WalterT Chen, what’s the difference between the parabolic vs hyperbolic curve way of focusing?
Long story, Walter, do you really want to know? :-)
Anyway, here goes...
If you go back to high school Physics, where they deal with thin lenses, Snell's law and stuff like that, the focus point of a simple thin lens looks like a triangle (i.e., rays converge right at the focal point).
If you look at this simple model, you will see (from using similar triangles) that the size of a star varies linearly with the distance the sensor is away from the focus point. If you draw star HFD versus focus error, you get a straight line. The minimum star size with this "geometric optics" model is infinitesimally small.
Notice that when you place yourself too far from the focus, because of similar triangles, the HFD of the again starts to increase linearly.
What you get as HFD from Geometric Optics is a "V" shaped curve. This is the "V curve" that you see written in astronomy circles.
But when we went to college, we learned that geometric optics is only an approximation of the real world. The actual spot size is not infinitesimally small but is the Fourier Transform of the aperture of the lens. For a perfectly circular aperture, the Fourier Transform is of course the Airy Disk right at the focal point. It is not infinitesimally small. This is also called diffraction limited optics.
Far away from the focus, the HFD of the star is predominately linear. But as you approach focus, the HFD does not keep reducing, but reaches the Airy Disk.
Interestingly, if you slice a Conic Section in a certain way, you get a hyperbola, which is predominantly linear away from the tip of the cone. If you slice the cone at a different angle from the hyperbolic slice, you get either an ellipse or a parabola. Or, in the extreme care, a circle. Hyperbolas, Ellipses, Circles, and Parabolas are all conic sections, and are all second order curves, i.e., in general something like,
y2 + x2 + a(xy) + b(x) + c(y) + d = 0.
Depending on a, b, c, and d, you end up with either a circle, parabola, hyperbola or ellipse.
From geometric optics, you know that that the HFD vs focal position is not a circle nor ellipse, and is actually close to a hyperbola. So, when you are estimating focus, you should use a hyperbola.
This is the result from a "V curve" program that I wrote for macOS that estimates the focus by using a minimum mean square match to a hyperbola (positive square root). The input data is from the average HFD at different EAF positions that is obtained using the ASIAIR Detect Star tool. Telescope was an FSQ-85.
Pretty much all the other software used in astronomy uses a hyperbola.
This is what ASIAIR's parabolic curve looks like:
The green curve is ASIAIR's parabolic fit The white lines are what I had overlayed to show what a geometic optics models the data as (i.e., V curve). Notice that ASIAIR's parabola does not pass through any of the actual data points (green dots).
Chen
