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Graphic: a global grid but not emanating from poles

  1. Nov 27, 2016 #1

    DaveC426913

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    This is for a fictional project, but the tools I need will more likely be found here in Earth forum, I think. Move to fiction if necessary.

    I want to render a map of Earth with the "poles" of the grid in a different place. Say, the South "pole" of the grid is at Tierra del Fuego (ish) in S. America. The North "pole" of the grid naturally will be at its antipodes. A Great Circle will define all points equidistant from these poles - a tilted equator.

    What I'm really after is not about an actual axis of rotation. I want to make a crude grid (just 30 or even degrees per segment or so) that will show - for any chosen point on the Earth - how far it is, and in what direction it is (a Great Circle) to Juan de Fuca and its antipodes.

    I've started by using Google maps to "measure distance" from point to point, which gives me partial Great Circles. These make crude "longitudinal" lines, but now I want crude "latitude" lines (which are essentially all points equidistant from the poles).

    Google maps is Mercator Projection-like so the lines form sine-like curves - very difficult to manually draw in latitudes.

    Can anyone suggest how I might do this with only a modicum of effort? Any online tools that might allow me to draw poles and lines on a map of Earth?
     
    Last edited: Nov 27, 2016
  2. jcsd
  3. Nov 27, 2016 #2

    Nidum

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    3D cad models of the Earth are available . You need suitable 3D cad software but if available you can manipulate the model in any way you like .

    Certainly you could overlay any grid system and set any axis as principle axis .

    Also possible are :

    Calcs of distance and surface areas , zoom and orientation control , cut planes , local area selection , flat plane projection .......and more .
     
  4. Nov 27, 2016 #3

    Nidum

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    Last edited: Nov 27, 2016
  5. Nov 27, 2016 #4

    Nidum

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  6. Nov 27, 2016 #5
    What are you trying to do? The latitude you cite is suspiciously near
    54.73561° = arccos[ sqr(1/3) ]
    which is related to the tetrahedron. That is, if you set down a tetrahedron at this latitude, it would function like a sundial. That is, one of the rising edges would run parallel to the axis of the Earth.
    Knowing this, the task is easy. Recalling that the dual of the tetrahedron is also a tetrahedron, we can make a gnomonic projection map, actually 4 gnomonic projections onto 4 triangles.
    This means that we put the Earth into a tetrahedron and make a gnomonic projection. The 4 points of this tetrahedron are:
    1) south pole
    2, 3) latitude 35.26439° north, longitude 60°, -60°
    4) antipode: latitude 35.26439° north, longitude 180°

    You can have this tetrahedron map unfold into a big triangle map of the entire Earth with your ground zero at the center. The north pole would be at the center in one of the sub-triangles. Sundial equations apply to the gnomonic projection. All great circles are straight lines. Your "tilted equator" is the horizon of ground zero. Direction is azimuth. Distance is altitude. Conic sections, circles, ellipses, hyperbolas, and a parabola would map out distances. These can be drawn on MS Paint and it's not really difficult.
     
  7. Nov 27, 2016 #6

    DaveC426913

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    Thanks but I've found the entry barriers and steep initial learning curves for 3D CAD software to be little too much for me.

    Maybe I'm thinking of 3DStudio Max etc. Maybe there are super-basic ones.
     
  8. Nov 27, 2016 #7

    DaveC426913

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    No, I simply want to have a quick dirty reference from any point on Earth, as to the distance and direction to my "special point".

    OK, fine, my special point is The Tunguska Explosion site, in Siberia.

    Don't judge me! I told you this was fiction. :cool:

    It is close enough to 60N 100E to use those coords.


    This would be perfect.
    It doesn't have to be accurate, just 'good enough'.

    How would I do this?
     
  9. Nov 27, 2016 #8

    DaveC426913

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    Your platonic solids reference gave me an idea that mgiht simplify things.

    If an actual graphic (containing lines) ends up being too much effort, I think I might settle for just a twenty, or merely a dozen, points, distributed around the Earth.

    So, if I were to calculate my coords at the vertices of a icosahedron or even merely a dodecahedron that would be sufficient.

    dodGn-s30.png
    This would be perfect, though I'd like to find one with Russia nearer the cetnre. (or rearrange this one).

    I'd still need to figure out the angles (through the solid) from vertex to vertex.
     
    Last edited: Nov 27, 2016
  10. Nov 27, 2016 #9
    At the latitude 60.916666° N, I don't think we can exploit the tetrahedron geometry.
    You can still make a gnomonic map with concentric circles for distance. Pick some angle interval (theta) and draw circles with radius = cot(theta). Each circle represents a distance R cot(theta), where R = 3959 mi = 6371 km (Earth radius).
    A helpful equation is
    cos( D ) = sin( Φ1 )sin( Φ2 ) + cos( Φ1 )cos( Φ2 )cos(δλ)
    gives the spherical distance between any two points given latitude Φ and longitude λ.
    One degree = 2 π R/ 360°
     
  11. Nov 27, 2016 #10
    https://en.wikipedia.org/wiki/Azimuthal_equidistant_projection
    "The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionately correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point.

    http://ns6t.net/azimuth/azimuth.html

    Request an Azimuthal Map
    I'm not recommending this particular generator, but it's an example of generators that are out there. You can pick Tunguska Impact site as the center of the projection and go from there. As with all projections, you lose something to get something.
     
  12. Nov 27, 2016 #11

    DaveC426913

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    Wow. That is absulotely perfect!

    Thank you!

    I have only one thing left to do now, which is determine the angle of declination at the latitudes.
    Angle X:
    picture-angle-tangent-chord.jpg
    http://www.mathwarehouse.com/geometry/circle/angle-tangent-and-chord.php

    There are only 3 azimuthal latitudes the mapmaker generates: 45, equator and -45, so this should be easy.
    So...
    A⊕C = 45°: X = ?
    A⊕C = 90°: X= ?
    A⊕C = 135°: X= ?
    (where ⊕ is the centre)

    Looking over the exercises, I think I can flip tje calculation.

    Is it as simple as halving the arc angle?
    A⊕C = 45°: X = 22.5
    A⊕C = 90°: X= 45
    A⊕C = 135°: X= 67.5
    ?
    Well, A⊕C = 90°: X= 45 works out correctly, since it would form a right triangle, so that's good.
     
    Last edited: Nov 27, 2016
  13. Nov 27, 2016 #12

    DaveC426913

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    Perfect!
    I still have to estimate both declination angle and direction but it's pretty straightforward now.
    Oh, and I guess I should put some instructions for doing so...
    Bigger-AzimuthalMap.png
     
  14. Nov 28, 2016 #13
  15. Nov 28, 2016 #14

    DaveC426913

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    Thanks but I've actually corrupted the compass roses. The roses - and all the red lines are not magnetic; they're actually geographic/true. (i,e, they point to the true geo poles)

    So, I can reference my new fabricated grid (the straight, radial grey lines) and say "at location X on Earth, a straight line running to "Tunguska North" is x degrees off from True geo North (the red curves)".

    Halifax is the only point referenced on the map. From Halifax, an arrow that points directly to "Tunguska North (the straight radial grey line) will actually point 11.25 degrees East of True North (the curvy red lines).

    Here is a closeup of Halifax:
    Closeup-AzimuthalMap.png
     
    Last edited: Nov 28, 2016
  16. Nov 28, 2016 #15
    You do know, of course, that the Canadian Security Intelligence Service had tracked the surviving alien family from the crash at Tunguska to Halifax and that you are not their offspring?
     
  17. Nov 29, 2016 #16

    DaveC426913

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    That's just what they're expecting you to think!
     
  18. Nov 30, 2016 #17
    Very interesting. I'll try 3D cad software.
     
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