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).
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...