Can a very small portion of an ellipse be a parabola?

Question

We can show that when a particle is projected from a certain height (from the surface of the Earth) with a speed lesser than the orbital speed and in a direction tangential to the surface of the Earth at that point, it follows an elliptical path with the center of the Earth as a focus. The process of deriving this result does not involve putting any lower bound on the speed of projection. Certainly, if the speed of projection is small enough then the trajectory would actually intersect the Earth but we can nevertheless consider the section of the trajectory until it actually hits the Earth to be elliptical. However, when we explicitly consider only those situations when the trajectory of the projectile does intersect the Earth, via approximations [1], we can show that if the height is sufficiently smaller (than the radius of the Earth) then the trajectory will actually be a parabola. In none of the two calculations, we have assumed the earth to be a point or a flat surface. So is it true that a very small portion of an ellipse is parabola even if both are entirely different conics by definition?

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[1]: Via "approximations", I meant treating an approximated gravitational field and solving for the trajectory anew rather than incorporating the approximation into the already evaluated original trajectory. Of course, it would give the same answer (as it should), but unaware of how to Taylor expand a curve, when I posted the question, I saw in this result a paradox as to how could two different conics be the same in some regime.

Answer 1

A small portion of any smooth curve looks the same as a small piece of a parabola in the limit. Choose a coordinate system so that the tangential direction in the middle of the segment is along the $x$ axis and choose a translation for the middle of the segment to sit at $(0,0,0)$, the origin of the coordinates. Then $y,z$ on the curve (ellipse etc.) may be viewed as functions of $x$ and these functions $y(x),z(x)$ may be Taylor-expanded. The first nontrivial term is $$ y(x) = a_2^y x^2 +O(x^3)$$ because the absolute and linear terms were made to vanish by the choice of the coordinates. But neglecting the $x^3$ and other pieces, this is just an equation for a parabola.

(A similar comment would apply to $z(x)$ and one could actually rotate the coordinates in the $yz$ plane so that $a_2^z$ equals zero.)

So a "supershort piece" of a curve is always a straight line. With a better approximation, a "very short" piece is a parabola, and one may refine the acceptably accurate formulae by increasingly good approximations, by adding one power after another.

But near the perigeum (the closest approach to the source of the gravitational field), we may actually describe a limiting procedure for which the "whole" ellipse – and not just an infinitesimal piece – becomes a parabola. As the maximum speed of the satellite (the speed at the perigeum etc.) approaches the escape speed, while the place of the perigeum is kept at the same point, the elliptical orbit approaches a parabolic trajectory – the whole one. Note that in this limiting procedure, the furthest point from the center of gravity goes to infinite distance and the eccentricity diverges, too.

So the parabola is a limit of a class of ellipses. In the same way, a parabola is a limiting case of hyperbolae, too. In fact, in the space of conics, a parabola is always the "very rare, measure-zero" crossing point from an ellipse to a hyperbola. This shouldn't be shocking. Just consider an equation in the 2D plane $$ y-x^2-c=ay^2 $$ which is a function of $a$. For a positive $a$, you get an ellipse; for a negative $a$, you get a hyperbola. The intermediate, special case is $a=0$ which is a parabola. One may also parameterize these curves as conic sections. The type of the curve we get will depend on the angle by which the plane is tilted relatively to the cone; the parabola is again the transition from ellipses at low angle to hyperbolae at a high angle.