Regarding the two dimensional figures we refer to as conic sections, those resulting from the sectioning of a three dimensional conical surface, the following offers a methodology of subsequent iteration to produce one of nature’s most enigmatic shapes.
Let’s begin with a straight line perpendicular to a conics section’s axis of symmetry and passing through its focus. This defines the section’s Latus Rectum.
(In the special case ellipse we call the Circle, the foci are coincident. As such, the orientation of it’s axis of symmetry is irrelavant and it’s Latus Rectum is it’s diameter.)
On either side of the axis of symmetry, the Latus Rectum intersects the conic section’s locus. Bisecting the section along its axis of symmetry, the halves are displaced inward, toward one another, such that their respective focal point locations become coincident with the other’s Latus Rectum intersection.
(Anecdotally, paraboloid surfaces are often displaced radially outward, creating a focal ring for axi-symmetric satellite antenna.)
Parallel to the original, and passing through the intersection of the loci for the two displaced halves, a new axis of symmetry is formed.
The two locus halves are trimmed at this mutual intersection and one half discarded. A three dimensional Surface of Revolution is generated using the new axis of symmetry and the remaining locus.
(Again for the instance of a Circle and likely the most recognizable for some, is the Vesica Piscis [vessel of the fish] figure. Its resulting three dimensional geometric surface is a “Lemon” or football shape [American].)
Regardless of the initial, two dimensional conic section used, the resulting open or closed three dimensional surfaces may undergo their own subsequent sectioning to produce an infinite variety of two dimensional “egg” shaped figures. Using the “egg’s” own axis of symmetry, these ovate shapes may be revolved to create the three dimensional surface of an ovoid or egg (most specifically that of an avian). In a sense, this process represents a second order sectioning of the conic section.
Just as the geometric construct of conic sections and their respective mathematical expression enjoy mutual validation, could the empirical locus produced by this method offer the same with any of the several mathematical derivation for the “egg”?
