Main curvature

Main curvature is a term used in differential geometry . Two main curvatures are assigned to each point of a surface in three-dimensional Euclidean space . ${\displaystyle \mathbb {R} ^{3}}$

definition

Given a point on a regular surface im . Every tangential direction, i.e. every direction that a tangential vector can assume at this point, is assigned the normal curvature : This is understood to be the curvature of the plane curve that results from a normal section , i.e. through an intersection of the given surface with that of the surface normal vector and plane determined by the given tangential direction. The minimum value and the maximum value of these curvatures are called the two main curvatures and . The associated tangential directions are called main directions of curvature . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$

Examples

Saddle surface with the normal planes in the direction of the main curvatures

• In the case of a sphere with a radius, the two main curvatures coincide at every point: ${\displaystyle r}$ ${\displaystyle k_{1}=k_{2}=1/r}$
• The curved surface of a straight circular cylinder with a base circle radius is given . In this case, the main curvatures at every point on the lateral surface have the values ​​0 (tangential direction parallel to the axis of the cylinder) and (tangential direction perpendicular to the axis of the cylinder). ${\displaystyle r}$ ${\displaystyle 1/r}$
• The same applies to cones and more generally to developable surfaces (torsos).
• Given an ellipsoid with the semiaxes , and . At the end points (vertices) of the semi-axis , the main curvatures are equal and . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle a}$ ${\displaystyle a/b^{2}}$ ${\displaystyle a/c^{2}}$

properties

• The two main curvatures are the eigenvalues ​​of the vineyard map .
• If the two main curvatures match, then each tangential direction is the main curvature direction. Otherwise there is exactly one main direction of curvature for each of the two main curvatures. The two are perpendicular to each other.
• If the second fundamental form is restricted to the unit circle in the tangential plane , then the resulting function has the main curvatures as extreme values.
• The Gaussian curvature is the product of the principal curvatures: ${\displaystyle K}$ ${\displaystyle K=k_{1}k_{2}}$
• The mean curvature is the arithmetic mean of the main curvatures: ${\displaystyle H}$ ${\displaystyle H={\tfrac {1}{2}}(k_{1}+k_{2})}$
• If the Gaussian curvature and the mean curvature are known, the main curvatures result as solutions of the quadratic equation ${\displaystyle K}$ ${\displaystyle H}$

${\displaystyle k^{2}-2Hk+K\,=\,0}$.

• For each tangential direction the normal curvature can be expressed by the two main curvatures: ${\displaystyle k_{n}}$

${\displaystyle k_{n}\,=\,k_{1}\cos ^{2}\epsilon +k_{2}\sin ^{2}\epsilon }$   ( Euler's theorem )

Here denotes the angle between the given tangential direction and the associated tangential direction. ${\displaystyle \epsilon }$ ${\displaystyle k_{1}}$

Classification of area points

A point on a surface is called

• elliptical point if is, that is, if both principal curvatures have the same sign ; ${\displaystyle k_{1}\cdot k_{2}>0}$
• hyperbolic point if is, i.e. the signs are opposite; ${\displaystyle k_{1}\cdot k_{2}<0}$
• parabolic point if exactly one of the two main curvatures is zero;
• Flat point if applies; ${\displaystyle k_{1}=k_{2}=0}$
• Umbilical point , if applicable. ${\displaystyle k_{1}=k_{2}}$

Umbilical points, which are also elliptical points, are also referred to as actual umbilical points, whereas the remaining non-elliptical umbilical points are called flat points.

In elliptical points, the Gaussian curvature is positive ( ). This is the case when the centers of the circles of curvature of the normal sections through both main directions lie on the same side of the surface, e.g. B. on the surface of an ellipsoid or, more clearly, in double-curved planar structures such as domes . In hyperbolic points, on the other hand, the centers of the two (main) circles of curvature lie on different sides of the surface, like a saddle surface . The Gaussian curvature is negative there ( ). In parabolic points such as B. on a cylinder surface, or in flat points, the Gaussian curvature is zero. ${\displaystyle K>0}$ ${\displaystyle K<0}$

The Dupin's indicatrix in an elliptical point is an ellipse (in an elliptical umbilical point a circle ), in a hyperbolic point a hyperbola and in a parabolic point a pair of parallel straight lines.

Connected regular surfaces, which consist entirely of umbilical points, are subsets of a plane or a spherical surface .

If there are two vector fields in an open neighborhood of a point , which are linearly independent, then there is a parameterization of a neighborhood of , so that the vector fields are tangential to the coordinate lines . If there is no umbilical point, there is a parameterization of an environment so that the coordinate lines are lines of curvature , i.e. H. are tangent to the main orthogonal directions. (In an umbilical point, every direction is the main direction.) In the vicinity of a hyperbolic point there is always a parameterization so that the coordinate lines are asymptote lines , i.e. have vanishing normal curvature. ${\displaystyle U}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle V\subset U}$ ${\displaystyle p}$ ${\displaystyle p}$

Individual evidence

Wolfgang Kühnel : Differential Geometry. Curves - surfaces - manifolds . 4th revised edition. Vieweg, Wiesbaden 2007, ISBN 978-3-8348-0411-2 .

1. ↑ Section 3B, 3.13 Definition, p. 49.
2. ↑ Section 3B, 3.14 sentence, p. 51.

Manfredo Perdigão do Carmo: Differential geometry of curves and surfaces . Prentice-Hall, Englewood Cliffs, New Jersey 1976, ISBN 0-13-212589-7 .

1. ↑ Section 3-2, Proposition 4, p. 147.
2. ↑ Section 3-4, Theorem, p. 182. Application to lines of curvature in Corollary 4 and to asymptote lines in Corollary 3, pp. 184-185.