tangent to the surface and a component which is orthogonal to the surface:

Where the tangent vector of the curve. The curvature is referred to as the normal curvature with respect to the surface. It is the curvature of that curve in the considered point that is formed by intersection of a tangent plane in the orthogonal plane. The normal curvature is therefore dependent on the direction of the curve, which is determined by the orientation of the cutting plane ( rotation about the normal vector of the surface ). , The extreme values ​​of the normal curvature are referred to as principal curvatures, the corresponding curve direction as the main direction of curvature.

For the space curvature of a curve is:

Denotes the angle between the normal vector of the surface and the principal normal vector of the curve, then:

Example

On the spherical surface with the parameter representation

Is the geodesic curvature of the meridians (). For the parallels () applies.

Properties

The geodesic curvature is a quantity of the intrinsic geometry of surfaces, that is, it depends not only on the shape of the curve only on the first fundamental form of the surface and their derivatives. Thus, it can be determined solely by linear and angular measurements within the area without the spatial shape of this surface must be known.

By specifying the geodetic curvature as well as a starting point and an initial direction a surface curve is clearly defined.

Of particular importance are surface curves with geodesic curvature 0 They are called geodesics and form the ( locally ) shortest distance between two points on the surface.

The geodesic curvature is signed. Reverses the orientation of one or the passage sense of order, changes sign.

The Gauss -Bonnet establishes a connection between the Gaussian curvature of a limited area of a surface and the geodesic curvature of the boundary curve of the surface.