A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Px vs. MOA/Px Graph for 50mm Prime Lens
From: George Huxtable
Date: 2010 Jun 27, 12:06 +0100
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 firstname.lastname@example.org 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