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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 curvature 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. For surfaces with genus 0, e.g. an ellipsoid, there must be at least four umbilics, a consequence of the Poincaré–Hopf theorem. 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. The three main type of umbilic points are elliptical umbilics, parabolic umbilics and hyperbolic umbilics. Elliptical umbilics have the three ridge lines passing through the umbilic and hyperbolic umbilics have just one. Parabolic umbilics are a transitional case with two ridges one of which is singular. Other configurations are possible for transitional cases. These cases correspond to the ''D''4−, ''D''5 and ''D''4+ elementary catastrophes of René Thom's catastrophe theory. Umbilics can also be characterised by the pattern of the principal direction vector field around the umbilic which typically form one of three configurations: star, lemon, and lemonstar (or monstar). The index of the vector field is either −½ (star) or ½ (lemon, monstar). Elliptical and parabolic umbilics always have the star pattern, whilst hyperbolic umbilics can be star, lemon, or monstar. This classification was first due to Darboux and the names come from Hannay. Image:TensorStar.png|Star Image:TensorMonstar.png|Monstar Image:TensorLemon.png|Lemon ==Classification of umbilics== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Umbilical point」の詳細全文を読む スポンサード リンク
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