Umbilic Point

Umbilic Point

 

An umbilic point on a surface is a point where all normal sections have the same nonzero curvature. A three-axis ellipsoid has four umbilic points, namely, the points of contact between the ellipsoid and planes parallel to planes that intersect the ellipsoid in circles. The only surface on which all points are umbilic is the sphere.

References in periodicals archive ?
A point p [member of] [M.sup.n] is called an umbilic point, if there exist real numbers [[delta].sup.1], [[delta].sup.2] such that
Then, p is an umbilic point if and only if rank[(df).sub.p] = 0 holds.
Since f has no umbilic point on U, the Gauss map v depends only on [u.sub.1] and[mathematical expression not reproducible] holds.
By Facts 3 and 5, q is not an umbilic point. By Proposition 8, we have that f is given by (3.10) on U := I x [R.sup.n-1].
A point p = X(u) is an umbilic point if all the principal curvatures coincide at p.
Let [[lambda].sub.0] [member of] [R.sup.n + 1.sub.2] and let M be a Lorentzian surface without any umbilic point satisfying Kp(u) # 0.
It can be shown that for a generic embedding of M, [K.sub.c](0) is a regular surface except by a finite number of points (u, v), which are singularities of type [[summation].sup.2, 0] of [L.sup.[+ or - ]] or equivalently umbilic points ([D.sup.[+ or -].sub.4]) of [h.sub.v] [16].