Testing telescope mirrors

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.

Knife edge testing

Analysis of the surface profile is done by means of a testing device, which can be made with fairly basic tools. The easiest is the knife edge test, invented by Leon Foucault (aka Foucault test). A point or slit shaped lightsource at radius of curvature (twice the focal length) is reflected by the mirror under test. The image of the lightsource is then analyzed by means of a knife edge.
What you need is a way to move the source and/or knife edge and measure its relative position in axial and preferably also in sideways direction. Then you need a proper source and a straight edge for analysis and a mirror stand which allows tilt adjustment. There are many sources on the web describing such devices. I made an X-Y stage from two old floppy drives: A Floppy Foucault Tester.
An exhaustive description and tutorial on the Foucault test is written by David Harbour: Understanding Foucault.
Alternatives similar to the KE test are for example a slit test, wire test and a Ronchi test.

Interferometric testing

A second method which is fairly easy to implement by amateurs is a Bath common-path interferometer. The basic idea is to split a laser beam, use one to illuminate the mirror and the other as a reference beam. The reference beam is also reflected from the mirror, and takes approximately the same route as the illuminating beam. Upon return both are mixed and from the interference pattern the surface shape can be deduced. There are two good information sources for starting with this way of testing: a very informative wiki and the yahoo interferometry group.
The coherence length of the laser source is not very critical, so cheap "key chain" laser pointer can be used. Further components are a cube beamsplitter, a flat first surface mirror and a small short focus lens. All is mounted on a X-Y-Z stage similar to the one used for knife edge testing.
I found a couple of micrometers on E-Bay, and built this Bath tester.

Test platform

Finally, I have upgraded the floppy drive tester base with a set of professional translator stages. This is now my generic test platform, used to carry all sorts of test heads.