The r squared / 2R formula can be a bit confusing. Return to calling page

In the old days, say before 1960 most people used a tester where the light source remained stationary and only the knife edge moved.  Now days most people use a tester where the light source, usually an LED, and the knife edge or Ronchi grating move together on the same carriage.  If you don't know which type of tester is being used in a description in a telescope making article then the wrong math will lead to a two to one error in calculations.

Confuse this with the 2 to 1 ratio from center to edge vs. center to 70 percent zone and there can be more 2 to 1 confusion.

Below I have collected four different values of difference between center ROC and the point on the mirror's center axis where the ray from the mirror's edge crosses the axis.
The results of calculation being for a tester where the  light source moves .
The test mirror is a 12.5 inch mirror with 165 inch focal length.

.118371 inch Classic r squared / 2R
.118455 inch 'Sky and Telescope' article
.118371 inch Derivation using trigonometry
.118415 inch Approximation derived from the conic constant formula

Classic r squared / 2R

This simple calculation is the one to use. 
The tester accuracy is most likely +/- .005 inch at best.  

 

'Sky and Telescope' article

The r sq / R formula is not exact. The exact formula is given in an article in Sky and Telescope, November 1992, page 574.    'The exact formula is:

 d  =  r2 / R + (  r4 / 2 * R3  )

" .5 * ( r ^ 2 / R + (r ^4 / 2 * R ^3 ) ) "  Appears as at left in the Java Applet parameters.

In the case of many amateur mirrors the second term is very small. Consider a 200 millimeter ( 8 inch ) f6 mirror. Its focal length is 1,200 mm, so R is 2,400 mm, and at the extreme rim of this mirror r = 100 mm. In this case the value of the second term is .004 mm.'

So when is this formula exact?  Apparently it is exact when using a fixed light source and compensates for the existence of unequal distance from the mirror of the knife edge and the light source.

 

Derivation using trigonometry

r_sq_2R_formula.gif (8080 bytes)This seems to be an exact calculation if the light source and knife edge are at the same place as in a slit-less tester.   The presumptions are that the angle of reflection on a parabola is equal to the angle of incidence.  That the light ray from space to the mirror edge is parallel to the axis of the center of the parabola.  That the derivation of the parabolic formula is correct.

" S + ( TAN( .5 * ( IVT( r / (( R / 2 ) - S ) ) ) ) ) "  Appears as at left in the Java Applet parameters.

The the height of the point on the mirror edge where the light ray strikes the mirror is described by  in the parabolic formula.  

 

 

coniceq3.gif (7444 bytes) Approximation derived from the conic constant formula

" ( R * ( ( 1 - A * r ^ 2 / R ^ 2 )^1.5 ) - R ) / 3 " 
Appears as above in the Java Applet parameters.

The basis-formula gives the distance from a point on the mirror surface to a point on the caustic curve.  I discovered that by subtracting R from the base formula and dividing the result by 3 a good approximation of r sq / 2R can be found if the 'conic constant' in the formula is set to minus 1 as for a parabola.
This is useful because the conic constant can be changed to generate a value for sphere, ellipse or hyperbola.
Unfortunately I don't have a mathematical proof as to why it works.

 

A Surface Formula for Conic Sections

And as an aside if we solve the surface formula for conic curves for S r using the 'quadratic equation' solution from a high school or college algebra book then we get the equation below which is undefined when the 'conic constant' = -1.  

"((-2 * R)-SQR(((R*2)^2) - 4*r*r * ( 1 + B )))/2 * (1+B)"
Appears as above in the Java Applet parameters.

The above formula will run in the Ronchi and Surf program's formula parser with 'conic constant'  set to -1 because the formula parser will sense a divide by zero and substitute a small number. 
This avoids a divide by zero which the computer cannot calculate.  With the 'conic constant' set slightly off of -1 the parser will produce the correct result although with some noise due to the limitations of the computer's floating point number processor.
The formulas for conic sections are derived from the formula for an ellipse in 'Astronomical Optics' by Schroeder.

 

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