While I wait for the missing part for my XL, I have been trying to set up projects for that day (sometime this decade?) that the assembly will be completed.
I have 100, 1.0 mm X 50.8 mm brass disks that I would like to engrave…but, there is a problem. The disks have been punched out of sheet brass and all of them are slightly domed to an average of about .35mm (±0.01 mm) from the edge to the center. The program that I am working with allows for carving on a dome, but I need to give it a spherical diameter. My trigonometry classes were almost 50 years ago and the old adage “if you don’t use it, you lose it.” is in play. I need a spherical diameter to plug into the program.
One of you [old man smacking gums sound] young whippersnappers [/omsgs] out there should still have it all fresh in you mind…yes?
Basically, you want the spherometer formula, which can be found with explanation at https://en.wikipedia.org/wiki/Spherometer
The version that is easiest to use without having the instrument is the circle-cup form, as that can be used with three points in line: The distance between the outer is equivalent to cup diameter, D, and the height of the center above these points is sagittal, h, measured at the center. The equation is: R=(h/2)+(D^2/8h).
I directly measure using a lens clock (https://en.wikipedia.org/wiki/Lens_clock), since I have one that also give radius of curvature and have never picked up or made a spherometer, but I have a wide variety of tools for odd purposes that most people won’t have lying around. If you do a bit of algebra, you will see that the lens clock equations for refraction and radius of curvature combine to give the above (unless I made a typo)
@enl_public exactly what the doctor ordered!
Hic dominus toto pendet. Dum Spiro, Spero.
@enl_public it works!
Too sweet! With this spreadsheet, domes will be no match for me! Thanks again.
Lazy me would have just measured the diameter, the height difference, and then modeled it in CAD…don’t bring math into this that I would have to do.
The problem is that what I was using demanded a diameter of the sphere. My object has a minuscule rise and the diameter was relatively large. So…I just built a spreadsheet that figures out the spherical diameter for any object by inputting object diameter and rise. Voilà!
Honestly, lazy me would and has done that, despite having the proper measuring tools and knowing the math. But, not everyone has good CAD available and knows how to use it, and, in some cases, the measurement can be rather a pain.
I have a few times needed to make several measures in different places and orientations to properly model a surface, and, if CMM or appropriate clearance for direct measure of offsets isn’t available, CAD won’t help. Gotta math it the old fashioned way. I actually learned this measuring railroad curves using chord-and-offset, but is has been useful for a number of other things, like corrections to a large reflector that couldn’t be removed from its mount and couldn’t be measured optically (should have been paraboloid in theory, but was shallow enough that is was nominally a spherical section, until damaged)