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  2. Umbilical point - Wikipedia

    en.wikipedia.org/wiki/Umbilical_point

    In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a principal direction .

  3. Umbilic torus - Wikipedia

    en.wikipedia.org/wiki/Umbilic_torus

    The shape also has a single external face. A cross section of the surface forms a deltoid. The umbilic torus occurs in the mathematical subject of singularity theory, in particular in the classification of umbilical points which are determined by real cubic forms + + +. The equivalence classes of such cubics form a three-dimensional real ...

  4. Focal surface - Wikipedia

    en.wikipedia.org/wiki/Focal_surface

    Away from umbilical points, these two points of the focal surface are distinct; at umbilical points the two sheets come together. When the surface has a ridge the focal surface has a cuspidal edge, three such edges pass through an elliptical umbilic and only one through a hyperbolic umbilic. [3]

  5. Ridge (differential geometry) - Wikipedia

    en.wikipedia.org/wiki/Ridge_(differential_geometry)

    The set of ridge points form curves on the surface called ridges. The ridges of a given surface fall into two families, typically designated red and blue, depending on which of the two principal curvatures has an extremum. At umbilical points the colour of a ridge will change from red to blue. There are two main cases: one has three ridge lines ...

  6. Differential geometry of surfaces - Wikipedia

    en.wikipedia.org/wiki/Differential_geometry_of...

    If a surface has constant Gaussian curvature, it is called a surface of constant curvature. [52] The unit sphere in E 3 has constant Gaussian curvature +1. The Euclidean plane and the cylinder both have constant Gaussian curvature 0. A unit pseudosphere has constant Gaussian curvature -1 (apart from its equator, that is singular).

  7. Geodesics on an ellipsoid - Wikipedia

    en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid

    A single geodesic does not fill an area on the ellipsoid. All tangents to umbilical geodesics touch the confocal hyperbola that intersects the ellipsoid at the umbilic points. Umbilical geodesic enjoy several interesting properties. Through any point on the ellipsoid, there are two umbilical geodesics.

  8. Sphere - Wikipedia

    en.wikipedia.org/wiki/Sphere

    Umbilical points can be thought of as the points where the surface is closely approximated by a sphere. For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property. The sphere does not have a surface of centers.

  9. Carathéodory conjecture - Wikipedia

    en.wikipedia.org/wiki/Carathéodory_conjecture

    The conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space needs to admit at least two umbilic points.In the sense of the conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of the umbilicus.