Thursday, August 7, 2008

The World Wide Web: So Many Answers, Some Right

The World Wide Web: So Many Answers, Some Right

I needed the formula for calculating the sagitta of a spherical mirror (which is the depth of the hole you have to gouge out to make a flat mirror into a spherical mirror). Go ahead: search for the formula. There many different formulas out there--some of them right, some of them wrong.

This one is correct:
Now you are almost ready to start grinding. Before you start you need to figure out how deep the hole is that we are going to need. The formula for this is pretty simple. You need to know the diameter of the mirror (D) and the Focal Length (F) that you want to make and you get the Sagitta (S) which is the depth of the hole that you need to carve into the glass. The formal formula is:

S = 2F - sqrt( (2F)2 - (D/2 )2)

The second way (an approximation) to calculate the sagitta with this formula which is probably a lot easier to calculate:

S = (D/2)2 / 4*F

"Sqrt" is the square root of the number inside of the brackets. Sagitta is how deep the curve of the mirror is going to be in the center of the glass. The more accurately you calculate and measure this dimension, the closer you will be to the Focal Length you want when you get done. Either formula should provide the same basic answer unless you're doing a fast mirror.
This one has the right answers in the example, but definitely the wrong formula:

SAGITTA This is the depth of the curvature of the primary mirror, measured at the center. The deeper the curve, the shorter the radius of curvature, and of course the focal length. The formula for sagitta is:

sagitta = R - Square root(R2 - d2/4)

R = Radius of Curvature
d = diameter of mirror

The following table ties it all together with some examples:

Mirror diameter F ratio Radius of Curvature Focal Length Sagitta
8" 7 112" 56" .071"
10" 7 140" 70" .089"
12.5" 6 150" 75" .130"
16" 5.5 176" 88" .182"
The World Wide Web is an amazing resource--but only slightly more authoritatively accurate than watching television news.

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