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POHow do I map a spherical triangle to a plane triangle?
 

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  1. POHow do I map a spherical triangle to a plane triangle?
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    <p>The gnomonic projection maps spherical triangles to straight-edged plane triangles.</p> <p>But I've heard that the Chamberlin trimetric projection has less distortion, so I'd like to use that instead. Alas, when I use my (extremely rough and probably buggy) implementation of Chamberlin trimetric projection to map the spherical triangle formed by its 3 base points to the plane, I seem to be getting a shape that is <em>almost</em> a triangle, but the three "lines" curve and bulge out. Is that a bug in my code, or is it supposed to do that?</p> <p>Is there some other way of mapping a spherical triangle to a straight-edged plane triangle that has less distortion than the gnomonic projection?</p> <p>EDIT: My goal here is to make a custom <a href="http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/projPoly.html" rel="nofollow noreferrer">"polyhedral map"</a> of Earth. If you print out something from the <a href="http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/Foldout/foldout.html" rel="nofollow noreferrer">"Map Fold-outs"</a> page, you will have something almost exactly like what I'm trying to make.</p> <p>I have 2 triangles. One is a spherical triangle drawn on a 3D globe. By definition, each edge of a spherical triangle is part of a great circle. When you look at that 3D globe, there are a bunch of cities, coastlines, etc. that are (hopefully) accurately plotted on that 3D globe, inside that spherical triangle.</p> <p>The other triangle is a flat, plane (2D), straight-edged, Euclidean triangle. On paper. At the moment the interior of that triangle is blank white paper, but eventually I want to draw a copy of all those cities, coastlines, etc. into that area.</p> <p>Every map projection will map that 3D spherical triangle to a 2D image. Then it's easy to (in 2D) rotate and slide and shrink, and perhaps skew, that image until the 3 corners exactly line up with the 3 corners of my plane triangle. If I stack the results of a bunch of different map projections on top of each other, even though I've forced the 3 corners to exactly line up, each projection will put the cities in a slightly different location. Unfortunately, many projections take cities that are slightly inside the spherical triangle and draw them slightly outside the flat 2D triangle. (One symptom of this problem is that the sides of the spherical triangle are mapped to plane (2D) curves whose endpoints match the corners properly, but they bulge outward slightly from perfectly straight lines drawn between the corners). That leads to the city being completely missing from my polyhedral map.</p> <p>I'd rather not run the risk of some Polish mathematician getting upset that Warsaw is inexplicably missing from the flat, plane (2d), paper map that I've made :-).</p> <p>So I'm looking for map projections that "keep it inside the lines". I need 3 specific great circles (the edges of the spherical triangle) to be mapped to straight lines on paper. I don't care about <em>other</em> great circles -- straight, non-straight, whatever.</p> <p>I hear that the map projection used for the <a href="http://en.wikipedia.org/wiki/Dymaxion_map" rel="nofollow noreferrer">Dymaxion map</a> meets that criteria; and someone told me that he <em>thought</em> it used the Chamberlin trimetric projection. But apparently either (a) I'm using the wrong equations -- so where can I find the right equations? Or (b) Dymaxion actually uses some other projection -- so where can I find the equations for that projection?</p> <p>What map projections meet that criteria? And can you give me a link to the (x,y) = f(lat, long) equations for that projection?</p> <p>I'm looking for answers of the form "The gnomonic projection meets your criteria. The <a href="http://en.wikipedia.org/wiki/Gnomonic_projection" rel="nofollow noreferrer">gnomonic projection equations</a>."</p>
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    1. USDavid Cary
    1. USDavid Cary
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      1. POHow do I map a spherical triangle to a plane triangle?
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    1. COIsn't some kind of curve inherent in using non-Euclidean triangles? I suppose intuitively, I'd expect each edge of a triangle to be part of a great circle - are these "bulges" relative to that?
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      1. POHow do I map a spherical triangle to a plane triangle?
      1. USSteve314
    2. COI won't be able to answer this regardless ;-) but for those of us that are curious is there any chance you can link in an image or two to indicate what you are getting vs. What you want to get?
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      1. POHow do I map a spherical triangle to a plane triangle?
      1. USscunliffe
    3. COI added a link to some pretty images to indicate what I want to get. Steve, I added (probably far too many) details to the question statement -- does that answer your questions?
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      1. POHow do I map a spherical triangle to a plane triangle?
      1. USDavid Cary
 

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