page_title: Umbilical point
section_title: Focal surface
hierarchical_section_title: Umbilical point / Focal surface
caption_attribution_description: English: A surface with an elliptical umbilic, and its focal surface. Here each sheet form a tricusp figure which come together at a single point.
context_page_description: 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. The name "umbilic" comes from the Latin umbilicus - navel.
  Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive.
  The sphere is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero Gaussian curvature. The monkey saddle is an example of a surface with a flat umbilic and on the plane every point is a flat umbilic. A torus can have no umbilics, but every closed surface of nonzero Euler characteristic, embedded smoothly into Euclidean space, has at least one umbilic. An unproven conjecture of Constantin Carathéodory states that every smooth topological sphere in Euclidean space has at least two umbilics.
  The three main type of umbilic points are elliptical umbilics, parabolic umbilics and hyperbolic umbilics.
context_section_description: The elliptical umbilics and hyperbolic umbilics have distinctly different focal surfaces. A ridge on the surface corresponds to a cuspidal edges so each sheet of the elliptical focal surface will have three cuspidal edges which come together at the umbilic focus and then switch to the other sheet. For a hyperbolic umbilic there is a single cuspidal edge which switch from one sheet to the other.
caption: A surface with an elliptical umbilic, and its focal surface.
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