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    Re: NG's "Midnight Fun"
    From: George Huxtable
    Date: 2010 Jun 11, 12:53 +0100

    I think there's been some confusion about these midnight Sun pictures.
    
    We're not talking about a particularly wide angle here. The Sun images are
    5 degrees apart (20 minute spacing) so the picture (whether cropped from
    the original or not) spans 50 degrees
    
    It's not necessary for there to be "distortion", as such, for the image of
    an object subtending a constant angle, such as the Sun, to vary in size
    when projected on to a flat plane, such as a photographic plate.
    
    A perfect lens, presenting no distortion whatsoever, will project the image
    of a rectangular object filling the field of view (such as a large sheet of
    graph paper or a squared-up office block) on to a rectangular film as an
    exactly similar rectangular image. A pinhole acts as such a perfect lens; a
    small thin lens will approximate it.
    
    A distorting loptical system would show either "pincushion", or "barrel"
    distortion, depending in the way that it reshapes such a rectangle, but we
    can concentrate on a non-distorting pinhole.
    
    The question is simply one of mapping. A pinhole maps such a flat-plane
    object undistorted on to a flat-plane film, but cannot do so with the
    spherical sky. It's just the same problem as mapping the polar regions of
    the Earth.
    
    If we put a spot on the film at the centre of symmetry of the optical
    system, then the radius of a point from that spot is not proportional to
    the angle A, subtended at the lens, but to tan A. And that implies an
    expansion, of radial lines, by a factor of 1/cos-squared A. So at the outer
    Sun images of that photo, radial lines will be magnified by 22%, compared
    with the centre of the picture. Circles around that centre spot must always
    correspond to 360-degrees, and to do that, all circumferential lines must
    be magnified by a different factor, Tan A /A (where A is in radians), which
    at that same radius works out as 7%.
    
    So towards the edge, just as an Earth-map of the polar regions distorts, as
    you get away from the Pole, images of the Sun, and any other object or
    consellation in the sky, are distorted, both in size and in shape. It's
    clear that the gaps between the Sun images, nominally 5 degrees, are
    increasing as they depart from the centre region (and in our case there may
    well be additional distortion, due the the individual design of the actual
    optics being used, which will in general, have been optimised for other
    properties, not for minimum distortion).
    
    This is one of the difficulties that I pointed to, sometime ago, when this
    list was discussing the use of digital cameras for extracting celestial
    information, and the observed diameter of the Sun was being proposed as a
    yardstick for angular calibration.
    
    As long as the optical centre point is known, and the function linking
    radius and angle of the sensor / lens combination has been specified, then
    remapping can always be done by software. Another possibility (but not for
    the likes of us) would be for a CCD sensing array to be produced on the
    inside of a sphere rather than a flat plane; I wonder if that's been tried?
    It might allow interesting new geometries.
    
    ========================
    
    Even after all this, the enhancement of the dimensions of the outer Suns on
    that picture seems even greater than the simple pinhole model would
    predict. Greg Rudzinsky suggests that it may be the result of increased
    distortion from a wide-angle lens, but I think the following reason is more
    likely-
    
    Sun images on any photo are ALWAYS vastly overexposed. I will believe that
    a Sun image is not overexposed, when someone shows me sunspots on such an
    image: so far, nobody has. The Sun disc edge-contour extends outwards to
    the radius where the overexposure ends. Any haze surrounding the Sun
    diffuses the light such as to increase the radius of that contour. That
    seems to have happened in the photograph, with increasing haze as the
    morning progressed, particularly in the last picture taken at 1:20 am.
    
    This was another of the problems with using Sun diameter for angular
    calibration that I alluded to in our earlier discussions about digital
    cameras.
    
    George.
    
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    
    
    

       
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