Here the action is concentrated when the mirror is near the edge of the tool. In that situation, the weight of the mirror is resting on its center, so the mirror gets worn away more there. Both the mirror and tool are rotated frequently so the action is evenly distributed (in terms of the angle, *but not the distance* from the center). Strokes that hollow out the center rather than lowering ("reducing") the edge seem to be more commonly used, probably because of the danger of getting a turned edge. This is one of the easier errors to get and one of the more difficult to cure. Using figuring strokes represents much of the *art *of figuring. Besides getting the right amount of effect on the right part of the mirror, the surface heights of different parts of the mirror need to be blended smoothly from one part to the next. Many variables can affect the efficiency of a given polishing stroke, such as the temperature, the hardness of the pitch lap, and the amount of water mixed into the polishing rouge. And before you can even start on parabolizing, you generally want to have the surface very nearly spherical, so other specialized strokes besides parabolizing strokes may be needed to make small corrections. Or sometimes it's better to just to use a steady standard polishing stroke to gradually reduce errors down to a spherical surface. Then when parabolizing you may get "nearly parabolic" with a small error somewhere, in which case you need to be able to get rid of that small error without interfering with the rest of the surface. And if that doesn't work and you get a distorted surface you need to know when to give up on parabolizing and temporarily go back to spherical. I (JM) was constantly amazed at how my inexperienced attemps would give unpredictable results, even giving different results if I tried the same thing twice, while WVAS' unofficial master optician Jim Sattler semed to get pretty close to the results he wanted most times.

Finally we can take a look at a nearly completed mirror...

Finally we can take a look at a nearly completed mirror...

Figuring is the art/science of fine-tuning the shape of a polished optical surface. After grinding and basic polishing, the surface is usually nearly spherical, and depending on what kind of optical design you're making, you will want it to be either spherical (but to probably a higher degree of accuracy than it already is) or paraboloidal, ellipsoidal, or hyperboloidal (say that three times fast) or sometimes some other shape. The difference from the starting near-sphere to one of the other shapes is usually very slight, perhaps one light-wavelength or less (a half micron or about 2 hundred-thousandths of an inch) for typical amateur scope dimensions, with a preferred accuracy of 1/8 wavelength or better (2.5 millionths of an inch!). In this case (Newtonian reflector) we need a paraboloid.

The way in which a paraboloid varies from a sphere depends on the radius of the sphere you compare it to. Alternatively, if we start with the sphere obtained after grinding and polishing, the variation depends on what type of parabola we want. Simplified to 2 dimensions (with the vertex at the origin), a parabola has the equation y=*a**x^2. Changing the value of *a* gives different parabolas. If we look for a paraboloid that matches our 8" f/5.5 sphere at the center and edge, the parabola will be higher than (a hypothetically perfect) sphere by up to 12 millionths of an inch in the intermediate zones. This means (in a sort of reversed sense) that we could obtain the parabola by polishing heavier in the center and edge of the sphere to remove (or at least redistribute) some of the glass there. But there are other options. Using a slightly different value of *a *we could have a parabola that matches the sphere at the center but gradually drops off as we move towards the edge (equivalently the parabola matches at the edge but rises in the center) . Alternatively, yet another value of *a* gives a parabola that matches the sphere in the center but rises as we go toward the edge (or equivalently matches at the edge but drops in the center).

So to produce a parabola we need to alter our polishing method to find a way to lower the surface of the sphere at the edge, or at the center, or both. This is done using polishing strokes that concentrate the pressure and action in one part or another of the mirror. Here is an example (adapted from Texereau's classic*How to Make a Telescope*, copyright Interscience Publishers, Inc. 1957).

The way in which a paraboloid varies from a sphere depends on the radius of the sphere you compare it to. Alternatively, if we start with the sphere obtained after grinding and polishing, the variation depends on what type of parabola we want. Simplified to 2 dimensions (with the vertex at the origin), a parabola has the equation y=

So to produce a parabola we need to alter our polishing method to find a way to lower the surface of the sphere at the edge, or at the center, or both. This is done using polishing strokes that concentrate the pressure and action in one part or another of the mirror. Here is an example (adapted from Texereau's classic