Toshimi Taki's work appeared in S&T april 96.

The calculations in this website are based on the table in that S&T on p76.

With the aid of the least square method the results can be approximately captured in the
following equation

W= X*a^2 + Y* a^4/t^2

The reason for this form of equation is partly due to Couder, namely the last term a^4/t^2 and is confirmed by Toshimi, but the first term is due to the following : in his article in S&T Toshimi is pointing out that shear effects are not negligible, certainly not when the distance between two supporting points gets relatively big, the shear effect has a a^2 effect on the sag, but Toshimi is also using refocusing, effectively determinating a new best fitting paraboloïd to the sagged surface, so the difference between two paraboloïds can also be written in the form of a^2. This should justificate the choice of the equation.

As this equation is not exact, the values in the table, nor the examples in the article can not be exactly duplicated. Do not forget that the FEM-method Toshimi used to obtain his results is also a method that is not exactly !! The lambda values do give an idea of the deformations and are used to determinate on how many points the mirror should be and that's all !

So far as I know, the work of Toshimi is both the most complete and accessible for amateurs as you and I, so till someone is doing better, this should do the job with sufficient accuracy. Recently David Lewis has done better, so take a look at his work. His results will be used for this site in a couple of weeks !

Remember not so long ago, everyone was using rules of thumb as thickness should be 1/6th of the diameter. Now it's possible for everyone simply calculating his own particular system. You don't even have to be on -line, just save the html-file on a local disk, open your browser and open it, you can even transport it on a disk and give it to others. This program is free to use for everyone, but should not be changed and it would be appreciated that someone using it mentions it in whatever publication !

Thanks Toshimi !