Geodesic curvature

The geodesic curvature is a term from the classical differential geometry and designated at a curve on a surface of that portion of the curvature of this curve, which can be measured in the surface. It is clearly illustrated by the curvature of the curve projected into the tangential plane .

The geodetic curvature is a property of the curve that depends on the area. It belongs to the internal geometry of the surface, i. That is, it can also be determined without knowing the curvature of the surface in space. Curves with the geodetic curvature 0 are called geodesics . They locally form the shortest distance between two points in the area.

definition

In three-dimensional space ( ) there is a surface with the normal unit vector and a differentiable curve parameterized according to the arc length . Then is called ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle S}$ ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {r}}(s)}$ ${\displaystyle s}$ ${\displaystyle S}$

${\displaystyle \kappa _{g}(s)={\frac {\mathrm {d} ^{2}{\vec {r}}(s)}{\mathrm {d} s^{2}}}\cdot \left({\vec {n}}({\vec {r}}(s))\times {\frac {\mathrm {d} {\vec {r}}(s)}{\mathrm {d} s}}\right)=\left[{\frac {\mathrm {d} ^{2}{\vec {r}}(s)}{\mathrm {d} s^{2}}},{\vec {n}}({\vec {r}}(s)),{\frac {\mathrm {d} {\vec {r}}(s)}{\mathrm {d} s}}\right]}$

the geodesic curvature of respect . ${\displaystyle {\vec {r}}(s)}$ ${\displaystyle S}$

Relation to normal curvature

The (spatial) curvature vector can be divided into two parts according to the derivative equations of Burali-Forti : a part which is tangential to the surface and a part which is orthogonal to the surface: ${\displaystyle {\mathrm {d} ^{2}{\vec {r}}(s)}/{\mathrm {d} s^{2}}}$

${\displaystyle {\frac {\mathrm {d} ^{2}{\vec {r}}(s)}{\mathrm {d} s^{2}}}={\frac {\mathrm {d} {\vec {t}}(s)}{\mathrm {d} s}}=\kappa _{g}(s)\cdot ({\vec {n}}({\vec {r}}(s))\times {\vec {t}}(s))\,+\,\kappa _{n}(s)\cdot {\vec {n}}({\vec {r}}(s)),}$

where is the tangent vector of the curve. The curvature is called the normal curvature with respect to the surface . It is the curvature of that curve in the point under consideration which is created by the intersection of with a plane orthogonal to the tangential plane. The normal curvature is therefore dependent on the direction of the curve in , which is determined by the orientation of the cutting plane (rotation around the normal vector of the surface in ). The extreme values ​​of the normal curvature are referred to as main curvatures , the associated curve directions as main curvature directions . ${\displaystyle {\vec {t}}(s)={\mathrm {d} {\vec {r}}(s)}/{\mathrm {d} s}}$ ${\displaystyle \kappa _{n}(s)}$ ${\displaystyle S}$ ${\displaystyle P}$ ${\displaystyle S}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P}$

The following applies to the spatial curvature of a curve:

${\displaystyle \kappa (s)=\left|{\frac {\mathrm {d} ^{2}{\vec {r}}(s)}{\mathrm {d} s^{2}}}\right|={\sqrt {(\kappa _{n}(s))^{2}+(\kappa _{g}(s))^{2}}}.}$

Designates the angle between the normal vector of the surface and the main normal vector of the curve, then: ${\displaystyle \psi }$ ${\displaystyle {\vec {n}}}$

${\displaystyle \kappa _{g}=\pm \,\kappa \sin \psi .}$

example

On the spherical surface with the parametric representation

${\displaystyle {\vec {r}}(\vartheta ,\varphi )=R{\begin{pmatrix}\sin \vartheta \cdot \cos \varphi \\\sin \vartheta \cdot \sin \varphi \\\cos \vartheta \end{pmatrix}}}$

is the geodetic curvature of the longitudes ( ) . For the parallels ( ) applies: . ${\displaystyle \varphi =\mathrm {const.} }$ ${\displaystyle \kappa _{g}=0}$ ${\displaystyle \vartheta =\mathrm {const.} }$ ${\displaystyle \kappa _{g}=1/(R\tan \vartheta )}$

properties

• The geodetic curvature is a quantity of the internal geometry of surfaces, i.e. In other words, in addition to the shape of the curve, it only depends on the first fundamental form of the surface and its derivatives. It can therefore be determined solely by length and angle measurements within the area, without the spatial shape of this area having to be known.

• A surface curve is clearly defined by specifying the geodetic curvature as well as a starting point and a starting direction. ${\displaystyle \kappa _{g}(s)}$

• Surface curves with the geodetic curvature 0. They are called geodesics and form the (locally) shortest distance between two points on the surface.

• The geodetic curvature is signed. If you reverse the orientation of or the direction of passage of , the sign changes . ${\displaystyle \kappa _{g}}$ ${\displaystyle S}$ ${\displaystyle {\vec {r}}(s)}$ ${\displaystyle \kappa _{g}}$

• The Gauss-Bonnet theorem establishes a connection between the Gaussian curvature of a limited area of ​​a surface and the geodetic curvature of the edge curve of this surface.

literature

• Manfredo Perdigão do Carmo: Differential Geometry of Curves and Surfaces. Reprinted edition. Prentice-Hall, Upper Saddle River NJ 1976, ISBN 0-13-212589-7 .
• Wolfgang Kühnel : Differential Geometry. Curves - surfaces - manifolds. Vieweg-Verlag, Braunschweig et al. 1999, ISBN 3-528-07289-X .