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| [[File:Lignes de courbure ellipsoide.jpg|thumb|right|Lines of curvature on an ellipsoid showing umbilic points.]]
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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 curvature]]s are equal, and every tangent vector is a ''principal direction''. The name "umbilic" comes from the Latin ''umbilicus'' - navel.
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| Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the [[Gaussian curvature]] is positive. For a surfaces with [[genus (mathematics)|genus]] 0, e.g. an ellipsoid, there must be at least four umbilics, a consequence of the [[Poincaré–Hopf theorem]].
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| 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 (mathematics)|plane]] every point is a flat umbilic.
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| The three main type of umbilic points are elliptical umbilics, parabolic umbilics and hyperbolic umbilics. Elliptical umbilics have the three [[ridge (differential geometry)|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''<sub>4</sub><sup>-</sup>, ''D''<sub>5</sub> and ''D''<sub>4</sub><sup>+</sup> elementary catastrophes of René Thom's [[catastrophe theory]].
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| 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 a vector field|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.<ref>{{cite journal|last=Berry | first=M V |last2=Hannay | first2=J H | title=Umbilic points on Gaussian random surfaces | journal=J. Phys. A | volume=10 | year=1977 | pages=1809–21}}</ref>
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| <gallery Caption="configurations of lines of curvature near umbilics" widths="150px">
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| Image:TensorStar.png|Star
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| Image:TensorMonstar.png|Monstar
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| Image:TensorLemon.png|Lemon
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| </gallery>
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| ==Classification of umbilics==
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| ===Cubic forms===
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| [[Image:Umbilic Torus.png|thumb|right|Computer representation of an Umbilic Torus]]
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| The classification of umbilics is closely linked to the classification of real [[cubic form]]s <math>a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3</math>. A cubic form will have a number of root lines <math>\lambda (x,y)</math> such that the cubic form is zero for all real <math>\lambda</math>. There are a number of possibilities including:
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| *Three distinct lines: an ''elliptical cubic form'', standard model <math>x^2 y-y^3</math>.
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| *Three lines, two of which are coincident: a ''parabolic cubic form'', standard model <math>x^2 y</math>.
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| *A single real line: a ''hyperbolic cubic form'', standard model <math>x^2 y+y^3</math>.
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| *Three coincident lines, standard model <math>x^3</math>.<ref name="Poston">{{Citation| title=Catastrophe Theory and its Applications | first2=Ian | last2=Stewart | first=Tim | last=Poston | publisher=Pitman | year=1978 | isbn=0-273-01029-8 | authorlink=Tim Poston | authorlink2=Ian Stewart (mathematician)}}</ref>
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| The equivalence classes of such cubics under uniform scaling form a three dimensional real projective space and the subset of parabolic forms define a surface – called the [[umbilic bracelet]] by [[Christopher Zeeman]].<ref name="Poston"/> Taking equivalence classes under rotation of the coordinate system removes one further parameter and a cubic forms can be represent by the complex cubic form <math>z^3+3 \overline{\beta} z^2 \overline{z} + 3 \beta z \overline{z}^2 + \overline{z}^3</math> with a single complex parameter <math>\beta</math>. Parabolic forms occur when <math>\beta=\tfrac{1}{3}(2 e^{i\theta}+e^{-2 i\theta})</math>, the inner deltoid, elliptical forms are inside the deltoid and hyperbolic one outside. If <math>\left |\beta\right |=1</math> and <math>\beta</math> is not a cube root of unity then the cubic form is a ''right-angled cubic form'' which play a special role for umbilics. If <math>\left |\beta\right |=\tfrac{1}{3}</math> then two of the root lines are orthogonal.<ref name="Port">{{Citation | authorlink=Ian R. Porteous | first=Ian R. | last=Porteous | year=2001 | title=Geometric Differentiation | pages=198–213 | publisher=Cambridge University Press | isbn=0-521-00264-8}}</ref>
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| A second cubic form, the ''Jacobian'' is formed by taking the [[Jacobian determinant]] of the vector valued function <math>F : \mathbb{R}^2 \rightarrow \mathbb{R}^2</math>, <math>F(x,y)=(x^2+y^2,a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3)</math>. Up to a constant multiple this is the cubic form <math>b x^3+(2 c-a)x^2 y+(d-2 b)x y^2-c y^3</math>. Using complex numbers the Jacobian is a parabolic cubic form when <math>\beta=-2 e^{i\theta}-e^{-2 i\theta}</math>, the outer deltoid in the classification diagram.<ref name="Port"/>
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| ===Umbilic classification===
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| [[File:Umbilic clasification.svg|thumb|200px|Umbilic classification, the <math>\beta</math>—plane. The Inner deltoid give parabolic umbilics, separates elliptical and hyperbolic umbilics. Cusps on inner deltiod: cubic umbilics. Outer circle, the birth of umbilics separates star and monstar configurations. Outer deltoid, separates monstar and lemon configuration. Diagonals and the horizontal line - symmetrical umbilics with mirror symmetry.]]
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| Any surface with an isolated umbilic point at the origin can be expressed as a [[Monge form]] parameterisation <math>z=\tfrac{1}{2}\kappa(x^2+y^2)+\tfrac{1}{3}(a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3)+\ldots</math>, where <math>\kappa</math> is the unique principal curvature. The type of umbilic is classified by the cubic form from the cubic part and corresponding Jacobian cubic form. Whilst principal directions are not uniquely defined at an umbilic the limits of the principal directions when following a ridge on the surface can be found and these correspond to the root-lines of the cubic form. The pattern of lines of curvature is determined by the Jacobian.<ref name="Port"/>
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| The classification of umbilic points is as follows:<ref name="Port"/>
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| *Inside inner deltoid - elliptical umbilics
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| **On inner circle - two ridge lines tangent
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| *On inner deltoid - parabolic umbilics
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| *Outside inner deltoid - hyperbolic umbilics
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| **Inside outer circle - star pattern
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| **On outer circle - birth of umbilics
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| **Between outer circle and outer deltoid - monstar pattern
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| **Outside outer circle - lemon pattern
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| *Cusps of the inner deltoid - cubic (symbolic) umbilics
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| *On the diagonals and the horizontal line - symmetrical umbilics with mirror symmetry
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| In a generic family of surfaces umbilics can be created, or destroyed, in pairs: the ''birth of umbilics'' transition. Both umbilics will be hyperbolic, one with a star pattern and one with a monstar pattern. The outer circle in the diagram, a right angle cubic form, gives these transitional cases. Symbolic umbilics are a special case of this.<ref name="Port"/>
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| ==Focal surface==
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| [[File:Elliptical umbilic focal surface.png|thumb|right|A surface with an elliptical umbilic, and its focal surface.]]
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| [[File:Hyperbolic umbilic focal surface.png|thumb|A surface with a hyperbolic umbilic and its focal surface.]]
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| The elliptical umbilics and hyperbolic umbilics have distinctly different [[focal surface]]s. 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.<ref name="Port"/>
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| == Definition in higher dimension in Riemannian manifolds==
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| A point ''p'' in a [[Riemannian submanifold]] is umbilical if, at ''p'', the (vector-valued) [[Second fundamental form]] is some normal vector tensor the induced metric ([[First fundamental form]]). Equivalently, for all vectors ''U'', ''V'' at ''p'', II(''U'', ''V'') = ''g''<sub>''p''</sub>(''U'', ''V'')<math>\nu</math>, where <math>\nu</math> is the mean curvature vector at ''p''.
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| A submanifold is said to be umbilic (or all-umbilic) if this condition holds at every point "p". This is equivalent to saying that the submanifold can be made totally geodesic by an appropriate conformal change of the metric of the surrounding ("ambient") manifold. For example, a surface in Euclidean space is umbilic if and only if it is a piece of a sphere.
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| ==See also==
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| *[[wikt:umbilical|umbilical]] – an anatomical term meaning ''of, or relating to the navel''
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| *[[Carathéodory conjecture]]
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| ==References==
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| * {{citation|first=Gaston|last=Darboux|authorlink=Gaston Darboux|year=1887,1889,1896|title=Leçons sur la théorie génerale des surfaces: [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0001.001 Volume I], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0002.001 Volume II], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0003.001 Volume III], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0004.001 Volume IV]
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| |publisher=Gauthier-Villars}}
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| *[http://deslab.mit.edu/DesignLab/Watermarking/NSF.htm Pictures of star, lemon, monstar, and further references]
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| <references/>
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| [[Category:Differential geometry of surfaces]]
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| [[Category:Surfaces]]
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