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# Parabolizing Question

Mirror Making
25 replies to this topic

### #1 -LEGEND-

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Posted 21 March 2023 - 03:17 AM

I've recently been reading into parabolizing mirrors as I am looking forward to start a dobsonian build. Most of the work seems to make sense, but I am still a bit confused regarding parabolizing. From what I read, parabolizing appears to involve "deepening the curve," usually at the center. However, when you inspect the corresponding cross sections of a parabolic and spherical mirror with the same focus/focal ratio, it is clear that the parabola never has a curve deeper than the sphere. So what do they mean that the curves need to be deepened? Thanks!

### #2 rutherfordt

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Posted 21 March 2023 - 06:05 AM

When grinding a mirror, its usually easier to take it from a sphere to a parabola by deepening the center-- although you could also reach a parabola by making the edges more shallow, that's a lot more work.  I think typically, that both of these things are what done-- the center is made a bit deeper and the edges are made a bit more shallow.

Take this with a grain of salt, however-- while I have encyclopedic knowledge of how to do this, I have never actually ground my own mirrors, but have always purchased commercial ones.  Someone who has actually done their own mirrors may be along shortly with more accurate advice.

### #3 TOMDEY

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Posted 21 March 2023 - 07:40 AM

It is made to match the "minimum removal best-fit sphere", not the starting sphere. That's where the ~point seven zone~ remains untouched, but the rest is ever so carefully figured down, feathered radially "just so". If you're into Zernike Circle Polynomials... this all makes sense! Here's my graphic showing that and other utilizations of the Zernikes, which helps explain why optical engineers and lens designers favor the mystical Zernike Circle Polynomials.  Tom

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Edited by TOMDEY, 21 March 2023 - 07:45 AM.

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### #4 Arjan

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Posted 21 March 2023 - 10:15 AM

" So what do they mean that the curves need to be deepened? "

You assume in your picture that the parabola and the matching sphere have the same paraxial radius of curvature. If that is what you want, you increasingly lower the outer zones to achieve the parabola.
You can also ditch that assumption and let the circle touch parabola at other points, ultimately at the edge of the mirror.
As Tom says, the optimum radius to minimize the amount of glass to remove is in between, at about 70% of the mirror radius.

### #5 Mike Lockwood

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Posted 21 March 2023 - 10:22 AM

That may be optimum, but it's not practical due to edge issues.

Generally the most practical way is to make the sphere match the radius of the parabola near the edge of the mirror, not at the 70% zone.

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### #6 Scott E

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Posted 21 March 2023 - 08:27 PM

If you're not clear yet, let me put it into slightly different words. You're entirely correct that the parabola has a longer focal length (flatter at the edge) than the center. By deepening the center of a sphere, you're creating a new shorter focal length in the center and leaving the edge behind so that it's now relatively longer. So if you have a very specific target focal length you would want to start from a sphere slightly longer than your final target.

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### #7 -LEGEND-

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Posted 21 March 2023 - 09:45 PM

Thanks for all the help, everyone! It definitely cleared up a lot of things for me. Although, I do still have a few questions.

That may be optimum, but it's not practical due to edge issues.

Generally the most practical way is to make the sphere match the radius of the parabola near the edge of the mirror, not at the 70% zone.

I'm actually trying to get the spherical curve pregenerated to save some time. Is there a specific formula I can use to find the focal ratio of a sphere that would meet my parabola at the edge of the mirror?

And secondly, how would the bevels toward the edge come into play while grinding these mirrors?

Edited by -LEGEND-, 21 March 2023 - 09:45 PM.

### #8 Dale Eason

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Posted 22 March 2023 - 12:10 AM

Most of the time we ATM's match the Telesccope tube and fittings to the mirror that we have.  Trying to hold a  focal length to a specific value is hard.  While you polish the mirror it will change.  Not by a lot usually less than an inch.  The difference between the sphere and the figured mirror's focal length is usually not very big so we just don't worry about it.  Now if you are trying to match an existing tube then that is a different story and most of us don't go there.  That's a job for a very experienced mirror maker.

The bevel on the mirror does not come into play except to narrow the diameter of the clear aperture and if it is not there then you are likely to chip the edge.

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### #9 brebisson

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Posted 22 March 2023 - 01:01 AM

Hello,

You have probably seen the picture bellow (or equivalent).

This picture tells you, for a sphere, how much material to remove, and where, to get from your sphere to your parabola.

Now, it looks like there is a lot to remove in the middle, and not a lot on the edges... But this is incorrect as this is a 1 dimentional cut. But your mirror is 2 dimentional. It is a disk. So the "center" is actually small in surface, and the outer are is actually rather big.

In fact, the volume of glass to remove here is the SAME inside the 70% zone and outside! What a surprise!!!

But it's normal! They are an infinity of very similar parabolas touching (tengeanting) the sphere that we have to get started.

But we we specifically decide to try for the one that minimized work (doing otherwize would be silly!)...

So, back to your question. Starting from a sphere, to get to the easiest parabola, what do you do?

Well, you have to "dig" in the center (to remove that center buldge) and on the side to remove the edges...

Removing from the center is relatively easy. Removing from the edges is much more touchy as the overhang from your tool can easely lead you to over remove and end up with edges too low... And then the only solution is to "re-dig" EVERYWHERE else to recreate your level...

Cyrille

### #10 Arjan

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Posted 22 March 2023 - 09:10 AM

" Is there a specific formula I can use to find the focal ratio of a sphere that would meet my parabola at the edge of the mirror? "

Sure, but I wouldn't worry too much about it; for most regular ATM mirrors the curve is so shallow that the difference between sphere and parabola is in the order of the wavelength of light. In other words, just generate and polish a sphere an only then polish in the required correction towards parabola.

What focal length and diameter are you considering?
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### #11 TOMDEY

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Posted 22 March 2023 - 11:46 AM

That may be optimum, but it's not practical due to edge issues.

Generally the most practical way is to make the sphere match the radius of the parabola near the edge of the mirror, not at the 70% zone.

Great point! The edge is critical, otherwise the whole effort can fall apart with a turned edge and right back to square one, just because the edge "dipped below" and now the entire surface has to be figured down to recover the low edge back to copacetic. Best to anticipate that ahead of time and not mess with the edge, but set it as the baseline. I'm not sure how our guys at work handled giant fast mirrors... but I did notice giant full sized laps making very short excursions toward the end stages of figuring. They called it a "smoothing run" and I got the impression that they did not want to screw up the (good!) edge.

### #12 Mike Lockwood

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Posted 22 March 2023 - 12:03 PM

A paraboloid has the same ROC as a particular sphere at its center/bottom, and then the ROC continuously increase as you move away from the center/bottom.

The slider below the surface error plot in FigureXP is nice - when you drag the ROC slider to the left, you shorten the ROC of the calculated paraboloid that you are comparing with the (reconstructed) shape of the mirror you are testing.

On a smaller, slower mirror, you can drag the slider far enough to make the outer zone "flat".  This means you have shortened the ROC of the mirror shape you are "shooting for", so it is necessary to remove material inside the outer zone to match this shape.  See image below, this is a 10" f/5 mirror that requires about 3.3 waves of center deepening.  )This size and f/# is just about perfect, because dragging the slider all the way left or right matches the ROC at the edge and the center of the mirror, respectively.)

Try it.  Pick some zones, and use the same ROC and a diameter of 10".

With respect to your original image that you posted, the sphere that you drew matches the ROC of the parabola at the center of the parabola.  To make it match at the edge, move the sphere downward and increase its diameter (ROC) so that the slope matches where it meets the parabola at the edge of the imaginary mirror.

Note the green text below the FigureXP error plot - when you first go to the (F4) Surface Error Analysis screen, FigureXP automatically calculates the best fit ROC, which is 0.06216" shorter than the ROC.  This makes the error plot flat at the 70% zone, or 3.5" radius, and this minimizes the P-V error when comparing a sphere to the ideal parabola.  If you drag the slider all the way to the left (for the 10" f/5 mirror I use), you shorten the ROC by another -0.058".  So, the total reduction is ~0.120", which is the same as the measured correction difference from center to edge when measuring with a moving source tester (r^2 / (2R)).  So, you have to reduce the ROC of the center of the mirror by 0.120" compared to the edge in order to fully correct the mirror.

This is what you would get if you could measure the ROC of the exact center of the mirror and the very outer edge, but we can't.  You measure at the center of zones with Foucault, so the difference beween readings between the center of the outer zone and the center of a zone that does not include the center will be less, I think it was ~0.093" for this mirror.

I've always wanted to explain this - think I might make a short article up about it......

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### #13 duck

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Posted 22 March 2023 - 01:55 PM

Stay away from the edge!  Lockwood echoed what Ceravolo told me many years ago.  Deepen the center and accept an FL slightly shorter than the sphere you started with.  Of course, you end up with an anomaly at the edge, but not a turned edge.  I've had a raised ring just inside the edge numerous times.  My approach is similar to what Tom Dey described.  Use a full sized lap as much as possible to deepen the center.  If you get the raised ring inside the edge, make a very small lap, perhaps 1/4D, and use accented pressure with tangential W strokes over the raised ring. Examine with the knife for roughness.  Go back with the full sized lap to smooth.  I don't have hundreds of mirrors under my belt, but this is the technique I've landed on after about a dozen.

Also, not wise to attack large errors (> 1/4 wave) with a small lap (Texereau), but I've done it.

### #14 -LEGEND-

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Posted 22 March 2023 - 02:38 PM

" Is there a specific formula I can use to find the focal ratio of a sphere that would meet my parabola at the edge of the mirror? "

Sure, but I wouldn't worry too much about it; for most regular ATM mirrors the curve is so shallow that the difference between sphere and parabola is in the order of the wavelength of light. In other words, just generate and polish a sphere an only then polish in the required correction towards parabola.

What focal length and diameter are you considering?

I'm considering an f4, around 12 or 13 inches in diameter - so not a small mirror, but not too fast either. Should I just have the f4 sphere pregenerated and polish the edges then?

### #15 -LEGEND-

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Posted 22 March 2023 - 04:04 PM

A paraboloid has the same ROC as a particular sphere at its center/bottom, and then the ROC continuously increase as you move away from the center/bottom.

The slider below the surface error plot in FigureXP is nice - when you drag the ROC slider to the left, you shorten the ROC of the calculated paraboloid that you are comparing with the (reconstructed) shape of the mirror you are testing.

On a smaller, slower mirror, you can drag the slider far enough to make the outer zone "flat".  This means you have shortened the ROC of the mirror shape you are "shooting for", so it is necessary to remove material inside the outer zone to match this shape.  See image below, this is a 10" f/5 mirror that requires about 3.3 waves of center deepening.  )This size and f/# is just about perfect, because dragging the slider all the way left or right matches the ROC at the edge and the center of the mirror, respectively.)

Try it.  Pick some zones, and use the same ROC and a diameter of 10".

With respect to your original image that you posted, the sphere that you drew matches the ROC of the parabola at the center of the parabola.  To make it match at the edge, move the sphere downward and increase its diameter (ROC) so that the slope matches where it meets the parabola at the edge of the imaginary mirror.

Note the green text below the FigureXP error plot - when you first go to the (F4) Surface Error Analysis screen, FigureXP automatically calculates the best fit ROC, which is 0.06216" shorter than the ROC.  This makes the error plot flat at the 70% zone, or 3.5" radius, and this minimizes the P-V error when comparing a sphere to the ideal parabola.  If you drag the slider all the way to the left (for the 10" f/5 mirror I use), you shorten the ROC by another -0.058".  So, the total reduction is ~0.120", which is the same as the measured correction difference from center to edge when measuring with a moving source tester (r^2 / (2R)).  So, you have to reduce the ROC of the center of the mirror by 0.120" compared to the edge in order to fully correct the mirror.

This is what you would get if you could measure the ROC of the exact center of the mirror and the very outer edge, but we can't.  You measure at the center of zones with Foucault, so the difference beween readings between the center of the outer zone and the center of a zone that does not include the center will be less, I think it was ~0.093" for this mirror.

I've always wanted to explain this - think I might make a short article up about it......

Wow, this is a lot of information! I'll try it out and see if I can get any new insight!

### #16 PrestonE

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Posted 22 March 2023 - 04:34 PM

I'm considering an f4, around 12 or 13 inches in diameter - so not a small mirror, but not too fast either. Should I just have the f4 sphere pregenerated and polish the edges then?

A 12 inch F4 for a First Mirror will be a Challenge!!!

Can it be done, yes...  With a Good Mentor, Lots of Time and Luck.

Better to start with something 6 to 8 inches at F6 to learn the basics.

Best Regards,

Preston

### #17 Arjan

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Posted 22 March 2023 - 04:38 PM

" Should I just have the f4 sphere pregenerated and polish the edges then? "

Not sure what you mean with pregenerate, it is usually a fairly course process.
You need to refine the rough spherical surface with increasingly finer grits and eventually polish with e.g. pitch and cerium oxide. Only when the surface is optically smooth you can start parabolization. Most of the work is in hogging the rough sphere and the polishing.
You skip the hogging when you have the blank pregenerated.

### #18 -LEGEND-

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Posted 22 March 2023 - 05:00 PM

" Should I just have the f4 sphere pregenerated and polish the edges then? "

Not sure what you mean with pregenerate, it is usually a fairly course process.
You need to refine the rough spherical surface with increasingly finer grits and eventually polish with e.g. pitch and cerium oxide. Only when the surface is optically smooth you can start parabolization. Most of the work is in hogging the rough sphere and the polishing.
You skip the hogging when you have the blank pregenerated.

Yeah, that's what I meant - I will have to fine grind every part, but I am just talking about the parabolizing process. I'm just a bit lost on what spherical curve (i.e. focal ratio) I should have pregenerated to make the parabolization process easiest for me

### #19 -LEGEND-

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Posted 22 March 2023 - 05:11 PM

Most of the time we ATM's match the Telesccope tube and fittings to the mirror that we have.  Trying to hold a  focal length to a specific value is hard.  While you polish the mirror it will change.  Not by a lot usually less than an inch.  The difference between the sphere and the figured mirror's focal length is usually not very big so we just don't worry about it.  Now if you are trying to match an existing tube then that is a different story and most of us don't go there.  That's a job for a very experienced mirror maker.

The bevel on the mirror does not come into play except to narrow the diameter of the clear aperture and if it is not there then you are likely to chip the edge.

Then is there a way to measure the actual focal length of the completed mirror? I can definitely wait off figuring the tube (cutting the holes, placing the components) until after I make the mirror, but I still need to know at some point what the focal length is to do so

### #20 drneilmb

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Posted 22 March 2023 - 05:22 PM

Yes, measuring the focal length of a completed mirror is quite easy. Waiting to manufacture the components until the optic is complete is the among the essentials of amateur telescope making.

Using a flashlight and an index card with a small hole in it, shine the flashlight onto the mirror and move the card and light around until the image of the hole is exactly focused on the card. That distance between the card and mirror is the radius of curvature. The focal length is half of that distance.

Edited by drneilmb, 22 March 2023 - 05:32 PM.

### #21 duck

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Posted 22 March 2023 - 05:26 PM

you need to read "HOW TO MAKE A TELESCOPE" by Texereau

### #22 Scott E

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Posted 22 March 2023 - 11:09 PM

I'm still getting the feeling you don't grasp the magnitude of variability of the ROC going from a pregenerated coarse-ground surface to a polished one. It's much easier for the ROC to vary, by as much as a few inches, whether by accident or on purpose, during the fine-grinding process than during the polishing or parabolizing process. Just get it pregenerated to the ROC you want in the finished mirror. Keep an eye on the sagitta with a feeler gauge or spherometer during fine grinding. If you can control it tightly enough, shoot for a fine-ground curve maybe a half inch (at most) longer than your target when you start to polish. That would make a good starting point for parabilizing.

My advice to you, though, is not to worry about hitting a small target like that with your first mirror. Just get a good figure, then build the telescope around it.

Having said all that though, there was mention of a simple formula for hitting that small target. That is, if I have a target focal length I need to hit, what ROC should I start with if I want to polish the center and end up with a parabola? It would seem as simple as adding the total center-to-edge knife-edge movement of a slitless tester. Is it that simple?  I ask for myself because the primary I'm planning for my 26" Cassegrain is f/2.5 with a conic of -0.8. That's a lot of glass to remove and I'm going to start parabolizing in fine grinding. And the target is very small!

### #23 -LEGEND-

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Posted 22 March 2023 - 11:32 PM

Yes, measuring the focal length of a completed mirror is quite easy. Waiting to manufacture the components until the optic is complete is the among the essentials of amateur telescope making.

Using a flashlight and an index card with a small hole in it, shine the flashlight onto the mirror and move the card and light around until the image of the hole is exactly focused on the card. That distance between the card and mirror is the radius of curvature. The focal length is half of that distance.

Well, the thing is I'm trying to measure it before the coating is complete, because I wouldn't be able to grind it once I apply the silver/aluminum coating... would this be the only way?

### #24 Dale Eason

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Posted 23 March 2023 - 12:14 AM

Yes you can easily measure the uncoated mirror just as he described.  Bare glass is reflective enough to do it.   In addition when you are testing the mirror during figuring you will also have the ability to measure the focal length then as well.  That usually is the where one gets the best measure of the focal length.  Using the Foucault or Ronchi test.

### #25 Dale Eason

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Posted 23 March 2023 - 12:17 AM

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