T3D telescope eyepiece distortion

[john bercovitz <bercov@xxxxxxxxxxx>]

Early in P3D, there was a fellow from IMAX on the list and he kept 
saying that an astro telescope eyepiece's _should_ have pincushion 
distortion; that it was desirable and that's why it was there.  
The reason the question came up was that I had suggested that 
telescope eyepieces would make good stereo viewer lenses*.  Off 
and on I understood what the guy was trying to tell me but it 
always seemed like an elusive, slippery thing.  Well, I'm 
presently reading an article on eyepieces and it's made the topic 
clear yet again so I'm going to write some notes for myself and 
for anyone else who is interested.  It's really easy but for some 
reason I have trouble holding on to it.

OK, let's look at our (stereo) situation first.  Let's use a 
pinhole camera because a real camera is the same thing with some 
complications.  A pinhole camera has an optical axis which is the 
line perpendicular to the film that goes through the center of the 
pinhole.  The height, h, of an image point from the optical axis is 
h = f*tan(theta)
where f is the perpendicular distance from the pinhole to the 
film.  When we view the film, we want to view from distance f and 
we want to use a lens (to help us) which doesn't destroy this 
relationship of f*tan(theta).  In other words, we want a lens 
which doesn't distort the image.  Notice that there is _no_ 
magnification here as there is in a telescope: the magnification 
in orthostereoscopy must be 1 to maintain perspective.

Now let's look at the astro telescope situation.  The sky is 
really a hemisphere as seen from earth.  Any part of the sky, no 
matter how small, is still part of that hemisphere.  And so the 
stars lie at _angles_ from the observer's position.  The 
telescope's objective forms a small image of a very small part of 
the sky for the eyepiece to magnify.  The objective also works 
like a pinhole but since the image is so small compared to the 
focal length of the 'scope, you would be hard pressed to tell if 
h = f*tan(theta) or h = f(theta) where theta is in radians.  Now 
here's the kicker: when the eyepiece magnifies the image, it would 
be nice if the image showed proper (but enlarged) angular 
relationships between the stars.  In other words, what came into 
the objective as theta, ought to come out of the telescope as 
M*theta where M is the magnification.  So now the eyepiece needs 
to gives us h = f*(theta), not h = f*tan(theta).  h = f*(theta) is 
pincushion distortion.

Make sense?  If not, please let me know how I can improve my notes 
because when I read them again, I surely won't know what I meant.  8-)

*I had just seem a Celestron 50 mm eyepiece which didn't have 
distortion but it cost $250 each.

Thanks,
John B


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End of TECH-3D Digest 264
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