# Generating a hollow of a desired radius of curvature



## savarin (Mar 26, 2022)

I need to hollow out two aluminium disks of 11" dia to a radius of curvature of 144"
Some years ago I'm certain I read somewhere about using a large grinding disk held at a shallow angle to do this but cannot find it.
I believe the grinding disk has to be larger in radius than the blank and a method of calculating the angle of the blade to the work to produce the desired radius of curvature was shown.
Does anyone know of this or how to work it out?
(I'm maths challenged so just the formula wont cut it for me) (pun intended)


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## woodchucker (Mar 26, 2022)

I don't know, but a question. can't you create a template?
take a board, a pencil, a string.
put  a nail in another board and measure out from the nail 144"
Then let the pencil scribe the radius on the template (board), cut it out and you have your template to match up to.  I'm sure it will be difficult, but not impossible.
I would make the template from hardboard / masonite/ tempered hardboard.   works well for templates, sands well.  if you need super accurate, kevlar does not stretch, or use wood joined together to avoid the stretch of the string...
I have some fishing line that does not stretch also... it's very fine, can't remember the name... it will cut through you...


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## savarin (Mar 27, 2022)

The template method will work after a fashion but the angled disk method appears to produce fine tolerance shapes.
So far I've found this but its not detailed enough for me to work it out.








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## homebrewed (Mar 27, 2022)

I did something like what you're describing but on a much smaller scale.  I speculated that the path of a cutter (whether it's an end mill or grinder) would describe an ellipse if it's tilted with respect to the work.  Maybe OK, maybe not -- depending on what you really need.

To form a surface with that shape, I installed a chuck on my rotary table, chucked up a round of aluminum (about 1 inch in diameter) and tilted the table slightly by putting a shim under one of the clamped sides. 

I installed an end mill and moved the mill table so the center of the aluminum round was directly underneath the lowest point of the end mill.   Turned on the mill and lowered it until the cutter encountered the work, then dropped it about .010" and rotated the round 360 degrees with my RT.  Repeated until the dished-out shape reached the OD of the aluminum piece.

It formed a nice dished-out surface but it's not spherical.  Technically, it's called a section of an oblate spheroid.

I don't think it's possible to use this approach to form a perfectly spherical surface, but maybe you don't need that.  Clearly, you can't use this method to make something with a 5.5 inch radius...unless you make a face-mill like thing yourself.  A big fly cutter might do it.

Another, much more laborious technique, would be to do it stepwise with flat-bottomed end mills or ball mills and then somehow smooth out the transitions:  but, again, without knowing exactly what you want with the final result, it's hard to say if the final result will work for you.

Having a CNC mill would make things much easier but you haven't said what you've got to form that surface.


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## savarin (Mar 27, 2022)

Heres one method I used for a much smaller radius of curvature but this has been extended to cut the 144" roc but I havnt tried yet.


the grinder is suspended from the centre hanger but for a 12 foot high tripod even with bracing bars its a wee bit over the top.
The pendulum swings back and forth and the disk is clamped to the turntable.
I will use diamond disk for glass and standard  grinding wheels for the aluminium (hoping they dont clog up).
I'm hoping I can get a more secure method.


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## Parlo (Mar 27, 2022)

Plot the path of a ball nose cutter the cutter radius above the finished profile. Divide half the path into equal stepovers, either on the X or Z. Note the coordinates relating to each step then using a rotary table cut each circle. A large diameter cutter would produce smaller cusps.


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## savarin (Mar 27, 2022)

sorry parlo, only have a lathe and drill press.


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## Parlo (Mar 27, 2022)

savarin said:


> sorry parlo, only have a lathe and drill press.


You can use the same method with a radiused end form tool set as a boring bar, remember that the cross slide movements are double.


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## T Bredehoft (Mar 27, 2022)

You're trying to make a Newtonian Telescope? That's beyond my tool room experience. Wow. 
My 'mentor' ground his 8" the traditional way, 'walking the stump'.


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## MrWhoopee (Mar 27, 2022)

This does not generate a true radius but is frequently close enough for loose tolerance work.  Radius of cutter divided by radius to be produced equals the sin of the tilt angle.


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## Parlo (Mar 27, 2022)

MrWhoopee said:


> This does not generate a true radius but is frequently close enough for loose tolerance work.  Radius of cutter divided by radius to be produced equals the sin of the tilt angle.


That's a handy formula, I'll make a note of that.

Does this only approximate a curve that is the required radius deep and it's diameter wide at the top or does it work at all depths of radius?


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## MrWhoopee (Mar 27, 2022)

Parlo said:


> That's a handy formula, I'll make a note of that.
> 
> Does this only approximate a curve that is the required radius deep and it's diameter wide at the top or does it work at all depths of radius?


I can't tell you. I've used it once or twice to produce a saddle for a cylindrical part. I recently attempted to analyze the math but quickly got lost, then lost interest. Perhaps someone with better math chops will come along.


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## Parlo (Mar 27, 2022)

MrWhoopee said:


> I can't tell you. I've used it once or twice to produce a saddle for a cylindrical part. I recently attempted to analyze the math but quickly got lost, then lost interest. Perhaps someone with better math chops will come along.


I seem to remember some charts that came with the Huron milling universal machine head for setting radii and angles.


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## woodchucker (Mar 27, 2022)

savarin said:


> Heres one method I used for a much smaller radius of curvature but this has been extended to cut the 144" roc but I havnt tried yet.
> View attachment 401927
> 
> the grinder is suspended from the centre hanger but for a 12 foot high tripod even with bracing bars its a wee bit over the top.
> ...


Is there a video of this?


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## Illinoyance (Mar 27, 2022)

It could be done with a radius rod between the cross slide and the tailstock provided your lathe bed is long enough.


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## gard (Mar 27, 2022)

There is a way to cut a spherical surface done with a rotary table on a milling machine, As I remember the stock is rotated on the rotary table, The mill head is tipped at an angle with a boring head. The head angle and boring diameter are adjusted to go between bottom center and edge of the dished out area.  I saw a cool utube of this years ago cutting a ball shaped part. To picture how this works take a large o-ring and set it on a larger diameter ball so one point of the O-ring is at the top of the ball, now rotate the ball under the O ring. The O-ring represents the boring head cutter path.
I think you can make any radius surface as long as the rotary table will rotate. The curvature of the spherical surface is determined by the angle of head and cutting diameter of boring head.
Perhaps something like a tool post grinder with a cutter that is 1/2 the diameter of the area you want dished out.
I would imagine you could do something along the line of a ball turner on a lathe where the single point cutter is slowly moved with the part spinning in the lathe.
This is not the video I was thinking of but does show the idea
spherical milling


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## RJSakowski (Mar 28, 2022)

I wa intrigued with the question of whether a grinding disk could cut a perfect spherical depression  in a disk.  I did some modeling in SolidWorks of a spherical surface with a radius of curvature of 144" and a diameter of 11" and a 24" disk.  The disk would necessarily be touching the spherical  surface at its center and at its edge.  It turned out that if the disk was inclined at a 85.9º angle to the line passing through the center of the sphere and the center of the radius of curvature, those conditions are met.  

SolidWorks has a diagnostic tool that shows where there is interference between two objects in an assembly which shows up as the red area in the picture below. As can be seen, the two surfaces are just touching at the center and at the lip of the spherical surface. but there is interference at all points between.  This would mean that the disk would cut deeper in those locations, creating a non spherical surface.   The deviation would be slight in most cases but not acceptable uf a true spherical surface was required. The deviation worsens as the diameter of the grinding disk decreases.  A 12" disk is depicted in the second picture.


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## gard (Mar 28, 2022)

That is very cool, what would happen if the cutting disc was only 5 1/2 or 6" in diameter?  The interference is interesting, is that due to the shape of the edge of the grinding wheel? I was under the impression it would create a perfect sphere if the cutter was a perfect point. I think the key point like you said, is the cutter needs to pass thru the center bottom of the spherical surface.


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## RJSakowski (Mar 28, 2022)

gard said:


> That is very cool, what would happen if the cutting disc was only 5 1/2 or 6" in diameter?  The interference is interesting, is that due to the shape of the edge of the grinding wheel? I was under the impression it would create a perfect sphere if the cutter was a perfect point. I think the key point like you said, is the cutter needs to pass thru the center bottom of the spherical surface.


The closer the diameter of the disk is to the diameter of the curvature, the better the approximation of a spherical surface.  A 6" disk will be even worse.  This representation is for a perfect cutting edge.  Here is a 6" disk.


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## WobblyHand (Mar 28, 2022)

Getting back to "hard machining" one _could_ use a single point diamond to turn the surface.  Now that would be an interesting rig.  As I understand it, single point diamond turning is used to make aspheric complex optical surfaces.  https://reynardcorp.com/diamond-turning/


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## Firstram (Mar 28, 2022)

I vote for a tripod with a pendulum, that would let you use the same set up for both operations. You could look for some used scaffolding for the tower and sell it when finished.


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## Mitch Alsup (Mar 28, 2022)

If you are grinding a disk into a sphere, what you need is the sagita of the center.

Set tangent to 144" set cosine to 11/2 = 5.5"

SQRT(144^2-5.5^2) = 143.895

144-143.895 = 0.10507

So, if you had a 5.5" diamond grinding disk you would tilt the head such that the inner edge was 0.10507" lower than the outer edge.


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## whitmore (Mar 28, 2022)

It may be tedious, but... I'd get a known-diameter round carbide lathe tool, center it carefully, and
make multiple lathe plunge cuts, guided by a distance-from-center and a table of depths.

It takes a bit of geometry, of course, to make that table.
First, make plunge cuts every 0.1 inch, to rough  it out; rotate the cutter to a fresh surface,
and reset the depth 'zero' and do a finish pass of plunge cuts every 0.010" from the center.

The diameter's 11", the radius range is 5.5, so it just takes 550 repetitions of cross-table set, plunge
according to a dial gage and table, retract.   It won't take much material off on the second pass,
so you can hope for heat buildup to be minimal.


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## RJSakowski (Mar 28, 2022)

If I were making this, I would use my Tormach CNC mill.  I would use an 11/16" ball end mill.  Setting the maximum scallop at .001", it would take about 45 minutes to cut.


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## Parlo (Mar 29, 2022)

RJSakowski said:


> If I were making this, I would use my Tormach CNC mill.  I would use an 11/16" ball end mill.  Setting the maximum scallop at .001", it would take about 45 minutes to cut.


The program looks complicated. Would you mill the radius from side to side, move forward and the repeat at greater depths until you reach the centre, then repeat in reverse? What stepover would you need for a 0.001" scallop using 11/16" ball nosed cutter? Is 11/16" the largest cutter you can use? larger cutters leave smaller cusps so less passes.


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## RJSakowski (Mar 29, 2022)

Parlo said:


> The program looks complicated. Would you mill the radius from side to side, move forward and the repeat at greater depths until you reach the centre, then repeat in reverse? What stepover would you need for a 0.001" scallop using 11/16" ball nosed cutter? Is 11/16" the largest cutter you can use? larger cutters leave smaller cusps so less passes.


The program is complicated but the CAM software takes care of it.  The software that I use is called SprutCAM and the particular method does mill from side to side, automatically changing the cut depth to follow the model.  I set the parameters for a maximum scallop of .001" and it calculated the required stepover.  I chose an 11/16 ball end mill because the largest one I have.  A larger end mill will result in fewer passes but a smaller end mill can be used with a resulting increase in machining time.  A 11/16" end mill will require a stepover of .05" while a 1/4" end mill will require a stepover of .03" to keep the scallop height to under .001".

I hadn't bothered optimizing feed rates or cutting speeds.  I arbitrarily selected a 20 ipm feed rate with a resultant calculated machining time of 103 minutes for the machining time for the curved surface.  Given the small width of cut, the deed rate can be significantly increased for a proportional decrease in machining time.  Here is the raw G code.


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## Parlo (Mar 29, 2022)

Blimey, that is a long code, good job you have the software. Without having anything so technical when making bowl moulds we have to manually program the radius path through the centre and copy rotate it around the centre.


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## RJSakowski (Mar 29, 2022)

Parlo said:


> Blimey, that is a long code, good job you have the software. Without having anything so technical when making bowl moulds we have to manually program the radius path through the centre and copy rotate it around the centre.


It is long but the CNC mill doesn't care.  The large number of lines is required because the G2 and G3 cuts have to continuously change the arc length and center when the cut doesn't pass through the center of the mirror.  Other strategies may be more efficient.  I could have mounted the work on my 4th axis and rotated the work as you did which would then require a single arc through the center.  It looks like 700 passes should give the the .001 scallop height.  At 20 ipm, that would be about 400 minutes of run time.

Fusion 360 has a strategy called adaptive clearing that may be more efficient.  I haven't tried the Fusion CAM yet though.


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## RJSakowski (Mar 29, 2022)

One benefit of using a CNC mill is that a true parabolic curve could be cut just as easily as a spherical curve.


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## Mitch Alsup (Mar 29, 2022)

RJSakowski said:


> One benefit of using a CNC mill is that a true parabolic curve could be cut just as easily as a spherical curve.



With his F/ratio, the difference between a sphere and a parabola is on the order of 2-3 waves of light (1 millionth).
I seriously doubt that the CNC is capable of that.


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## WobblyHand (Mar 29, 2022)

Mitch Alsup said:


> With his F/ratio, the difference between a sphere and a parabola is on the order of 2-3 waves of light (1 millionth).
> I seriously doubt that the CNC is capable of that.


Ordinary CNC, no.  CNC controlled by optical (interferometer) feedback, yes.


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## AllenHendey (Mar 29, 2022)

T Bredehoft said:


> You're trying to make a Newtonian Telescope? That's beyond my tool room experience. Wow.
> My 'mentor' ground his 8" the traditional way, 'walking the stump'.


I had the same thought: telescope mirror grinding techniques sound like the place to look.


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## Parlo (Mar 29, 2022)

RJSakowski said:


> It is long but the CNC mill doesn't care.  The large number of lines is required because the G2 and G3 cuts have to continuously change the arc length and center when the cut doesn't pass through the center of the mirror.  Other strategies may be more efficient.  I could have mounted the work on my 4th axis and rotated the work as you did which would then require a single arc through the center.  It looks like 700 passes should give the the .001 scallop height.  At 20 ipm, that would be about 400 minutes of run time.
> 
> Fusion 360 has a strategy called adaptive clearing that may be more efficient.  I haven't tried the Fusion CAM yet though.


Unfortunately I don't have a 4th axis.
Fortunately, the XYZ Protrak program only needs 2 events. 1. The arc through the centre + 2. The amount of times I want it rotated around the centre.


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## RJSakowski (Mar 29, 2022)

Mitch Alsup said:


> With his F/ratio, the difference between a sphere and a parabola is on the order of 2-3 waves of light (1 millionth).
> I seriously doubt that the CNC is capable of that.


That is correct. The difference at the rim is 38 microinches.   A wavelength of visible light is on the order of .5 microns or .20 microinches.  I was looking at expanding the process to much shorter focal length mirrors for something like a solar furnace.

The CNC won't make a specular mirror without additional work but I think it will be more accurate than the other suggestions.  I used to make aluminum mirrors 2" in diameter with 3.75" focal length by turning on the lathe but for a mirror with a focal length of a foot or more, that process is getting unmanageable.


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## homebrewed (Mar 30, 2022)

These days highly precise optics are made using a process called Deterministic Polishing, basically a specialized type of CNC machine.  It's an iterative process where a measurement of the surface is made and regions that need to be corrected are identified.  Those regions are selectively polished using a so-called subaperture polishing machine (that's the CNC part of the deal).  Multiple passes are made, the end result achieving a surface that is accurate to something like lambda/50 or better.  The metrology tools needed for this are highly precise and highly expensive.  

I have wondered if it might be possible to use some of the tools that amateur astronomers use, like the Fizeau Interferometer, to implement a less-exalted version of deterministic polishing.


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## gard (Mar 31, 2022)

RJSakowski said:


> The CNC won't make a specular mirror without additional work but I think it will be more accurate than the other suggestions.  I used to make aluminum mirrors 2" in diameter with 3.75" focal length by turning on the lathe but for a mirror with a focal length of a foot or more, that process is getting unmanageable.


I have thought about various design of ball turners on a lathe and made a couple, all are based on the idea that there is a pivot point (or axis) at the center of rotation of the convex or concave shape. From a practical sense this limits the radius if curvature. I think the method of cutting spherical surfaces on a milling machine could create any radius of curvature


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## gard (Mar 31, 2022)

gard said:


> I have thought about various design of ball turners on a lathe and made a couple, all are based on the idea that there is a pivot point (or axis) at the center of rotation of the convex or concave shape. From a practical sense this limits the radius if curvature. I think the method of cutting spherical surfaces on a milling machine could create any radius of curvature


A search of lathe ball turner with boring head will result in several techniques where the boring head is mounted on a rotating shaft either on the tool post or directly to the compound slide. The shaft is adjusted perpendicular to the lathe axis at the same height. The radius of curvature of the  ball (or concave pocket) is limited by the maximum cutting diameter of the boring head. If the shaft of the boring head was adjusted to be almost parallel to the lathe axis, it could be made to make a spherical surface with a very large radius of curvature. I think the boring head shaft would need to be powered by a separate motor and lathe rotation slowed way down to keep the cutter moving in the right direction relative to the stock.  Perhaps a flycutter or boring head mounted to a tool post grinder if it could be adjusted to the correct angle to the lathe axis.


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## RJSakowski (Mar 31, 2022)

@JimDawson made a ball turner using a boring head about five years ago.  There were some boring heads being sold super cheap on eBay back then and a number of them were turned into ball turners. https://www.hobby-machinist.com/threads/new-ball-turner.61524/


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