Don’t be a Square; Be a Sphere!

It’s all well and good to say I’m going to grind a perfect sphere into the surface of my Pyrex disc and measure how deep it is, but how do I actually know if it’s a sphere? Luckily, a sphere is the natural consequence of rubbing together two flat surfaces, but there are some ways I can check. Just in case.

OLYMPUS DIGITAL CAMERA
I guess it looks like a sphere?

Grinding will not only shape the front of the mirror into a sphere, but also the front of the tool. The only difference between the two is that the mirror will be concave (lower in the middle) and the tool will be convex (lower at the edges). More material is ground away where there is contact between the Pyrex and the tool, and this is why I started grinding with the Pyrex hanging over the edge of the tool. By overhanging there is more contact between the centre of the mirror and the edges of the tile tool, and this means more material is removed from these areas. I then gradually reduced the overhang until the mirror was directly on top of the tool. The rest pretty much sorts itself out as low points on the Pyrex and the tool are not ground away, and high points are ground away first. The eventual result is that all the high and low points are taken off both the mirror and the tool until they are both spheres that perfectly match each other.

GrindSphere1

By grinding away with the Pyrex and tool at different orientations, the only shape that will have perfect contact between the two is a sphere. Pretty much a spherical peg into a spherical hole. So to test that I have a sphere I just have to check that I have good contact, and this is quite simple. It’s called the ‘Sharpie Test’ for very unimaginative reasons, and to do it I drew a grid on the surface of my mirror with a sharpie and then did a few minutes of grinding. If the surfaces are in good contact then the sharpie should be worn away evenly across the whole mirror. If the sharpie is worn away unevenly then it means some areas (low points) are not making contact while other areas (high points) are being ground away. If the sharpie is worn away unevenly then it just means grinding must continue until the high points are all ground down and you can get a good result from the sharpie test.

SharpieTest
Two attempts at the ‘Sharpie Test’ to see if I have a sphere. A bad result (left) where the edge and middle have not been worn away, and a good result (right) where the whole surface has been worn away evenly.

So now that I’m confident that I have ground a sphere into the surface of my mirror another question might be: how big is that sphere? Up to this point all my diagrams show very exaggerated curves, but the truth is that the curve is so slight that it’s hard to see at all. The deeper the depth of the curve (also called sagitta), the smaller the sphere that it describes. So the curve depth of 2.45mm in my mirror is part of a sphere with a radius of 2,359.64mm! A schematic sketch shows how all the measurements are related to each other, but a diagram to scale shows how massive the size difference is between mirror depth and sphere.

SphereSagitta

Finally, I’m confident that I’ve got a perfect sphere of the right depth in my Pyrex. Getting to this point has removed almost all the material that I need to, and has been pretty quick. The next stages will take only a tiny fraction more Pyrex, but will take much longer as I finely grind and polish the surface to a ridiculous smoothness.

SagittaTester
Measuring the depth of the curve across the whole mirror.

 

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