# Umbilical point

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 (Geom.) an umbilicus. See Umbilicus, 5.

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Since it is beyond the scope of this paper to delve into the theory of cubic surfaces, the reader is asked to accept that there is one important reference point that conveniently serves the purpose of a local origin, otherwise known as an umbilical point. This point is indicated in Figures 3-5 and is a localised point of inflection that can exist on a three dimensional surface; what makes this point unique, and hence a convenient reference point, is that it has zero Gaussian curvature in any direction (Harvey Mudd, n.d.).
With reference to Figures 3(d) and 4(d), when viewed from above, Surfaces A and B both share a common umbilical point located in the Argand plane at G = -1, H = 0.
Figure 5 also shows the coincidence of the umbilical point on Surface A with the Pol on the original curve; these points represent the same notion of zero curvature in three- and two-dimensions respectively.
* Surface A is free to translate vertically along a normal to the GH (or Argand) plane through its umbilical point on account of the constant term d in Equation (6).
* The cubic form of surface B may also be identified as a general monkey saddle because a monkey sitting astride an axis through the umbilical point parallel to the H-axis of surface B has a place for its two legs and tail.
Failure of the retraction of urachus and umbilical components and failure of the closure of abdominal aperture at umbilical point results in congenital prolapse.
Umbilical region was otherwise normal except the prolapsed organ at the umbilical point and at the level of abdominal aperture.
(c) The equality case of (3.9) holds for all X [member of] [T.sup.1.sub.p]M if and only if either p is a totally geodesic point or n = 2 and p is an umbilical point.
(4.4)) is true for all unit vectors in [T.sub.p]M if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point.
(c) The equality case of (4.9) holds for all X [member of] [T.sup.1.sub.p]M if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point.
Since [xi] [member of] TM, therefore from Lemma 5.1, each umbilical point is a totally geodesic point; thus (d) is correct.
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