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George, you wrote:

> Two alternative fitted equations are shown, one linear, one second-order,
> with little to choose between them, but it looks to me as though only one
> line-of-fit has been drawn, the second-order fit.

My old version of Excel can only show the equation for the fit if the
fitted line is also drawn. This figure shows therefore both lines but
they are so close to each other that they can't be distinguished.

further on you tried the following:

> I have used that graph to check out my suggested model, in which radius in
> the array varies as the tan of the subtended angle from the centre point.
> In this case, because each measurement has been equally disposed about that
> centre point (a sensible practice which minimises distortion) the relevant
> angle is half the altitude.
>
> So I have tried a function as follows-
>
> H(pixels) = 18140 Tan (altitude / 2) (in degrees)
>
> or, for those that prefer it the other way,
>
> Altitude in degrees = 2 arc-tan ( pixels / 18140)
>
> And I can report that that function fits exactly on top of the data and the
> fit-line that's already drawn in, so that it's just not possible to see any
> difference.

Yes, the tangent function correlates in this case better than
polynomials. I fitted your suggested function the following way to the
original data:

Altitude in degrees = 2 C arc-tan ( pixels / D)

Setting first C=1 (constant) results in a best fit for
D = 18161.80 (compared to what you estimated from the figure 18140).
The resulting standard deviation is only marginally greater than the
one obtained for the linear fit having two parameters (0.56 moa vs
0.52 moa).

Fitting then the parameters C and D results in
C = 1.116192
D = 20308.33
The standard deviation is about the same than the one obtained for the
fitted second order polynomial having three parameters (0.22 moa).

For his 50mm lens Greg could therefore also use instead of the two
equations already provided the following ones:

either
Altitude in moa = 120 arc-tan ( pixels / 18161.80)
or, more precise
Altitude in moa = 133.94 arc-tan ( pixels / 20308.33)

Marcel

   

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