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    Re: Graphical Solution to Double Altitude Sight using Stereographic Projection
    From: Robin Stuart
    Date: 2010 Jun 17, 16:53 -0700

    Hanno,
    Thank you very much for pointing out the reference to J. B. Breed’s work. I have ordered but not yet gotten my hands on a copy of his "Navigation without Numbers". The method is summarized in J. A. Russell's ION article you refer to in which he talks about "constructing the triangular pyramid whose vertex is the center of the Earth". This and his diagrams show that Breed uses the gnomonic projection, seen on great circle charts, rather than the stereographic. Under gnomonic projection great circles on the sphere become straight lines on the plane hence spherical triangles become plane triangles. However the gnomonic projection is not conformal (angle preserving) and so the angles at the vertices of the spherical triangle are not the same as those of the plane triangle. Under gnomonic projection small circles on the sphere become conic sections (ellipses, parabolas, hyperbolas) on the plane.

    The graphical solution to the double altitude sight I described involves finding the intersection of small circles making essential use of the property of stereographic projection that circles, great and small, on the sphere map to circles on the plane. It would not work under the gnomonic projection.

    Thanks too for the link to "The Stereographic Projection and its possibilities from a graphical standpoint" by S.L.Penfield. Coincidentally, I was shown a hard copy of this document from the Mystic Seaport archive at the Navigation weekend. The author has a background in crystallography, a field that makes extensive use of stereographic projection, and is concerned with the mechanics of plotting and measuring great circles on a stereographic grid. (Amusingly in this 1901 publication he bemoans the fact that "Students come to the universities with altogether too little knowledge of how to do things accurately...".) He does indeed describe the same construction for drawing small circles on the stereographic grid as I had used. His application to navigation does not go far beyond plotting great circle courses and measuring distances and bearings. For that Penfield uses what he calls a "Stereographic Protractor". He does not really deal with celestial navigation or sight reduction explicitly but if one were in possession of a stereographic chart (say for the graphical reduction of double altitude sights!) his methods might be used to extend it’s the uses to which such a chart could be put.

    Regards,
    Robin Stuart

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