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    Re: NG's "Midnight Fun"
    From: George Huxtable
    Date: 2010 Jun 14, 22:02 +0100

    I had written about a distorted image of an object on the celestial sphere-
    "Why must the scale distortion be the same in one direction (largely 
    circumferential, about the centre of the image) as it is radial? Far from 
    And Frank responded- Well, no, not "far from it". That could easily be the 
    No. Frank is wrong. It's a matter of geometry. It's only possible for the 
    radial scale factor and the azimuthal scale factor, at any part of the 
    image, to remain equal to each other if there's no distortion of the image. 
    That happens in a pinhole camera, and is approached with a "rectilinear" 
    lens, optimised to reproduce straight lines on an object-plane as 
    straight-lines on an image plane.
    In any normal camera, there must be complete symmetry about the optic axis. 
    A piece of polar graph paper, placed on a plane and centred on the optic 
    axis, will create a plane image, as another polar graph, also centred on 
    the optic axis. The radial lines will be reproduced exactly. Any distortion 
    shows up in the spacing of the imaged circles. If they are equally spaced, 
    there's no distortion. With any lens system, there will always be a 
    deviation from that perfection. A plot of the "transfer function" of y (the 
    radius of an image circle) against x (the radius of the object cirle that 
    produces it) tells us all there is to know about the distortions of that 
    system. If that plot is a straight line, there's no distortion. If it 
    curves upward at increasing radius, this is pincushion distortion. If it 
    droops, it's barrel distortion.
    At any radius, just as a result of geometry, the radial scale and azimuthal 
    scale are equal only if, in that transfer function y = f (x), dy/y = dx/x. 
    Which implies that the transfer function is a straight line pointing toward 
    the origin. That corresponds to the transfer function of a no-distortion 
    system, in which y = k x.
    There's a useful wikipedia article, on distortion (optics), at 
    http://en.wikipedia.org/wiki/Distortion_(optics) , which shows the effects 
    of different types of distortion.
    What we have been considering is the imaging of a plane object on a flat 
    plane, the sort of job a camera for document copying or aerial survey was 
    spefically designed to do precisely, but a job that any other camera copes 
    with pretty well.
    Next, consider instead portraying an object, such as the Sun, which always 
    occupies a known angle, on to a flat image plane. Even in the absence of 
    any lens distortion (that is, using a pinhole or a perfect rectilinear 
    lens), the transfer function of angle A (measured from the optic axis) to 
    radius y will not be a straight line, but y = k Tan A. This in itself gives 
    rise to gross distortion near the edges of the photo in question, when the 
    angle A becomes 20º. If the camera lens added no distortion of its own 
    (that is if it was a rectilinear design) then that tangent factor would 
    cause radial dimensions , around those outer Sun images 1, to be expanded 
    by a factor of 1.13 to 1 (compared with the centre) and azimuthal 
    dimensions to be enhanced by a very different factor of 1.04 to 1. The 
    image is then very far from being "orthomorphic"; in other words, it fails 
    to preserve the Sun's circular disc, and turns it into an ellipse, pointing 
    radally. That was the situation with optics that are nominally "rectlinear" 
    with a plane object, and any deviation from that ideal will combine with 
    the distortion we have considered, enhancing or reducing it depending on 
    the sign.
    Frank seems to be postulating that some sort of special lens has been used 
    which somehow corrects for that lack of orthomorphism (and yet has failed 
    to linearise the radial scale) when he states- "Trouble is, cameras and 
    other optical systems are designed with various sorts of distortions built 
    in which go around the theoretical argument you've proposed". Does he know 
    if such a system exists, then, which might conceivably rectify that 
    distortion, or is he simply imagining it? Does Frank think it likely that 
    someone taking photos of Arctic sunsets is likely to be emplying a lens 
    system which is intended to create distortion, and to deliberately depart 
    from the rectilinear norm?
    Frank called in aid some comments from another posting, which he ascribed 
    to Mike Burkes but were actually from an earlier contribution by Peter 
    Fogg. As that posting confused distortion with perspective, it had little 
    relevance to the matter in hand.
    This question takes on a bit of importance because postings appear on this 
    list, quite often, from proponents of the use of cameras for making 
    celestial measurements. The geometrical distortions discussed here, that 
    arise from portraying a spherical surface on to a plane array, add serious 
    complications to interpreting measurements, and deriving scale factors; 
    complications which are often neglected. It would indeed be possible to 
    calibrate the transfer function of a lens system and remap images on a 
    corrected grid, though this would call for a lot of work if a zoom lens is 
    being used.
    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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