To grind a mirror is relatively easy, except for the final stages of figuring. The whole grinding and polishing process the mirror and tool have a tendency to converge to a spherical shape. After all, this is the only shape where full contact is possible while the surfaces are moved. High or low regions will wear faster and slower respectively, until the sphere is attained. The stroke that will get you there is the infamous standard stroke; approximately 1/3 diameter stroke length with only a little sideways movement.
During figuring however, the strokes are targeted to deliberately deviate from the spherical equilibrium. The purpose is to aspherize the mirror so that it gives the best image in its designed system. For a simple Newton telescope the asphere must be a paraboloid in order to get the best (on axis) image of a distant object.
The Floppy Foucault tester|
Knife-edge or Foucault testing has long been the ATM way of analyzing mirror surface. It gives a good qualitative view on the overall surface quality in a single shot. For quantitative analysis at least the displacement in paraxial direction needs to be accurately measured.
This page describes a tester platform based on the slide bearings of a 5.25" floppy drive.
For the definitive treatise on Foucault testing, buy Understanding Foucault by David Harbour.
Any ATM has access to a more accurate and less subjective way of testing: interferometry. The Bath interferometer is relatively easy to build, similar to a Foucault tester, and the open source DFT-Fringe software package (by Dale Eason) gives a rich toolbox for igram processing and analysis.
Also have a look at the Interferometry group, featuring a very informative wiki and for further discussion a mailing list.
This page describes construction and use of a Bath interferometer.