First, allow me the following characterization of the Parabola and conical surfaces:
There is only “the” Parabola, not “a” parabola in the sense that “a” implies variety of shape. “The” Parabola, like “the” Circle, has only a single shape representation. It can be depicted in any dilation, but its contour remains the same. In contrast, conical surfaces (if you like, right circular conical surface or cone) have infinite shape representation as consequence of their conic generator angle.
Therefore, as an infinity of conical surfaces can arise from the Parabola (as detailed below), and whereas only the Parabola can come from an infinity of specific sectioning’s of any conical surface, does it stand to reason that the Parabola is a “parent” rather than “child” of a conical surface?
A line parallel to the Parabola Directrix and midway between the Vertex and Focus is used as an axis for a mirrored Parabola. The mirrored is then rotated 90° about the Axis of Symmetry. An extension of the mirrored Parabolas Vertex (now coincident along the Axis of Symmetry with the original Parabolas Latus Rectum) contains the centers of all conic Dandelin or focal spheres. Within the locus of the mirrored Parabola are all conic vertices at points tangent to each conic axis. Each conic generator angle will be equal to the angle between its axis and the Axis of Symmetry.
Depicted below are three, in an infinite succession, of conics pivoting, or rotating, about the Parabola Vertex.
GeoGebra animation can be viewed here: The Parabola as a Parent of the Conical Surface – GeoGebra
F = Parabola Focus
α = Conic Generator Angle
θ = Angle between Conic Axis and Parabola Axis of Symmetry
RD = Dandelin Spherical Radius
Xn = X Axis Value at Tangent Point on Mirrored Parabola, Conical Vertex
Yn = Y Axis Value at Tangent Point on Mirrored Parabola, Conical Vertex
RD = ( 2F / tan α ) / 2
Xn = ± ( 2F / tan α )
Yn = ( ( F / 4) ( – ( 2F / tan α ) ) 2 ) – 1 (or if you like, simply Yn = Xn2 )
