* The umbilical point of surface B is fixed in the vertical sense at B = 0 which in turn fixes surface B.
* The principal vertical planes of symmetry of surfaces A and B are always at right angles to one another and, when viewed directly from above, always intersect through the umbilical point.
The manner in which the principal planes of symmetry of surfaces A and B intersect each other at right angles through the umbilical point helps explain why the complex conjugate roots have to occur in pairs equidistant from the G-axis; they can now be visualised occurring behind and in front of the original Cartesian x-y plane.
Surfaces A and B still share a common umbilical point when viewed from above in the Argand plane, but this is now located at:
Instead, when viewed from above, this plane is rotated about a normal to the GH plane passing through the now offset umbilical point such that it makes an angle [alpha] with lines drawn parallel to the G-axis.
Although surfaces A and B are both free to rotate about a normal to the GH plane passing through their common umbilical point, their principal planes of symmetry are 'locked' together such that a fixed right angle is always maintained between them.
Hence surfaces A and B, whilst locked in a rotational sense, are free to move independently (i.e., up and down) along their common normal to the GH plane passing through the umbilical point. This allows for more varied intersections--and hence a greater possibility of root combinations--to occur than is possible when compared with the real coefficient case.
Equations (14) and (15) locate the umbilical point at [G.sub.U] = 1 and [H.sub.U] = 5/3.