# Producing Concave Surfaces With A Vertical Mill



## randyc

_I posted a method of producing small ball configurations on a vertical mill  in a previous thread regarding various setups for a rotary table.  This method might be useful to a few people (telescope guys, maybe).  I'm not sure that this is the correct place to post this but I couldn't think of a more appropriate one..._

This is an experiment that I tried a few years ago following another interesting experiment to learn how to produce large concave surfaces on a lathe or a vertical mill.

Tilting the head of the mill (or the rotary table) to a specific angle and adjusting the “stickout” of a flycutter, one can produce a fairly precise concave surface.  Errors in the radius caused by runout in the mill spindle or rotab bearing are averaged out with multiple passes.





The flycutter is SLOWLY advanced into the work with minimal DOC and the rotary table is then rotated through a full turn(s) until the full depth is nearly reached.  Final cut should be multiple revolutions to average errors.





It is an exercise in descriptive geometry to determine the starting parameters.  One of these days I’ll make a spreadsheet to determine them.  I suspect that making a CAD layout would be lots faster 

PS  I'm too impulsive about spelling and construction otherwise I wouldn't have to edit !


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## Billh50

There is a formula for cutting a large radius such as 32" in a 3" wide part with a 6" cutter as an example. The formula tells you how much to tip the head of the mill. I would assume if you used that formula and put your cutter at it's lowest point on center of a round piece it would create a concave the same way when turning the part in a rotary table.
The formula is  this.  Divide 1/2 radius of the cutter by the radius to cut. The answer will be the Tangent of the angle to tip the head of the mill.

What you are really cutting is the bottom of a parabola.


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## Billh50

example of what will be cut.


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## randyc

We'll have to agree to disagree.  The flycutter, rotating in a circle cannot produce a parabola as the work is also rotated in a circle - this can only generate a spherical surface when set up as shown.

The concave surface is a generated one just as the ball produced on a vertical mill was generated (mentioned above, referencing the post).  In fact, except for the cutting tool orientation, the two processes are identical.

PS:  I neglected to mention that your sketch shows the spindle axis and the rotary table (and the work) axis as being coincident.  That's not the case, see the first photo.


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## chips&more

I use the mill and rotary head all the time to turn round balls on the mill. Can’t remember the last time I did it on the lathe? You must have the spindle datum center in line with the rotary head datum center or you will not produce a round ball. Rather, it will look like Stewie’s head! IMHO the concave machining example in this thread will produce a concave segment. And if the example always has a perfect center with no tit, then it will be a true arc of a circle…Nice Job, Dave.


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## randyc

chips&more said:


> ...You must have the spindle datum center in line with the rotary head datum center or you will not produce a round ball. Rather, it will look like Stewie’s head! ...



Hi Dave, I'm probably misinterpreting your statement but to produce a ball in a vertical mill the spindle axis and rotary table axis can't be coincident.  Consider the following sketch, which I posted in the "Rotary Table Tricks and Tips":



As you can see, the angle between the rotary table axis and the spindle is 105 degrees (15 degrees off vertical) in this case.  This isn't particularly critical, the angle is mainly determined by the desired dimension of the "neck".  In the following photo, I used 15 degrees offset:




If the axis of rotary table and mill spindle were coincident, then the vertical mill would be a vertical lathe unless I am not understanding your statement.

Cheers and thanks for your comments !


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## chips&more

randyc said:


> I'm probably misinterpreting



You misinterpreted, but in fact your drawings show the two datums in line as I explained! I did not say in line with the arc of the boring head, I said in line with the datum center of the boring head/spindle. Sorry, typing and explaining are not my first love...Dave.


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## Billh50

I was talking about the concave part not the ball. It will appear as a sphere because you are only seeing the lower part of the parabola. But unless the cutter is at 90 degree to the part you are cutting a rotating parabola. If you take a flat round disc. Hold it in front of your eyes and slowly tilt it away from you. you will see the parabola start to appear at edges until the disc is 90 degrees when it becomes a circle again. But tipping the head or part the radius will become smaller and smaller causing more of a dish. That is the parabola effect.


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## randyc

Billh50 said:


> I was talking about the concave part not the ball. It will appear as a sphere because you are only seeing the lower part of the parabola. But unless the cutter is at 90 degree to the part you are cutting a rotating parabola. If you take a flat round disc. Hold it in front of your eyes and slowly tilt it away from you. you will see the parabola start to appear at edges until the disc is 90 degrees when it becomes a circle again. But tipping the head or part the radius will become smaller and smaller causing more of a dish. That is the parabola effect.



I see your point about rotating the disc but it would be elliptical, not parabolic.  I definitely agree that the cutter axis should be at 90 degrees and the correction is appreciated !


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## randyc

chips&more said:


> You misinterpreted, but in fact your drawings show the two datums in line as I explained! I did not say in line with the arc of the boring head, I said in line with the datum center of the boring head/spindle. Sorry, typing and explaining are not my first love...Dave.



Dave, my memory is not so good anymore.  I thought that you posted a sketch and the axis of the sketch was labeled indicating that the spindle and the rotary table were coincident.  Maybe I'm thinking of something else.  Anyway we're in agreement now


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## Billh50

Randy,
your right....I was thinking 2D again instead of 3D. If the tipped cutter was just passed along a part without the part turning it would then be the bottom of a parabola.


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## randyc

Billh50 said:


> Randy,
> your right....I was thinking 2D again instead of 3D. If the tipped cutter was just passed along a part without the part turning it would then be the bottom of a parabola.



Bill, are you sure ?

Seems to me that it would still be elliptical since the path of the tool would be similar to your example of examining a disc that is slowly rotated.

As the angle of the spindle axis is changed from 90 degrees (with respect to a non-rotating workpiece), the cutting path would gradually evolve from a circular arc to an ellipse to a straight line segment. Or so it seems to me.

Cheers -

edited to add:  Not that this is really of importance to most of the forum, right


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## Billh50

I am sure....it's how it was all explained to me by an engineer 50 yrs ago when I first started using the formula. In fact he used the same sketch I used to show it. The tool would be running as an elliptical but the cut if made straight through the part would be a parabola. As far as cutting a turning a piece while tipping the cutter I think it would appear more as an elliptical than a parabola. But then what do I know I only stayed at a Holiday Inn once.


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## randyc

Hi Bill,

I definitely agree with your first statement but I think that the second one is incorrect.  But who knows, I’ve never even stayed in a Holiday Inn, LOL, so my education is incomplete 

Tilting the axis of a cutter that rotates in a circular path will always produce an elliptical contour in my opinion.  I can’t even think of a way to create a parabola with manual machinery except maybe with a tracer mill or some kind of pantograph.  Anyway, this is the way that I see it:





On the right is a cutting tool whose axis is vertical.  It rotates in a circular path that is indicated by the grey dashed line.  The green object represents a stationary workpiece that has been machined by the cutting tool as it passed through the work.

On the left, the axis of rotation has been adjusted to about thirty degrees, producing a thirty degree ellipse, according to my CAD program.  The grey dashed line is the path of the cutting tool and the green object again is the stationary workpiece that has been machined by passing the cutting tool through the workpiece.  The contour of the resulting cut must follow the path of the cutting tool as shown, right?

Cheers,
randyc


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## randyc

Bill, I got confused, what I meant to say was: I disagree with the FIRST statement and agree with the SECOND.


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## ScrapMetal

I'm going to throw this out there for the sake of clarity.  I can see how a parabola could be cut but I think it would require a more sophisticated setup.




-Ron


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## Billh50

All I know is it was explained that way to me by someone with a much higher IQ than me and was a wiz in all kinds of math disaplines. He had even corrected college physics books when he was in high school. So I trust what he says about things like this.
But either way it worked for cutting a 32" rad in a part with a 6" cutter almost 50yrs ago.


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## Billh50

I was just looking at scrapmetals post and think I can explain it better. looking at the picture he posted. An ellipse is a closed figure with no open ends where as a parabola and a hylerbola have open ends.


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## randyc

Bill: but the concave surface we're discussing is only part of a geometric shape - an ellipsoid segment to be nerdy about it - so it doesn't make any difference whether the configuration is open or closed.

It's only logical that the contour of the part being milled has to be identical to the path of the tool milling it.  A circle tilted at an angle (which is the tool path) cannot be any other shape except an ellipse.

(You've mentioned your engineer friend a couple of times and he might be way smarter than me but FWIW, I have a BSME and a BSEE along with five decades of experience.  I'm not claiming that this makes me right, just sayin'.)

ScrapMetal, that's the classic example I was taught with in high school drafting except that you can't obtain a hyperbola from a conic.  A hyperbola is two curves that are mirror images.

Also I can't visualize how a concave surface can be obtained from a cone, help me out


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## Billh50

you know what...we can debate this forever...I give up


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## randyc

Billh50 said:


> you know what...we can debate this forever...I give up



I hope that I didn't offend you - I certainly didn't intend to !  You pointed out an important point that I'd missed (about having to tilt the mill 90 degrees to produce a true spherical surface) and I was attempting to also make a correction, not to be argumentative.

randyc


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## Billh50

No problem......My only thing as simply as I can put it was that an ellipse is nothing more than a stretched circle where as a parabola would always be open and never closed. And if you look horizontally at the bottom half of the cutter you will see a parabola and if you came up with the part the sides would eventually be straight so would not create an ellipse. 
I know it is hard to explain on here so I am just not going to try anymore. But am not offended at all. Everyone has their opinion on things and sometimes it is hard to explain well enough to change them.

So anyway,
how's your day going?


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## randyc

You're right - everyone has an opinion and the rest of us should respect it.  I believe that you and I have reached that point and there's little point in continuing so peace, bro 

My day is going OK.  Since my wife died two months ago, I've had very little ambition to get out in the shop (or do anything, really) but I'm determined to change that and it was this thread that stimulated me, LOL.  I have a short, simple lathe project in mind that I'm going to document and post.

Thanks for asking !


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## Billh50

Randy I can sympathize with you about your wife.....my first wife died 19 yrs ago this past valentine's day. not easy to loose someone.


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## chips&more

randyc said:


> ScrapMetal, that's the classic example I was taught with in high school drafting except that you can't obtain a hyperbola from a conic.  A hyperbola is two curves that are mirror images.



IMHO, Scrapmetals example of a solid angle with its identified cut-outs, i.e.; the hyperbola, is in fact a hyperbola…Dave.


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## randyc

What an awful, awful day to lose a loved one !

(Joyce died of cancer, both my parents died of cancer and I have cancer although I'm in remission.  I always thought that I'd go before Joyce because I was 12 years older.  Sometimes I get angry about this but then I'm grateful for the time we had.)

Thanks for your empathy -


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## randyc

chips&more said:


> IMHO, Scrapmetals example of a solid angle with its identified cut-outs, i.e.; the hyperbola, is in fact a hyperbola…Dave.



Nope, see this (or do your own search for a hyperbola).  The example Scrapmetal depicted was two parabolas of different size.

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRw&url=http://www.intmath.com/plane-analytic-geometry/6-hyperbola.php&ei=1BfpVMvCL8KcNuregYgG&bvm=bv.86475890,d.eXY&psig=AFQjCNHigU2wn236zyqj-TwhNWWtR0uurA&ust=1424648530985348

Scroll down far enough in the above link and you'll find that a hyperbola requires two cones stacked together to be able to achieve a "sliced" cross-section with the correct configuration.


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## Billh50

My wife died of a strain of viral pneumonia that caused her lungs to fill with fluid so fast she couldn't breathe. I found her too late to help and it hurt for almost a year. But then I realized we had had a great day that day and she was happy just before it happened. And life does go on for the rest of us.


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## randyc

Billh50 said:


> My wife died of a strain of viral pneumonia that caused her lungs to fill with fluid so fast she couldn't breathe. I found her too late to help and it hurt for almost a year. But then I realized we had had a great day that day and she was happy just before it happened. And life does go on for the rest of us.



That's a bad way to go and will be my demise as well.  One lung has already been removed (I have non-small cell lung cancer caused by chemical exposure when I worked in the semiconductor industry - I'm not a smoker) so it's difficult to exert myself already.

I'm so sorry about your wife.  It's been many years but I'm sure that there are still times when you feel the same pain as you did then.

My wife's parents and I were with her when she went.  She was dozing off and on and then she just didn't wake up.  Here we were on our wedding day.  I keep this picture in my living room so that I can look at her every day.  She was so beautiful and so happy ... Need to make a frame one day.


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## brav65

Guys I think you are missing the point. It's all ball bearings today... Now get me some gauze pads and 3 in 1 oil so I can check it (Chevy Chase in Fletch Lives) ;-)

I posted this before I read about your wife,  sorry for your loss.


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## randyc

Not to worry, you didn't know.  We got way off topic and yours was a funny yet tactful post


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## DKD228

Billh50 said:


> There is a formula for cutting a large radius such as 32" in a 3" wide part with a 6" cutter as an example. The formula tells you how much to tip the head of the mill. I would assume if you used that formula and put your cutter at it's lowest point on center of a round piece it would create a concave the same way when turning the part in a rotary table.
> The formula is  this.  Divide 1/2 radius of the cutter by the radius to cut. The answer will be the Tangent of the angle to tip the head of the mill.
> 
> What you are really cutting is the bottom of a parabola.


Do not believe that is correct, any mathematic proof ?


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## Tony Wells

What you are doing is truncating a cylinder at an oblique angle. You obtain an ellipse (or section of one). No debate about it, really.....it's geometry.

http://www.matematicasvisuales.com/english/html/geometry/planenets/cylinderobliq.html


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## Billh50

DKD228 said:


> Do not believe that is correct, any mathematic proof ?


well it was good enough for aircraft work. we used it all the time when making aircraft parts.


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## jmh8743

I worked on this topic for several months off and on last year. I had a client that wanted a spherical bearing with a radius of 12".  Came to the same conclusion. I did not have equipment or tooling that would accomplish task. (surfaces must mate.) But many surfaces can be made to approximate another as I have done with a parabola and a circular arc thousands of time in my checkered and dissipated career. A 90 deg circular path must be rotated to achieve a spherical segment. its geometry.


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## Paul Thompson

I used this technique 30 years ago to put a raised arc along a plate. Having it be higher in the center means cutting on the upside of the tool path so for a 6" long plate the flycutter needed  more than a 6 inch radius. Otherwise it would start cutting a dish on the backside. I always assumed it cut an ellipse because it would put a smaller radius at the edges if you made the swing radius only slightly bigger than the width of the part. This trick is often used with table saws by tilting the blade and pushing the wood across sideways to make a shallow depression, like a serving dish. The part I designed was a slightly curved backing plate for a silicone rubber pad. It was used for making microfiche (film) duplicates by pressing the original and film together and flashing light to cast a shadow of the master onto the film. The curved foam made line contact down the center and squeegeed any air bubbles as it compressed.


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## Billh50

The best way I can explain it is this. Hold a round plate horizontal. Now slowly turn it toward you and look at the  bottom half of the rim as you turn it. It is slowly becoming becoming a parabola with narrower legs and smaller radius at bottom. Now if you are just raising the table into the cutter as you cut across the surface you are creating a parabola which if you go past center of cutter it will have straight sides. But by using just the lower part of a large cutter on a small width part you will be using the very bottom of that parabola, which even though not a perfect radius, is so close to one that it would be very hard to tell the difference.


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## tcarrington

Look up how spherical lens surfaces are generated. This is essentially the same thing. The lens people use a specialized tool and diamond cutters, but they make spherical lenses and other more exotic surfaces with a similar setup. The key is the cutter must sweep through the center of rotation as it has been explained to me. 

Google "generating spherical surface"


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## wrat

I can't believe I'm wading into this, but here goes.

There is no Parabola.  There is no part of a parabola.  There is nothing parabolic about any of it.

With two rotating axes, there can only be circular (spherical) and ellipse (ellipsoidal) surfaces produced.  There cannot even be radii produced larger than the sweep of the cutter.  The cannot be any hyberbolics.

What is produced is an ellipsoidal (swept 3D ellipse) that can be used to *approximate* a spherical surface.  Close enough for even aircraft work (that is typically .03 at best.  Careful probing on your local CMM will reveal this.  Yes, I've seen the charts and "formulas", but the geometry is not up for opinion.

This is because all of these fall into a class of curve known as 'conics'.  Someone posted a cone that had been cut up.  That's why the cone matters.

Analytically, these curves all follow what is known as the "2nd degree general equation, or <deep breath> Ax^2+By^2 +Cxy+Dx+Ey+F=0
So when you have, say, A,B,and F coefficients, you have a circle. (where F = radius^2).  
Mathematically, without a distinct Cxy component (and a particular proportion at that) no parabola can exist.  If you're working with circles, like rotating things are, no Cxy component enters into the equation.  Even if they're eccentrically rotated, the resultant intersection is still just elliptical.
So, returning to the cone, we can see that when the cone is sliced exactly perpendicular to the axis, a circle results.  Only then.  When sliced exactly perpendicular to the base, a parabola results.  Only then.  All the infinite other slicings that are not normal to base or axis are ellipse and hyperbola.
I got into this both as an Engineer and as a Lofter before that.  I lofted portions of some small planes you've heard of.  Everything subsonic, including airfoils and propellers, are lofted to 2nd degree equations or combinations thereof.  As an Engineer, i investigated 'best-fit conics' for surveillance algorithms.  (There are no true circles in nature, so finding one with your UAV means you found something manmade... like a truck.)
In defense of anyone i might have offended, the word "parabolic" is like "aircraft aluminum" or "surgical steel" and is thrown around with reckless abandon.. often by folks that really ought to know better but choose to go with the flow instead of correcting a roomful of people (BT;DT).  So it's no one's shortcoming if they haven't ventured deep into GeekLand to sort all this out.
Also, very often the differences between these geometric distinctions are microscopic.  Indeed, unless your cutter is *perfect*, which none can be, you're cutting an ellipse, anyway.
All i'm offering is analytical theory.  Not how to cut stuff.

Wrat


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## randyc

Great description, much more eloquent than my feeble efforts -


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## DKD228

wrat said:


> I can't believe I'm wading into this, but here goes.
> 
> There is no Parabola.  There is no part of a parabola.  There is nothing parabolic about any of it.
> 
> With two rotating axes, there can only be circular (spherical) and ellipse (ellipsoidal) surfaces produced.  There cannot even be radii produced larger than the sweep of the cutter.  The cannot be any hyberbolics.
> 
> What is produced is an ellipsoidal (swept 3D ellipse) that can be used to *approximate* a spherical surface.  Close enough for even aircraft work (that is typically .03 at best.  Careful probing on your local CMM will reveal this.  Yes, I've seen the charts and "formulas", but the geometry is not up for opinion.
> 
> This is because all of these fall into a class of curve known as 'conics'.  Someone posted a cone that had been cut up.  That's why the cone matters.
> 
> Analytically, these curves all follow what is known as the "2nd degree general equation, or <deep breath> Ax^2+By^2 +Cxy+Dx+Ey+F=0
> So when you have, say, A,B,and F coefficients, you have a circle. (where F = radius^2).
> Mathematically, without a distinct Cxy component (and a particular proportion at that) no parabola can exist.  If you're working with circles, like rotating things are, no Cxy component enters into the equation.  Even if they're eccentrically rotated, the resultant intersection is still just elliptical.
> So, returning to the cone, we can see that when the cone is sliced exactly perpendicular to the axis, a circle results.  Only then.  When sliced exactly perpendicular to the base, a parabola results.  Only then.  All the infinite other slicings that are not normal to base or axis are ellipse and hyperbola.
> I got into this both as an Engineer and as a Lofter before that.  I lofted portions of some small planes you've heard of.  Everything subsonic, including airfoils and propellers, are lofted to 2nd degree equations or combinations thereof.  As an Engineer, i investigated 'best-fit conics' for surveillance algorithms.  (There are no true circles in nature, so finding one with your UAV means you found something manmade... like a truck.)
> In defense of anyone i might have offended, the word "parabolic" is like "aircraft aluminum" or "surgical steel" and is thrown around with reckless abandon.. often by folks that really ought to know better but choose to go with the flow instead of correcting a roomful of people (BT;DT).  So it's no one's shortcoming if they haven't ventured deep into GeekLand to sort all this out.
> Also, very often the differences between these geometric distinctions are microscopic.  Indeed, unless your cutter is *perfect*, which none can be, you're cutting an ellipse, anyway.
> All i'm offering is analytical theory.  Not how to cut stuff.
> 
> Wrat


Thank you so much for finally putting forth a proper explanation of this procedure.


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## Pstovall

I know this is an old post but I just joined and thought I would share a mathematical proof for anyone interested, please see attachment.


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## higgite

My head hurts.

Tom


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## savarin

So does mine.
I was really hoping a simple solution would be forthcoming to get the real parabolic surface.
I will stick to sweeping my pendulum grinder across the rotating glass to get my spherical concave surface.


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