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    Re: Px vs. MOA/Px Graph for 50mm Prime Lens
    From: George Huxtable
    Date: 2010 Jun 27, 12:06 +0100

    I've been looking at Greg's graph of slope of the calibration curve of his 
    camera's image, in terms of arc-minutes per pixel, as the angular span 
    varies between two measurement points. Presumably these points are horizon 
    and a Sun limb, equally disposed about the centre of the array.
    
    Greg hasn't explained how, or how well, he gets that span to be symmetrical 
    about the centre; perhaps simply by eye-estimation as the shot is being 
    taken.. It's clearly a good thing to do so, to minimise distortion. 
    However, by understanding the shape of the calibration curve, it may be 
    possible to correct for the effects of any asymmetry, after-the-event, just 
    from the two pixel readings of limb and horizon, if they happen to be 
    different distances from the array centre. Then, that could allow both 
    limbs of the Sun to be used, in a single shot.
    
    One thing we need to know is the overall size of the array, in pixels, in 
    that vertical direction, which looks as though it will be somewhere near 
    4000. Then we can know the pixel-number corresponding to the array-centre. 
    Can we take it that each picture has been taken along the centre-line of 
    the frame, in the horizontal direction? That can be checked from the 
    pixel-count in that horizontal direction, if we know haw many pixels the 
    array spans along that axis also. I don't expect it to be very sensitive to 
    deviations from that centrality.
    
    The calibration curve of pixels versus angle from the centre, will be 
    somewhere near a straight line passing through zero, but will have a bit of 
    an S-shape to it, depending on which way up it's plotted. Greg appears to 
    have plotted, not the actual slope of that line, but (in effect) the slope 
    of a "chord", a straight line drawn from the centre to each individual 
    point. This could be fitted by a polynomial curve, with its origin at the 
    array centre, and that seems to be the basis on which his Greg's graph has 
    been plotted. If the calibration was exactly linear, the slope would be 
    constant. If quadratic, the slope would change linearly with the angle from 
    the array-centre, which very nearly describes what his plot shows. But not 
    quite; it looks to me that a significantly better fit to his data points 
    would be found with a slightly curved line, convex-up according to the way 
    his graph has been plotted. So higher-order terms might usefully provide a 
    better fit. That's one reason why I suggest fitting, instead, with a 
    different function, that of tan(angle from the centre), or, in accord with 
    the way Greg measures, 2 tan(half-angular-span). There must be a 
    multiplier constant, which needs to be fitted from the observations: it 
    depends on the array spacing and the focal length, so ideally, should not 
    then change.
    
    The other reason for fitting a tan function is that it's exactly how you 
    would expect the calibration with angle to vary, if the lens itself does 
    not distort planar images: that implies that it's a "rectilinear" lens. Any 
    additional distortion at wide angles, from the lens itself, will add to 
    that geometric tangent effect.
    
    It could well be instructive for us if Greg provided, not those deduced 
    slopes, but the actual values of corrected altitude and the pairs of pixel 
    numbers that correspond to top and bottom of the observed span. It's clear 
    that Greg has made a very careful set of observations, and we can depend on 
    him to have made the correct allowances for dip, refraction, and 
    semidiameter. The actual values from which were deduced those 17 points 
    shown on his graph are all that are needed.
    
    I may have been somewhat dismissive of claims made about these photographic 
    methods, but that does NOT imply that I deplore them or think they are of 
    little use. Indeed, it's a technology that's worth pushing to its possible 
    limits, to understand any limitations, and to discover where applications 
    lie. Greg's careful observations are certainly useful, in that respect.
    
    What I have argued with is unrealistic claims, such as, from Greg-
    
    "Accuracy close to that of a metal sextant.", but later "Precision 0.4' vs 
    0.1' of a sextant."
    "Please try out the digital camera SLR with a fixed lense as I have 
    described the settings and technique. It works as good as a sextant."
    
    And I still await real observations made from at sea, not at anchor in a 
    cove, under conditions in which a sextant remains usable
    
    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.
    ----- Original Message ----- 
    From: "Greg Rudzinski" 
    To: 
    Sent: Saturday, June 26, 2010 1:48 AM
    Subject: [NavList] Px vs. MOA/Px Graph for 50mm Prime Lens
    
    
    The linked graph shows pixels (Px vertical) vs. minutes of arc per pixel 
    (MOA/Px horizontal) for a 50mm Pentax prime lens. Data for the graph was 
    figured from a set of timed setting Sun observations between 24° and 12° of 
    altitude. Each image data point is generated by working an Hc backwards to 
    get an Hs. The Hs in minutes of arc is divided by observed image pixels to 
    obtain the MOA/Px.
    
    Subsequent image altitudes can be derived with the graph by multiplying an 
    image altitude in pixels by the MOA/Px from the graph to get the altitude 
    Hs in MOA.
    
    Greg Rudzinski
    
    
    File:


       
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