A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Tibor Miseta
Date: 2021 Mar 13, 15:12 -0800
You wrote: "Do the “twilight” lines of the altitude/azimuth grid at 5 degrees latitude look like the one I’m attaching"
You gave me a puzzle! :-D I had to think the position of these twilight lines over. Several times. For many days :-D
My conclusion is, that the problem here is that the simple ortographic projection (that is used by the 2102, by your app, and by mine too) is not suitable for negative altitudes at low latitudes. To better understand your example: at 180° azimuth on 5°N latitude Hc=0 altitude means a great circle crossing the points (LHA=0°; Lat=-85°) and (LHA=180°; Lat=+85°) on the globe. If we consider an altitude let's say -12°, the small circle of this altitude will fall completly on the other hemisphere; at the most southern point its latitude should be less than -90°, causing ambiguity (e.g. for -12° twilight line: 5°Lat -90°-12°= -97°. Such latitude does not exist.). The simple orthographic projection (x=r*sin(fi), y= r*cos(fi)) can handle only the 180° half circle, and fails in this case. But it is very interesting, that not the whole small circle is ambiguous, only its southern part, and where does it change from ambiguous to unambiguous is a bit complex.
Therefore the CP300 must use a different perspective. My guess is that they have scaled the half circle (multiplying the polar distances and altitudes by 180/190), so it is unambigous in the (+90 ... -100) range. But this scaling shifts the equator a bit North, so they might use a projection from a near side perspective, perhaps from a point when it tangets the scaled equator. Whitout knowing the exact transformation, it is impossible to regenerate the alt-az grid. And the starwheel must use exactly the same transformation, too!
To better understand, have a look at the azimuthal equdistant projection on Wikipedia: https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection.
And yes, my twilight lines looks exactly like yours. But I clip them only for the sections between Capricorn and Cancer (Lat +/- 23.5°), and I hope that this sections lie on the unambiguous part. The most affected latidude is 5° and the most problematic is around solstice, so I have built one, and made several tests. I could get very good results for twilight times, so I left it this way. The sections either really lie on the unambiguous part or do not cause noticable error. And latitudes above 12° are not affected at all. (Anyway, the whole twilight time calculation is more an estimation than a calculation with this tool, 4-5 minutes error must be acceptable!)