Rounding point

Rounding point ( circular point , umbilical point or umbilical ) is a point on a smooth regular surface in Euclidean space in which the normal curvatures in all directions are equal.

The name " umbilica " comes from the Latin "umbilicus" - "navel".

Rounding points and a network of lines of curvature of the surface around them. In the general case, there are three different topological types of features, often called "lemon", "star" and "monstar" ^[1]

Content

1 Properties
2 Examples
3 Hypothesis of Caratheodory
4 Summary
5 Literature
6 notes

Properties

At rounding point:

The main curvatures of the surface coincide.
The first quadratic form and the second quadratic surface shape are proportional.
any tangent direction is the main direction .
The contacting paraboloid is a paraboloid of revolution .
Indicatrix Dupin is a circle .
The network of lines of curvature (ie, lines touching at each point of one of the main directions of the surface) has a feature ^[1] .
Any rounding point is either an elliptic point of the surface (if the main curvatures are not equal to zero, and therefore, the Gaussian curvature of the surface at this point is positive), or the so-called flat rounding point (if the main curvatures are not equal to zero, and therefore, the Gaussian curvature and the average curvature of the surface at this point are equal to zero). In the first case, in a small neighborhood of the rounding point, the surface resembles a sphere, and in the second, it resembles a plane.

Examples

Round points on a triaxial ellipsoid

In Euclidean space with metric $ds^{2}=dx^{2}+dy^{2}+dz^{2}$ ${\ displaystyle ds ^ {2} = dx ^ {2} + dy ^ {2} + dz ^ {2}}$ :

The sphere consists entirely of elliptical rounding points.
A three-axis ellipsoid (with pairwise different axes) has exactly four rounding points, all of them are elliptical and are of the lemon type.
The whole plane consists of flat rounding points.
The monkey saddle has an isolated flat rounding point at the origin.

Hypothesis of Carathéodory

Caratheodory hypothesized that on any sufficiently smooth closed convex surface M in three-dimensional Euclidean space there are at least two rounding points . This hypothesis was subsequently proved under the additional assumption that the surface M is analytic ^[2] ^[3] .

Summary

Let be $M$ ${\ displaystyle M}$ - smooth manifold of arbitrary dimension $n\geq 2$ ${\ displaystyle n \ geq 2}$ in Euclidean space of greater dimension. Then at every point $x\in M$ ${\ displaystyle x \ in M}$ identified $n$ ${\ displaystyle n}$ eigenvalues $\lambda _{1},\ldots ,\lambda _{n}$ ${\ displaystyle \ lambda _ {1}, \ ldots, \ lambda _ {n}}$ pairs of first and second quadratic forms defined on a tangent bundle $T M$ ${\ displaystyle TM}$ . Point $x\in M$ ${\ displaystyle x \ in M}$ called umbilical if it has a set $\lambda _{1},\ldots ,\lambda _{n}$ ${\ displaystyle \ lambda _ {1}, \ ldots, \ lambda _ {n}}$ contains at least two matching numbers. The set ombilik has codimension 2, i.e. set on $M$ ${\ displaystyle M}$ two independent equations. ^[4] Thus, the umbilical points on the surface of a general position are isolated ( $\dim =2-2=0$ ${\ displaystyle \ dim = 2-2 = 0}$ ), and on a three-dimensional manifold in general position they form a curve ( $\dim =3-2=1$ ${\ displaystyle \ dim = 3-2 = 1}$ )

Literature

Toponogov V.A. Differential geometry of curves and surfaces. - Fizmatkniga, 2012 .-- ISBN 9785891552135 .
Rashevsky P.K. The course of differential geometry, - Any publication.
Finikov S.P. The course of differential geometry, - Any publication.
Finikov S.P. Theory of surfaces, - Any publication.
Porteous IR Geometric Differentiation for the intelligence of curves and surfaces - Cambridge University Press, Cambridge, 1994.
Struik DJ Lectures on Classical Differential Geometry, - Addison Wesley Publ. Co., 1950. Reprinted by Dover Publ., Inc., 1988.

Notes

↑ ¹ ² Remizov A.O. The multidimensional Poincare construction and features of raised fields for implicit differential equations, - CMFD, 19 (2006), 131–170.
↑ Zbl 1056.53003
↑ Ivanov V.V. Analytical hypothesis of Carathéodory, - Sib. mate. Zh., 43: 2 (2002), 314–405.
↑ Arnold V.I. Mathematical methods of classical mechanics, - Any publication. (Appendix 10. Eigenfrequencies and ellipsoids depending on parameters).

[autogenerated1-1] ¹ ² Remizov A.O. The multidimensional Poincare construction and features of raised fields for implicit differential equations, - CMFD, 19 (2006), 131–170.

[2] Zbl 1056.53003

[3] Ivanov V.V. Analytical hypothesis of Carathéodory, - Sib. mate. Zh., 43: 2 (2002), 314–405.

[4] Arnold V.I. Mathematical methods of classical mechanics, - Any publication. (Appendix 10. Eigenfrequencies and ellipsoids depending on parameters).