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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: NG's "Midnight Fun"
From: George Huxtable
Date: 2010 Jun 11, 12:53 +0100
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.