Shape of a "V-curve"

I examined the results of AF runs of three nights plotting and analysing the data in a spreadsheet. Thereby I observed that the “V-curves” obtained with my Takahashi FSQ 106N are not V-shaped but are parabolic. So I made a regression of the curves with 2nd order polynome and got the coefficient of determination (R^2). For decent AF runs the fit was very good, R^2 being in the range of 0.96 to 0.99. Here is an example:

I would like to learn whether the parabolic shape of the “V-curve” is a specific property of the FSQ’s modified Petzval design or if other lenses behave similarly. What is your experience?


My SN 10 also produces a parabola.

That’s interesting, a Schmidt-Newtonian - a completely different construction. Thank you for sharing.

I would like to get some more experiences with different scope types, if possible. Or is it a matter of course, that the HFR values will fit to a parabola?


This is similar to a thread during the period where the focus routine was being revised, where I suggested one could fit a quadratic to three points. It only works close to focus though. I was using my Tak85 last night with a wider range of focus points. The linear approach was very good and my HFR was just over 1.0. I have also seen some Taks produce a slight bump in the middle too. I think the linear approach is probably the most robust.

Hi Buzz,

thank you for your input, I appreciate it very much. I am a beginner with SGP and the presented plot was one of my very first AF tries, but it is typical.

I started with a step size of 17 (= 113 µm) and got a HFR ratio of about 2.5, the plot above shows one of these. Later on I increased the step size to 25 (= 167 µm) and the HFR ratio increased to 3.2. This is in the proposed range (the manual says: “… until the HFR reading is ~3-5x greater than your ‘at-focus’ HFR”). With step size of 25 the 2nd order polynome fit was again great, with R^2 > 0,98 reproducibly. So there was no linear range detectable either.

When I increased the step size further, the first focus shot didn’t afford a HFR value any more because NO stars could be identified (3 tries, all failed).

Perhaps I am doing something wrong with Auto Focus? I am aware of the following variables: AF binning, exposure time and step size. I use a DSLR and did most of my focus runs with an AF binning of 1x1. My questions are:

  • Is it advisable to set AF binning to 2x2 in order to be able to further increase the step size and use a wider out-of-focus range?

  • What typical number of stars should I go for? (this number strongly varied with changing focuser position, the maximum not situated at best focus position, but at left and right side besides)?

I have done an evaluation of 19 focus runs and compared the results. My conclusion from that is:
If I cannot manage to use a larger step size (and wider out-of-focus range) a 2nd order polynome fit affords significantly better focus results than SGP’s evaluation. I can present my data if desired.


If you are using a DSLR, I would advise not to bin the exposures. I think your focus range is about right - I typically use 9 steps too and with a refractor, enable smart focus, which will extend the range if required. I find the SGP routine very good with refractors and works well too with my RCT, but only if it is collimated well. (The V-curve is accidentally a very sensitive indicator of collimation!)
With a Tak, you can include focus stars close to the image boundaries, with scopes with field curvature, I suggest using 10 to15% exclusion zone around the periphery to avoid the elongated stars.

Focuser response is indeed quadratic, which is to say that there’s only a weak response of a focus metric (HFM or FWHM) to small changes in focus position near the “nose” of the parabola, but a strong response away from critical focus.

Once you’re out on the “limbs” of the HFM response function, the response becomes quasi- linear. That’s just how parabolas work. Furthermore, if you fit two lines to the HFM response (one on each limb), the intercept of the two lines is very close to the focal position of the “nose” of the parabola.

This is the central insight of the FocusMax software that was popularized a decade ago. It’s much easier and faster and more robust to measure the slopes of the lines with a handful of exposures on the limbs, then estimate the intercept, than it is to measure the insensitive shape of the nose near critical focus.

Isn’t that, strictly speaking, a hyperbola?

OK, fine …

But over a handful of points on either side of focus, yet not close enough to get into the “flat part,” the response is very close to linear.

The point is that it’s MUCH easier with a few measurements to estimate the best focus position as the intersection of the two lines than it is to measure the curvature of the flat part of the curve near the apex. That’s why we do V-curves.

When I have fitted empirical curves I also get parabolas, but the V-curve approximation sure seems to work well. Maybe you have to go out farther to get nonlinear again.

There are also approaches that don’t use HFD or FWHM at all, but rather measure contrast of extended objects (@focus3).

The ideal curve, as I understand it, is not a conic section. Outside of the CFZ, the HFR should vary linearly with distance from focus. So the “wings” ideally should be linear. This makes sense if you think about it. You are moving the image plane up or down the light cone, and the diameter of the light cone varies linearly with distance.

When you get near the CFZ, the curve tends to act more parabolic, but within the CFZ, the data get very noisy. The intent, again - as I understand it, of the SGP focus routine is to fit lines to the linear “wings” and avoid the noisy region around the CFZ entirely.

As for the Petzval design affecting this, I have not noticed with my Tak FSQ106EDX4. I get a curve that is just as described above - very linear wings. SGP fits lines to these wings and the focus point is interpolated at the intersection. However, I use a significantly larger step size to get this effect. Bernd’s focus is about HFD = 3, and his max HFD is less than 8. That’s less than 3x the focus point HFD. I shoot for the maximum to be no less than 4 times the HFD at the focus point. I find that closer to 5 works best. This always produces very linear “wings” and perfect focus every time.


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Yes Euclid!

Thanks to all for your contributions. Since focusing correctly is one of the most demanding challenges in astrophotography, I really want to achieve it best possible.

I didn’t want to question that the focuser response is nearly linear if you only are away from focus sufficiently.

My problem seemed to be, that I didn’t manage to get HFR values in that region far away from focus because then no stars were found. Therefore I continued to do AF runs, following Buzz’s advice not to use AF binning with a DSLR. (Perhaps important, so I repeat it: I use a refractor FSQ 106N and a Canon EOS 600D.)

Extending the exposure time at ISO 800 from 8 s to 15 s at an AF binning of 1x1, I succeeded in getting decent AF runs with a step size of 27 (= 180 µm). Beyond that value star detection always failed. However, the HFR ratio of AF runs with 15 s decreased again to 3.0 (I already had reached 3.4 with ISO 800, 8 s, AF binning 1x1 before). So by extending the exposure time it even went worse. At a step size of 27 there was no indication of the response becoming linear either. Typical R^2 for the quadratic regression was in the region of 0.970 to 0.985.

In these focus run trials I didn’t crop autofocus frames. (Buzz proposed an exclusion zone of 10 to 15 %.) The ‘Minimum star diameter at 1x1 (px)’ was left at the default of 6. Nevertheless SGP does downsample the focus frame (excerpt from the logfile):

AF frame was too large… downsample = 0,25…
Star detection using min star size of 2px…
Star detection using max star size of 40px…

On further inspection of the very few (only 4 of 28) focus runs that were taken with an AF binning of 2x2, I detected that these delivered a higher HFR ratio than the shots with 1x1 binning, step size (= 25) being equal.

  • Does this mean that the Bayer Matrix of the DSLR disturbs the HFR calculation?
  • Have I to use 2x2 binning to get it right?


Hi Buzz,

Showing my ignorance here, would you be kind enough to explain how you might make use of the ‘V’ curve as a collimation check. I use a good old meade plus a f/6.3 focal reducer which together can famously give artistic comets around the especially if one’s collimation is slightly off. I’d therefore be very interested to see how to add a new tool to the set to crack this proverbial nut.

Hi Dave - unfortunately it is only an indicator, as you realize, picking up weird comet shaped stars away from focus. I have not found a way to use focus HFD to determine collimation, only as a friendly pat on the back that you are good. That being said - the goldfocus mask system uses Bahtinov-like diffraction spikes to provide numerical analysis of collimation. Not the easiest system to use in practice on a RCT but considerably easier with systems with one moving mirror.