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Milling – how to make – spherical surfaces

      If the workpiece is mounted on a rotary table and the head of the milling machine is tilted it is possible to machine various concave and convex surfaces using a boring head. It would be possible to do this using a fly cutter but it is easier to set up with a boring head.

      In all of these cases it is essential that the axis of the rotary table is in the same plane as that of the spindle in the x/z plane. The axis of the sphere that the surface being cut is part of will be coaxial with the axis of the rotary table the movement of the cutter to the workpiece changes because the milling table is raised.

Cutting a convex surfaces

      This method can be used to make any spherical surface up to a whole hemisphere. It is possible to mill even more of a sphere but by a different method which is covered later.

      The geometry of a typical setup for a convex surface can be seen in fig. xxx.

Fig.  geometry for machining a convex surface 1003


      radius of the surface  = r

      radius of sphere = R

      tilt of the vertical head = ?

      diameter of the fly cutter =d

      height of the surface =x


       sin ? =(d/2)/R

       sin ? = x/D

If the axis of the boring head is co-planar with the axis of the rotating table then, if the boring head rotates in a circle each point on this circle is the same distance from the intersection of the axis of the boring head and the rotating table regardless of the rotation of the rotating table.

      The surface from A through CB to C is part of a sphere whose radius is R.

      The diameter of the sphere is solely determined by the diameter of the cutting circle, that is, the diameter of the boring head and the distance from the top (or bottom) of the cutting circle and the point where the axis if the boring head and the axis of the rotary table. This is determined by the angle of tilt of the vertical head.


8 milling a convex surface 1

8 milling a convex surface 1

 machining a convex surface – 8

      A key feature of this is that the boring head is cutting the outside of a circle on each revolution of the head. The cutting edge is on the outside of the circle. Using the conventional boring tool this means the tool is round the “wrong” way. This will only cut with the boring head rotating in reverse. There is, of course, only one cutting edge but it is shown in both the top and bottom position.

      If the tip of the cutter in the top position is not in line with the axis of the rotary table the surface produced is not part of a toroid as one might expect.

572 finished convex surface

572 finished convex surface

Fig.     finished surface 572

      An example of a part of a hemisphere like this would be a smoke box door. This is part of a sphere. How this works is easily seen from the drawing.

It is worth turning (on the lathe) the workpiece so it is the right diameter and thickness. It is probably not worth trying to remove any more metal, for example, on the lathe.

      It is not usually possible to clamp the workpiece directly to the rotary table. In the example shown the workpiece has some holes drilled and tapped into the back of it. These are used to mount the workpiece on a plate which is then clamped to the rotary table.

      It might be necessary to have some sort of spacer between the work piece and the plate that is fitted to it and bolted to the milling table.

Milling a hemisphere

      An example of a whole hemisphere would be a dome covering a regulator on a boiler. This is really just a special case of milling a convex surface.

Fig.449 – geometry for milling a hemisphere – 1004

      If the height of the surface equals the radius, i.e., X = R, then the surface is a hemisphere. In this case ? will be 45º.

695 cutting a hemisphere

695 cutting a hemisphere

Fig.  milling a hemisphere 695


      The workpiece is centered on the rotary table

      The head is tilted to 45°.

      In this case it is not obvious where the cutter is relative to the axis of rotation of the workpiece. But it is possible for the workpiece to be moved towards the boring head till enough is cut away to see when the cutter is close to the axis of rotation.

Milling a convex surface – alignment

      The boring head is set to the diameter D.

      see     setting the outside diameter of a boring head

      The rotary table is mounted on the milling table. It is aligned with the axis of the vertical head.

      see   centering a rotary table

      The x and y-axes are locked.

      The plate with the workpiece is mounted on the rotary table. The workpiece is aligned with the spindle using a DTI on an arm device.

      see    centering a round shape

      The vertical head is tilted to the required angle.

      The x axis is unlocked.

      The milling table is moved along the x axis till the tip of the boring head at its highest point is over the middle of the workpiece.

      The x axis is locked.

      The spindle is turned on. The workpiece is raised.   The cutter will first touch the workpiece at a point Z at one place on the edge of the workpiece. But if the workpiece is raised a small amount and then rotated a thin ring will be cut all the way round the edge.

      After each full turn of the workpiece it is raised again and the ring gets wider.

      Finally the width of the ring covers the whole radius of the workpiece.

      It will be noticed that the diameter of the boring head was significantly larger than the diameter necessary but it does show the nature of successive cuts.

      All of the cutting is done with the x and y movements locked. The workpiece is raised till it is being cut. While this is happening the workpiece is rotated till is has done a complete circle.

      The cut will form a ring at the edge of the workpiece. Each time the workpiece is raised this ring will get wider. The cutting is complete when the cutter finally cuts to the middle of the workpiece.

103 Odd case

      An odd case occurs if the diameter of the boring head passes over the axis of the rotary table.

      In this case what is cut is still part of a sphere but it is the surface between heights A and B. This is easy to see if one considers that height A is the highest the cutter ever reaches and B is the lowest.

Fig. Odd case – 1057

      It might seem that if the vertical head is less tilted the shaped formed would be the surface round from A to B. But it is easy to see that if the head is vertical it does not cut the surface round from A to B.

      What happens is that the surface cut is the surface between the vertical height of A and the vertical height of B.

Cutting a concave surface

      This is in many respects similar to the concave case. It is set to the pointer touches at A and B. The difference is that the position of the cutter is used it starts from position A dash and b dash

Fig  milling a concave surface – geometry

632 machining a concave surface

632 machining a concave surface

Fig. Milling a concave surface 632

  1. Jim Marsh said:

    I used to do ball milling on a vertical mill using a dividing head. I was trying to tell on old friend about this on the phone and he nearly understood!

    I also used a rotary table to produce hemispheres using a similar technique.

    I seem to have lost my notes/diagrams, it has been 30-35 years since I was involved.

    It has taken a lot of searching to find you site.

    I am continuing to search to see what else I can find.

    More power to your elbow.

    Kind regards

    Jim Marsh
    P.S. I am in Surrey England.

    • johnf said:

      I think you will find on this website how to make fairly accurate balls using a dividing head or an accurate hemisphere or less of a true sphere.
      I think if i wanted to make a complete or nearly complete sphere I would do it on a lathe though it is not as easy as one might imagine since with just one cutting tool the amount of the surface that can be made is very limited.
      I live in london w8 but my workshop is near thame in oxon. I am currently trying to finish my 7 1/4 gauge Jenny Lind.
      I have lots of unpublished pages but I have found wordpress totally inadequate for managing my pictures I am a bit discouraged.
      I was thinking of turning my stuff on capstan techniques into an e-book
      john f

  2. Jim Marsh said:

    Thanks for your comments John. My involvement is historic in as much that I used the technique to teach\show my apprentices what could be done without resorting to an expensive form tool for the lathe. We use to produce a 7\8″ spherical end on a threaded steel shaft for a small vice. Accuracy was important as we had to satisfy EITB criteria. I am no longer involved in manufacturing, but still have an interest. Sorry to hear of your discouragement and I wish you luck with you e-book. I have only recently learned how to send text to my kindle!!!

    Kind regards
    Jim M

  3. Cibin kurian Mathew said:

    Hello it’s cibin Mathew from india
    Actually I want to makr
    Hemisphere in cnc milling
    Plz tell me the formula

    • johnf said:

      I have written about this somewhere on this website. It is not so much a formula as a way of setting it up
      correctly. I have no idea of how to do this on a cnc machine.


      • James Marsh said:

        Only done this on manual vertical milling machine. At least 25 years. Don’t remember much!

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