Re: ATM Spheres, Parabolas etc.

[Aplanatic@aol.com]

Michael Perata asks:

> Given a ROC of 100" how can I plot a parabola and a hyperbola, against a 
> sphere so I see the differences as if looking through a cross section of 
the 
> curves/mirror. I know when dealing with optics, the differences a minisule, 
> but I do have the ability to plot the curves (Autocad) and zoom in to see 
the 
> differences.

Perhaps the easiest method is to use a spreadsheet program like Excel.  The 
general conic formula is given by:

Z = (x^2/R) / ( 1 + sqrt( 1 - alfa*(x^2/R^2) ) )

where R is paraxial radius of curvature of the mirror, Z is the surface 
height at radius x and

alfa = 1+b

where b is the conic deformation constant.

For a sphere b=0, for a paraboloid b = -1, for a hyperboloid b < -1, for a 
prolate ellipsoid -1 < b < 0 and for an oblate ellipsoid b > 0,

To see the difference between two surfaces in Excel, make a column of x 
values ranging from zero to the radius (half diameter) of your mirror.  Make 
two more columns using the above formula but having separate parameters for 
both R, the radius of curvature, and b, the conic constant.  Finally, make a 
fourth column that is the difference in the surface heights.  Plot this 
column against column 1, the zonal radius.  Now, you can adjust the conic 
constants and the RoC of each surface independently and see what happens.  It 
is quite interesting and will give you a good feel for where glass must be 
removed to go from a sphere to some other conic, for example.  Changing the 
RoC of one of the surfaces shows how one can minimize the amount of glass to 
be removed by allowing the RoC of the mirror to change during figuring.

Dave Rowe
Torrance, CA
Medium Format Astrophotography:
http://members.aol.com/aplanatic/photos/astro.html