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## How are umbilical points used in differential geometry?

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.

## How many umbilics does a genus 0 surface have?

This classification was first due to Darboux and the names come from Hannay. For surfaces with genus 0 with isolated umbilics, e.g. an ellipsoid, the index of the principle direction vector field must be 2 by the Poincaré–Hopf theorem. Generic genus 0 surfaces have at least four umbilics of index ½.

## Where are the umbilics of an ellipsoid found?

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. An ellipsoid of revolution has only two umbilics.

## When is a submanifold said to be umbilical?

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.

## When is a point P in a Riemannian submanifold umbilical?

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 p(U, V) ν {displaystyle nu } , where ν {displaystyle nu } is the mean curvature vector at p.

## How to calculate the curvature of a curve?

Theorem 1. Given a plane curve C shown in Fig.1. Let T be the tangent to the curve at point P, Q be a point on the curve near P and let QM be a line perpendicular to the tangent. Let h= QM and l= PM as shown in the figure. Let kbe the curvature of the curve at point P.

## Is the umbilical point a flator planar point?

At an umbilical point a surface is locally a part of sphere with radius of curvature . In the special case where both and vanish, the point is a flator planar point. Alternatively we can derive the principal directions by solving a quadratic equation in

## How many ridge lines are in an elliptical umbilical?

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.