420 replies to this topic

[MKV]

I think the test is prone to systematic errors, it is based on an over-simplified view of a complex situation, and there is a pre-existing better way to do it.

Eevery measurement is subject to systemic errors, simply because every measuring device has an inherently limited accuracy. This is why equipment needs to be calibrated, i.e. one needs to establish the "uncertainty" range, expressed as a "± error" margin in the readings.

In other words, just because a spherometer reads a sagitta of -0.0025 doesn't mean it's -0.0025 ± 0.0000. If one really needs to know the sagitta to four decimal places accurately, then the spherometer better have a systemic error smaller than four decimal places. This is not to say that a careful worker will not consistently get -0.0025 readings (which is precision, not accuracy).

Also, I believe commercial lenses are usually listed as having a clear aperture (CA) equal to 90% of their physical diameter, so if a bino OG has a usable aperture of 25 mm, it's physical diameter should be 28 mm.

regards,
Mladen

[Jon Isaacs]

Yes I agree. It isn't my concern if the 19.5 is a problem or not - I am only talking about the accuracy of the test. In fact - I am also one of the people asking - Who cares? and Why are people being encouraged to measure aperture in the first place?

I do think that aperture should be stated accurately and not "skimmed" from the consumer - but that is not my primary interest in this test. I think the test is prone to systematic errors, it is based on an over-simplified view of a complex situation, and there is a pre-existing better way to do it.

Frank

I think people are encouraged to measure the effective aperture for the reason I do it, to know if their equipment is operating at full aperture. With fast achromtic refractors, it is relatively common the drawtube clips the light cone and does not permit operating at full aperture. Likewise, an undersized Newtonian secondary can mean the scope is not operating at full aperture.

Both these situations are easily discovered though not necessarily measured using simple techniques like looking through a pinhole at the focal plane. For the Newtonian, simple analysis can be done using programs like Newt for the Web... This can even be used for the drawtube clipping issue in a refractor.

I personally find these techniques easier than the flashlight test. I think the flashlight test is most useful is situations where the eyepiece is not easily removed, ie binoculars.

Jon

[freestar8n]

Here is a worked example for "my" test to determine aperture when there is known restriction near the focal plane. I calculate the impact of the measurement errors on the error in the result based on the assumed formula. This contains some simple math - so for people who avoid math it is time to look away - but it shows that for my 20" f/15 mak with a restriction f/10 from focus, and measured carefully with a micrometer to 0.1mm, the resulting error in aperture is around 1mm.

Frank

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[MKV]

I think Glenn's method works, it's applicable to all systems with eyepieces, and the results are a good estimate of the aperture. From the Hubble Space Telescope fiasco, we know that even the titans with huge budgets make colossal mistakes, despite their advanced degrees -- because we're human.

Whether Glenn's test is accurate to 1mm or not is or should be irrelevant to an ATM. If you have a 200 mm diameter optic and your drawtube or some other element masks 2 mm, you're operating at 99% of aperture. That may for some reason be significant to NASA, but I seriously doubt an ATM will or should lose any sleep over it. Operating at 99% of anything is pretty darn good for mortals. This is the practical aspect of it.

The theoretical aspect of Glenn's test can be easily settled by raytracing -- no not sketches -- at different iris positions. Otherwise, all we are discussing is how many angles can fit on the head of a pin. And thus far i have not seen any raytracing, except as schematic drawings.

regards,
Mladen

[roscoe]

I asked this question a couple of days ago, but don't believe anyone answered it.....
Normally, I eyeball ascertain if there is a clear full-arpeture light path through a refractor by setting the focuser at infinity focus, replacing the EP with a small-opening ep body with the glass removed, and looking to see if I can see the foil spacers in the objective. Would I be smarter to use an empty EP body with a large opening (1/2" dia for instance)? This wouldn't be a 'pinpoint' test any more, but would certainly show any obstruction even near what a pinpoint test would show....
Would the same apply to the flashlight/laser test described above?
Thanks,
Russ

[Asbytec]

Russ, I think that will work. It's a version of the pin hole test. The pin hole is just to ensure your eye is on axis, so the hole could be larger as long as you keep your eye in the center or close to it. If the objective has very tight fit, having a larger pin hole might allow you some freedom to move off to one side (then back to center) to spot the edge.

I think Frank's example above makes sense. If the smaller triangle is not known to a great accuracy, then the larger triangle projected from it will be off. The error will be amplified. But, light takes no prisoners and does not make an measurement error when it strikes an obstruction - provided it's a parallel beam. It is what it is (accepting diffraction) while a rule of some sort can be off.

On the Gaussian behavior, though, it might be important. A wave from a point source at infinity is sampled by the aperture and forms a point (diffraction image, actually) in the focal plane. Then it exists the system as a wavefront of equal dimension of the exit pupil - just as it did entering the entrance pupil. It seems to me a laser approximates this behavior (in reverse) when it is shown into the exit pupil.

Actually, in this case isn't the eyepiece, then, a very small and fast objective while the primary objective becomes a very long focal length eyepiece with a huge field lens - in effect? If so, then it appears a laser is Gaussian.

[GlennLeDrew]

The pinhole test should be done with as small a hole as possible, so that parallax error is minimized. The combination of a large-ish hole and your eye's sizeable iris would introduce the same kind of uncertainty, or 'fuzziness' as the flashlight projecting a too-large spot of the light at the focus.

That's why a laser spot is so good. It behaves exactly like placing a very tiny pinhole at the focus, then shining a bright light through it. But this bright spot is so tiny, the shadows cast by anything even within a few cm are very sharp. No parallax error of note. No need to know the obstructor diameter. No need to know how far from the focus the obstructor lies. If you know the spot is at the focus, the light path must necessarily follow the same path through the system as from an optically distant point source.

It must be conceded that if the pinhole-at-focus method is deemed trustworthy, the laser spot (a near diffraction limited source) at the focus must be at least equally as good.

I get the impression that folks put more stock in the visual sighting through a sizeable hole than they do for what is really the identical test, but with a far tinier and hence less uncertain source for the tracing of the fielded light envelope.

[freestar8n]

I think Frank's example above makes sense. If the smaller triangle is not known to a great accuracy, then the larger triangle projected from it will be off.

My main point in that example is to demonstrate a very well defined measurement, with well defined measurement techniques, and with a well defined formula describing the phenomena being modeled. This allows a direct calculation of the propogation of errors, and lets you know how the error behaves under different conditions. In fact it behaves extremely well for very long focal length systems with the entrance pupil very far from the front of the scope - which is where the flashlight method loses accuracy due to error in the taper and fuzziness of the shadow edge - combined with the long distance the shadow is thrown. For the mak with the flashlight method, under a very optimistic assumption of 1 arc minute taper and no other errors, the error is 20mm, which blows Glenn's 1mm budget. The direct measurement near focus gives 1mm.

All I have seen about the flashlight method is a claim that it is accurate to 1mm if done carefully and it doesn't matter how far away the pupil is because everything is perfectly parallel. If people still don't think the error depends heavily on the pupil distance and can become arbitrarily large - then I'm not sure how people can be reached. This is experimental practice 101 - how to do a good measurement and estimate its errors.

Frank

[freestar8n]

Normally, I eyeball ascertain if there is a clear full-arpeture light path through a refractor by setting the focuser at infinity focus, replacing the EP with a small-opening ep body with the glass removed, and looking to see if I can see the foil spacers in the objective.

The closer any of the baffles in the refractor are to focus, the more critically you need to set everything to make a reliable measurement. If you are only concerned about near the objective, then a simple flashlight test would work. If you are concerned about very close to focus - then it becomes much more challenging unless you can physically measure the baffles.

In all these tests, you need to factor in the location of the suspect aperture because when it is near focus, slight errors in focus and baffle size will magnify the error in aperture.

Frank

[roscoe]

Maybe I'm asking the wrong question, or maybe I'm not clear on how a scope focuses.......
Does the objective (or mirror) focus the image to a very tiny 'pinpoint' image, which is then magnified by whatever EP is used, be it a (for instance) 4mm or 40mm, or does the 40mm EP, which shows a much wider field, magnify a small but wider image - say 1/4" or more wide, produced by the primary optics?
But now that I'm drawing pictures in my head, if that is indeed the way a lower magnification ep works,would that little 'disc' of image be a bit toward the primary, meaning the pinpoint is still there, so if a baffle wasn't in the fov at pinpoint focus, it wouldn't be in the way with a bigger glass ep?
Russ

[Asbytec]

But now that I'm drawing pictures in my head, if that is indeed the way a lower magnification ep works,would that little 'disc' of image be a bit toward the primary, meaning the pinpoint is still there, so if a baffle wasn't in the fov at pinpoint focus, it wouldn't be in the way with a bigger glass ep?
Russ

Russ, if I understand the pictures in your head, I think you may be mixing on and off axis behavior. The "pinpoint" at the center FOV would be on-axis and no baffles would be in the way if you can see the entire objective through a pin hole. Eventually, as you move off axis, baffles and draw tubes, etc., begin to vignette. At some point low power eyepieces will pick up that vignetting as the FOV expands. But, the on-axis pinpoint remains fully illuminated - provided there is no on-axis constriction reducing aperture.

This is what the pin hole at focus tests for when you can view the entire objective. If you can see beyond the objective - off to the side - that's an indication of off-axis performance. As long as you can see the entire objective, you have a fully illuminated FOV and no constrictions.

Frank, thanks again. Glenn, sure, the point on paralax is taken. Mine was so tight, though, it was helpful to view off axis, spot the meniscus edge, then move back on axis to see if it remained within the primary baffle opening. It just did with very little room to spare. That's one tight baffle system - worse with the secondary in place. LOL It would have been better with a pin hole, but it is so dark in there, already, it's more difficult. (Maybe I'll start an illustrated thread....)

[roscoe]

Russ, if I understand the pictures in your head, I think you may be mixing on and off axis behavior. The "pinpoint" at the center FOV would be on-axis and no baffles would be in the way if you can see the entire objective through a pin hole. Eventually, as you move off axis, baffles and draw tubes, etc., begin to vignette. At some point low power eyepieces will pick up that vignetting as the FOV expands.

OK, I'm clear on the pinpoint part, it's what I thought it was....
So, for a big fat low-power EP, the baffles would have to have a larger inner diameter (bigger holes in them) so the wider FOV of that EP would not vignette? Yes?
Vignetting, if it happened, would dim the edges (but not the entire image), yes?

[GlennLeDrew]

Here's something to ponder. I just put laser and expander up to a 10mm eyepiece (no scope involved.) When I place the edge of a sheet of paper within 17mm of the field stop (as close as the barrel allows), The shadow 4 meters away is so sharp (with diffraction) that I see in highly magnified form the raggedness of the edge! And this is verified by sliding the paper and seeing the pattern of roughness move as a unit.

This indicates the spot of light is probably less than 0.01mm across. Which makes sense, given the diffraction seen.

If such a sharp-edged shadow is cast by an obstructor a mere 17mm from the focus, for any conceivable obstructor in any telescope it will be sharply projected. And so edge blurring with the laser is a non-issue.

I also poked the corner of an envelope up the barrel, so as to get it even closer to the field stop. I could magnify the little fibres making up the paper so that at just 1m distant some were projected as long as 30cm. Talk about a microscope! So even in nearly the plane of the focus a coherent, magnified image is seen, although by that point diffraction is *really* prominent.

[GlennLeDrew]

Now that we know that a more than suitably sharp shadow is available, the remaining matter to address is confidence in the diameter of the projected circle of light.

Frank would have us obsess over the unknown dimension and axial location for the relevant aperture stop. It doesn't matter. We need only ensure parallelism of the exiting beam, which must be cylindrical when the light source us at the focus.

If one finds the same circle diameter just in front of the system as at 2m, or 5m or 10m, we know the emergent light is parallel, and this diameter must be the working aperture.

Frank,
Can you explain how the emergent light bundle can be of constant diameter (parallel), and yet *not* be representative of the working aperture.

My claim if 1mm accuracy arises from this simple fact; that one can adjust the focus position of the tiny light source--whose shadow edges so produced are blurred to less than 1mm--until the emergent bundle is measured as parallel.

Again, and again, and again... No need to know any dimensions whatsoever. We are exactly enough replicating the path of light as from an on-axis star that the full envelope accommodated by the system has the same dimensions. The light going through in reverse is the same as that making it through the front end.

All your red herrings regarding error propagation based on unknown dimensions are inapplicable because we are using *light itself* to trace out the *path of light the system can field*. And we have the means to check for error directly, by measuring for parallelism on the output.

Or do you suggest that light behaves differently going one way vs the other?

[Asbytec]

So, for a big fat low-power EP, the baffles would have to have a larger inner diameter (bigger holes in them) so the wider FOV of that EP would not vignette? Yes?
Vignetting, if it happened, would dim the edges (but not the entire image), yes?

Yes a constriction (baffle, small diagonal, etc.) will dim the edges and it can dim the center if it's severe. Any "limiting aperture" could be at the eyepiece field stop, though. It all depends on how well it's baffled and how large the field stop is. Most, if not all, telescopes will vignette toward the edge of the field and often it's so very minimal until you get out to a FOV for wide field imaging. The trick is to get full aperture illumination and resolution over some area at the center for visual.

My own scope was fully illuminated, but only at a slightly reduced aperture. This is what the flashlight and other tests showed. Now, it uses full aperture illumination over a small field of view but still vignettes toward the edge. That's normal, and it's okay for small objects like planets. The FOV easily contains all of Jupiter and then some at full illumination. It does not need a full degree of full illumination. A couple dozen arc minutes is plenty for higher power lunar and planetary work. And double stars. Small stuff.

[Nils Olof Carlin]

In an early post in this thread, Frank asked:
1) What do you mean by the term, "aperture".

In the first post, Glenn wrote:

The flashlight aperture test is an easy and direct way to determine the actual working aperture of an optical instrument which employs a positive eyepiece.

Note "the actual working aperture", which I take as the diameter of the circle of (star)light (assuming circular symmetry of course) that contributes light to the on-axis image. Any disagreement?

In a later post, Frank wrote:

"if I have a 20" f/15 system with a small obstruction 750mm, or 1/10 efl, from the focus, then the entrance pupil will pushed back 9xf away, or a distance of 67.5m. If there is a taper of only one minute of arc in the beam, projected over that football field distance, the error in the measurement will be 19.6mm"

The entrance pupil taken as the image by the objective of the small obstruction. In this case it will be a virtual image 67.5 m behind the telescope, and not useful for determining the actual working aperture. Why indeed should it?

Consider a Newtonian with the limiting aperture at the primary, and the shadow in Glenn's test picked up at the tube opening, approx. one focal length away. If the focus of the EP is a distance x from the focus of the primary mirror, there will be a taper to the light path, causing a mismeasure of the actual aperture of approx. x/f-number - a defocus of 5 mm will cause an error of 1 mm in a f/5 instrument, and a fuzziness comparable to the width of the light source. This must be easily avoided if you are aware of it. The fuzziness might be an advantage - the fuzzier the circle, the nearer to the focal plane the obstruction is.
With a refractor, the aperure is measured close to the objective and this error is not likely to matter.

I just checked my good old 8.5x44, and a cheap 10x50, using a non-focusing high intensity LED of <1.5 mm dia. The actual working diameters read were 40 mm and 42 mm respectively, in both cases determined by an internal baffle between the objective (of nominal size) and first porro prism. A bit surprising perhaps, anyone else who have tried?

Nils Olof

[GlennLeDrew]

Nils Olof,
Kudos! You clearly understand the geometry.

I first came up with this flashlight test because so many binos suffer a restricted aperture. I wanted to devise a test which others could easily do, and which requires no interpretation of a visual inspection, which moreover is not easy to quantify, like I had been doing for years.

I promulgated it back then it in the Bino Forum, and a good number of folks have been using it and reporting results. Indeed, I have a feeling it could well have inspired some bino makers to refine at least a product or two. Take for example the 70mm right-angle Mk II binoscope, which supplanted its predecessor which worked at a 60mm aperture. The modification to accept a larger pentaprism has brought the working aperture up to 69-70mm. I like to think that my raising awareness of this issue of common underperformance to a level not so well known of before has been a catalyst.

Flashlights are good enough in many cases, but the refinement of the laser makes it very much more reliable in all scopes. And don't let the notion of the eyepiece put one off; it's a good way to deliver a miniscule point of light at the focus.

[Asbytec]

If the focus of the EP is a distance x from the focus of the primary mirror, there will be a taper to the light path, causing a mismeasure of the actual aperture of approx. x/f-number...and a fuzziness comparable to the width of the light source.

This seems to support what Frank is saying. If the EP focus is any distance from the primary focus, isn't this a condition of not being at infinite focus? This is why we perform the test focused at infinity.

Glenn, how accurate does the laser diameter have to be in relation to the exit pupil? One might imagine it has to be at least as large...flooding the entire light path. But, what if it is smaller, say 5mm with a low power eyepiece and 7mm exit pupil? Will the projected cylinder diameter to the full aperture be proportional to the laser to the exit pupil diameters? In other words, if the laser is smaller than the exit pupil, will such a condition mimic a reduced aperture? Or does the idea a laser will, "deliver a miniscule point of light at the focus" make the diameter of the input beam irrelevant? I am sure this is why you employed a beam expander.

[GlennLeDrew]

Norme,
Yes, the fraction of the exit pupil filled equals the fraction of the objective filled.

The taper which results from an incorrect placement of the focused light is corrected simply by adjusting the focus until parallelism is achieved. Parallelism is checked fir by measuring near the scope and at least one focal length farther away (the farther, the more readily can the non-parallesim be detected.) Then you know the light source is correctly at the focus.

[Jon Isaacs]

I get the impression that folks put more stock in the visual sighting through a sizeable hole than they do for what is really the identical test, but with a far tinier and hence less uncertain source for the tracing of the fielded light envelope.

Glenn:

A pinhole at the focal plane is simple and tells me what I want know. If I can see the entire objective, then I know I am operating at full aperture. If I cannot see the full objective then I need to do something.

To use this test to look at fully illuminated field size, it seems to me a pinhole could be used off axis. A piece of paper drawn with a cad program with points for the pinholes. Pierce the points and start looking...

Seeing is believing.

Jon

[freestar8n]

The entrance pupil taken as the image by the objective of the small obstruction. In this case it will be a virtual image 67.5 m behind the telescope, and not useful for determining the actual working aperture. Why indeed should it?

Ugh... Unfortunately that is optics 101 and gets to the crux of all this. That image way back there doesn't just represent the entrance pupil - it IS the entrance pupil - and it lives in object space. That hole way back there is what is seen by photons emitted by the object - and they travel all they way back there in object space - and if they go in, they go in. Otherwise they don't.

The shadow method is based on measuring the size of that thing, which is far away, based on a shadow cast by it. That is extremely problematic to do accurately when the pupil is far away.

The shadow is created by projection optics that have nothing to do with the entrance pupil in the first place - even if they use optics critical to the system itself. Any optics behind the limiting stop plays no role in the size of the entrance pupil or its appearance from object space.

There are numerous other sources of systematic error that I have not mentioned. But the simplest form of error in this measurement is that the method of "collimation" is based on setting the focus so the beam appears to have the same size at two different locations. For a distant pupil, the separation of these measurement points will need to be very great - and any fuzziness of the edge due to diffraction and aberration within the projecting optics will make it hard to match those sizes without introducing taper in the beam. Even if this is achieved, an error of 1' will create an error of 20mm for my mak. If the offending aperture is any closer to focus, the error will begin to explode because the pupil distance increases dramatically.

If the beam is tapered at all, and if it happens to be even slightly divergent, then you will get a clear shadow of the front of the objective - and you won't even see the true pupil diameter because its shadow is slamming into the back of the objective. You will get a false impression the objective is the problem - when it isn't. If the true aperture is only slightly smaller than the objective, which is often the case, then whether or not the shadow is seen depends critically on how the beam happens to have been prepared.

This means that the measurement is completely ambiguous and sensitive to the beam preparation. And this is a case that is common for people trying to measure the aperture: The aperture is potentially slightly undersized, and the obstruction is near focus.

Frank

[freestar8n]

I just checked my good old 8.5x44, and a cheap 10x50, using a non-focusing high intensity LED of <1.5 mm dia. The actual working diameters read were 40 mm and 42 mm respectively, in both cases determined by an internal baffle between the objective (of nominal size) and first porro prism. A bit surprising perhaps, anyone else who have tried?

This is a description of the original "flashlight test" - and it only got improved after my complaints about it. In the original description, you "focus the system to infinity" with your eye, and then hold a flashlight to the back. The "system is perfectly collimated" and the image of the flashlight is "a tiny spot of light at the focus" that "makes a sharp shadow." Since the "beam is collimated" and "focused at infinity" this method has "zero error."

Only later did caveats to the measurement get added - so that now it is the "collimated laser test." In this test, you create a beam expander and you "collimate" it and shine it into the eyepiece. Then you adjust the output beam so it is "collimated" by making sure its diameter "does not change" with distance.

It's not clear to me if the earlier "flashlight test" measurements were ever redone with the "collimated laser/projected shadow" test - but any measurements caused by an aperture near focus should be revisited - particularly if they claim 0.5mm accuracy.

What I am requesting now is that additional caveats be added, and the term "flashlight test" not used anymore. You need to consider the stop that is causing the restriction - and use a different test if it is near focus.

Frank

[Asbytec]

OK - for people who want pictures, here is the main issue I am pointing out.

Frank, using the equations s/d < D/f and s/d*f = Aeff seems pretty straight forward for a single optic coming to focus with an obstruction. Doing the math, using D=162mm/f=324mm = 0.5 and s=53mm/d=120mm = 0.43 indicates a reduced aperture. The result s/d=0.43*f=324mm shows a reduced effective aperture of 140mm as measured by the flashlight test - strike that - the collimated laser test. I find that interesting the math and the test are consistent. I treated the mirror singularly without regard to the diverging light coming from the meniscus understanding it adds slightly to the primary's effective focal length. Doing so adds a bit of a fudge factor and indicates the difficulty of doing the math in a compound scope.

That's really my question. In a compound optic, such as a CAT, how does one treat the system? As a whole or in parts? What I mean is, is it okay to use D as the corrector diameter and f as the effective focal length even though the primary is larger and the light cone changes speed within the system? Or is it best to break the system in parts and use, say, the usable secondary area and the primary baffle diameter to determine if vignetting is occurring. If so, then some of that data seems hard to come by. It easy enough to measure the secondary diameter (42mm), but not how much of that diameter is actually in use. One might assume all of it, but the super fast focal ratio of the primary seems to say otherwise.

When I use the system as a whole, I get no indication of vignetting and that appears consistent with the pin hole test leading me to believe you can use the figures for the whole system. However, breaking the system into parts the results vary depending on what figures are used, such as full secondary diameter or part of it. One figure suggest vignetting, the other does not...nor does the system as a whole. It's confusing when some assumptions are made using figures that are hard to measure. I am sure about 32mm of the secondary is in use based on it's distance and the primary's focal ratio. In that case, there is no vignetting evident - but just barely. That, again, is consistent with visual inspection using a pin hole.

Is it safe to use the system as a whole?

[freestar8n]

Hi Norme-

I'm not clear which measurement this is, but if the f is 324 and d is 120, then the pupil is very close to the front of the system and isn't pushed back much. That is exactly the case where the flashlight test will be robust - and the agreement you get is consistent with that. Even then, you should make sure the output beam is collimated if you want 1mm accuracy.

For a compound system where the stop is not known, and it is in between lenses/mirrors in the front and lenses/mirrors in the back - it becomes much harder to do the measurement, and to know how far back the pupil is pushed. A rough guide is to take the ratio of the expected aperture to the stop diameter and multiply that by the focal length. If the stop in question is very small then you know its pupil is very distant - but often you don't know the physical size of the stop when it is inside.

The idea of using "one focal length" as a relevant length over which to determine collimation makes no sense because the distance the shadow is thrown is arbitrarily scaled relative to the focal length.

Frank

[GlennLeDrew]

Frank,
From the very first post about the flashlight test, I specified a minimum distance at which to place the flashlight. I did not stress so much the edge blurring aspect then because I was addressing the bino crowd, for whom the test was originally intended. Smaller instruments of short focal length are rather less affected by the blurring. But I did mention the importance of a not too large light source due to the chance of its projected image being not tiny enough.

I think you might be taking more credit than deserved. It was realized quite well and early by myself the limitations, and when extending the method to larger scopes independently came up with the laser variant. You certainly did little to inspire anything here, if for no other reason than your opposition to the technique from the get to.


An *image* of a pupil arises only when located by more than just the light from a point source on axis. In the restricted case of the *on-axis-only* source as used here, there is NO image of the pupil. There is only an arbitrarily long cylinder of light entering and an arbitrarily long cylinder if light exiting. By definition there can be no pupil, for a single image point is optically dimensionless; it requires an array of points to form an image.

This is why I keep pointing out that the consideration of the pupils is unnecessary; they do not factor in.

Tell us how, in this special case of a single image point, the pupil is located. Let's see a diagram.