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Okay I haven't gotten Dutton yet but I'm looking at my copy of Bowditch (1977 ed.) and I found, in article 1432, similar diagrams. They're called diagrams on the plane of the celestial meridian, and it is indeed an orthographic projection. The outer circle is the local celestial meridian.
The book first shows two diagrams that each contain a part of the diagram shown in the OP. One has the body located in equatorial coordinates, and one has the body located in horizontal coordinates.
In both diagrams, the circle is the (local) celestial meridian. We therefore draw both diagrams on the same circle, and by convention place the horizon horizontal with the north point on the horizon on the left.
Converting DeclLHA to AltAz works as follows:
1. You plot the parallel of declination by measuring the angle around the circle away from the equator either towards the north or south pole, and draw a chord parallel to the equator at this point on the circle.
2. You plot the hour circle by measuring an angle equal to the LHA around the circle, from the point where the equator meets the upper branch of the celestial meridian, clockwise. The upper branch is the right one in the northern hemisphere. You then draw a chord parallel to the axis at this point. This chord will cross the equator at some point. This new point, along with the north and south poles, are all located on the hour circle in this orthographic projection. The hour circle's projection should really be an ellipse but Bowditch says you can approximate it with a circle. The center of the circle will be someplace on the equator. You can use a bit of compassandstraightedge geometry to find this center, or you can use trial and error as suggested in Bowditch. Draw a circular arc through the intersection of this chord and the equator, all the way to the north and south poles.
3. The point where the hour circle from step 2 intersects the parallel of declination from step 1 is the position of the body, which the OP diagram marks as M.
4. Draw a chord parallel to the horizon through M. Measure the angle up from the horizon to this chord that is the altitude of M.
5. Again a tricky arcdrawing step. This draw a circle, centered someplace on the horizon, that intersects the zenith, nadir, and M. Again, you can use geometry or trial and error. This is M's vertical circle. Draw a chord tangent to this arc where it meets the horizon (i.e. perpendicular to the horizon). Measure the angle from the horizon to where this chord meets the celestial meridian (i.e. outer circle). If LHA is less than 180 degrees, the azimuth is 180 + the angle, otherwise it's 360  the angle. I think. I always get confused here.
So indeed this method can be used to compute altitude and azimuth from almanac data, albeit with some serious caveats:
1. I don't know about your protractor, but mine does not measure minutes of arc. So you can expect errors of up to 30 nm in position. I don't know whether this is acceptable for ocean navigation, but certainly not for coastal.
2. Partly errors are from insufficient precision in the measuring tools, but another big part comes from lack of skill in plotting (like my equator and horizon didn't intersect exactly at the center of the circle, my arcs didn't exactly meet the zenith/nadir, etc.)
3. Another source of errors is the fact that an ellipse is approximated with a circle. This is probably not a problem when M is near the celestial meridian, but for large LHA the errors can probably get quite large (analysis to come later).
Still, if you had a large chart table and could draw big circles and you practice enough, and especially if you have a setup that permits plotting elliptical arcs, this seems like it could be a cool way of doing all your CN without electronics. At the very least, you can use this in conjunction with noon/Polaris sights to minimize electronics usage.
Or you could just use a slide rule and do lots of errorprone arithmetic. I feel like this diagram destroys a lot of the intuition you get from drawing "3D" diagrams, so unless it's computing something for you, I don't see the point.
The book first shows two diagrams that each contain a part of the diagram shown in the OP. One has the body located in equatorial coordinates, and one has the body located in horizontal coordinates.
In both diagrams, the circle is the (local) celestial meridian. We therefore draw both diagrams on the same circle, and by convention place the horizon horizontal with the north point on the horizon on the left.
Converting DeclLHA to AltAz works as follows:
1. You plot the parallel of declination by measuring the angle around the circle away from the equator either towards the north or south pole, and draw a chord parallel to the equator at this point on the circle.
2. You plot the hour circle by measuring an angle equal to the LHA around the circle, from the point where the equator meets the upper branch of the celestial meridian, clockwise. The upper branch is the right one in the northern hemisphere. You then draw a chord parallel to the axis at this point. This chord will cross the equator at some point. This new point, along with the north and south poles, are all located on the hour circle in this orthographic projection. The hour circle's projection should really be an ellipse but Bowditch says you can approximate it with a circle. The center of the circle will be someplace on the equator. You can use a bit of compassandstraightedge geometry to find this center, or you can use trial and error as suggested in Bowditch. Draw a circular arc through the intersection of this chord and the equator, all the way to the north and south poles.
3. The point where the hour circle from step 2 intersects the parallel of declination from step 1 is the position of the body, which the OP diagram marks as M.
4. Draw a chord parallel to the horizon through M. Measure the angle up from the horizon to this chord that is the altitude of M.
5. Again a tricky arcdrawing step. This draw a circle, centered someplace on the horizon, that intersects the zenith, nadir, and M. Again, you can use geometry or trial and error. This is M's vertical circle. Draw a chord tangent to this arc where it meets the horizon (i.e. perpendicular to the horizon). Measure the angle from the horizon to where this chord meets the celestial meridian (i.e. outer circle). If LHA is less than 180 degrees, the azimuth is 180 + the angle, otherwise it's 360  the angle. I think. I always get confused here.
So indeed this method can be used to compute altitude and azimuth from almanac data, albeit with some serious caveats:
1. I don't know about your protractor, but mine does not measure minutes of arc. So you can expect errors of up to 30 nm in position. I don't know whether this is acceptable for ocean navigation, but certainly not for coastal.
2. Partly errors are from insufficient precision in the measuring tools, but another big part comes from lack of skill in plotting (like my equator and horizon didn't intersect exactly at the center of the circle, my arcs didn't exactly meet the zenith/nadir, etc.)
3. Another source of errors is the fact that an ellipse is approximated with a circle. This is probably not a problem when M is near the celestial meridian, but for large LHA the errors can probably get quite large (analysis to come later).
Still, if you had a large chart table and could draw big circles and you practice enough, and especially if you have a setup that permits plotting elliptical arcs, this seems like it could be a cool way of doing all your CN without electronics. At the very least, you can use this in conjunction with noon/Polaris sights to minimize electronics usage.
Or you could just use a slide rule and do lots of errorprone arithmetic. I feel like this diagram destroys a lot of the intuition you get from drawing "3D" diagrams, so unless it's computing something for you, I don't see the point.
s/v Laelia  1978 Pearson 365 ketch
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Originally Posted by AdamLein
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Or you could just use a slide rule and do lots of errorprone arithmetic. I feel like this diagram destroys a lot of the intuition you get from drawing "3D" diagrams, so unless it's computing something for you, I don't see the point.
What are you pretending not to know ?
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I think this says a lot, so many people seem to start out trying to learn celestial navigation with a sextant, but would probably be better served starting out with a globe, some string, and one of those clear celestial spheres, and just move the things around until you see what is really going on.
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I know that I learned a lot just by going outside and watching the sky throughout the year, learning constellations, guessing which stars were going to be up at approximately what times of the night from month to month, etc. Once you kind of have the model in your head and see your place in it then it all makes a lot more sense.
s/v Laelia  1978 Pearson 365 ketch


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Greetings,
I am sorely tempted to give an explanation of the celestial coordinate diagram. After spending a good part of today at it, I have come to the conclusion that it is the height of vanity to think that I could condense in a few paragraphs what Dutton needed in 34 pages.
But in summary; what you will do in completing such a 360 tick marked diagram is to find:
Pn; Ps; P; Q; z; Na; M; L; Lo; h; Dec; LHA; Z; Zn; z; coL; p; t;
And that my friends is the heart and soul of Celestial Navigation, or so my task masters
Insisted on.
Dick
I am sorely tempted to give an explanation of the celestial coordinate diagram. After spending a good part of today at it, I have come to the conclusion that it is the height of vanity to think that I could condense in a few paragraphs what Dutton needed in 34 pages.
But in summary; what you will do in completing such a 360 tick marked diagram is to find:
Pn; Ps; P; Q; z; Na; M; L; Lo; h; Dec; LHA; Z; Zn; z; coL; p; t;
And that my friends is the heart and soul of Celestial Navigation, or so my task masters
Insisted on.
Dick
Senior Member
Dick, you misunderstand: we don't need an explanation of the coordinates, but rather, and explanation of how to produce this diagram.
I suggest this website where you can download Bowditch as PDFs. Get chapter 15; the explanation of these socalled diagrams on the celestial meridian are explained in detail there in only five pages. Assuming they already have a basic understanding of the different coordinates, this should be well within the grasp of most readers of this thread.
I suggest this website where you can download Bowditch as PDFs. Get chapter 15; the explanation of these socalled diagrams on the celestial meridian are explained in detail there in only five pages. Assuming they already have a basic understanding of the different coordinates, this should be well within the grasp of most readers of this thread.
s/v Laelia  1978 Pearson 365 ketch
Crazy Woman Boat Driver
Quote:
Originally Posted by wind_magic
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I think this says a lot, so many people seem to start out trying to learn celestial navigation with a sextant, but would probably be better served starting out with a globe, some string, and one of those clear celestial spheres, and just move the things around until you see what is really going on. I know that I learned a lot just by going outside and watching the sky throughout the year, learning constellations, guessing which stars were going to be up at approximately what times of the night from month to month, etc. Once you kind of have the model in your head and see your place in it then it all makes a lot more sense.
Melissa Renee
Moondance
Catalina 445, Hull #90

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