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Through the Looking Glass

Formula for Correction of Angles

I can look at far-away objects like the moon and the horizon through my crystal ball, but if I'm not looking directly though the center of the ball, then the positions that I see don't represent the true anglular distances between objects.  They are distorted a little bit.  Out of curiosity I decided to calculate what the difference is, so I've examined the case where one object is directly in the center of the view and another object is at an angle.  Here are my results.

The goal of these calculations:  to calculate the true angles between objects seen in the sky, such as two planets, by first marking the angle on the ball (with a wax pencil or dry-erase marker) and later correcting the result with the forumla.  To measure the angles marked on the ball, we can roll the ball along a ruler and scale by the circumference of the ball.

To simplify the calculations, I found a polynomial that approximates the correction for a particular ball and it is no worse than ten minutes from the exact formula.  The polynomial is the cubic  f(x)=76x(x-76)(x+62)/1000000 where x is given in degrees (between 0 and 80 degrees for best results).  That is assuming that the radius of the ball is 4 cm, the distance from the eye is about 30 cm and the index of refraction is 1.5.

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