Why do objects follow curved trajectories under Earth's gravity?

[JesseM]

And as I've said, the fact that the path of a tossed ball looks like a parabola has nothing to do with air resistance. Again, if we were to actually calculate the difference between an elliptical path and a parabolic path for a ball tossed above the surface (not continuing the paths past the surface) at ordinary velocities, I am confident that the difference will be microscopic.

And as I've said well within the microscopic range of a circle and a hyperbola as well

Nonsense. Again, as far as I know there is no well-defined limit in which the curve of a ball tossed a small height would approach a circle or a hyperbola (again, only looking at the section of the curve above the surface, not extrapolating it further), but there is such a limit for a parabola. And if I come up with an actual numerical example, calculate the exact heights at different times using the exact elliptical solution, then calculate the heights using the parabolic approximation and find only a microscopic difference, you will not be able to give me the equation of a circle or hyperbola that fits the elliptical curve so well that there is only a microscopic difference between them.

Are you willing to actually look at such an explicit numerical calculation and reconsider your position if it turns out I am right?

Sure if you can work out the numbers that translate these curve shapes to a flat surface I'll look at them.

I didn't ask you whether you'd look at them, I asked you whether you would reconsider your position if it turns out I'm correct that in a typical numerical example there is only a microscopic difference between the exact Newtonian prediction and the parabolic approximation (and if it turns out I'm also correct that it's not possible to find a circle or a hyperbola that fits the exact solution with such accuracy). Will you or won't you reconsider your position if this is the case? And are you in fact predicting that I am wrong that this is what will happen when we look at a numerical example, or are you unwilling to commit to a definite prediction?

Why not include what you expect the curve shape with air resistance to be as well, if it’s not a parabola it be worth knowing what it matches up best with, ellipse at perigee, circle, ellipse at apogee, hyperbola.

I'm not sure I'll know how to solve the differential equations for the path when air resistance is included, but I can give it a try.