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B Calculating a straight line within a sphere

  1. Nov 27, 2016 #1

    DaveC426913

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    This is a fiction project, but the question is about 3D geometry (a bit beyond my HS math skills).

    Given two points on a sphere (oh say, hypothetically, the Earth), draw a line from point A through the Earth to Point B.

    If I were to stand on point A and point directly at point B, what geographical direction (eg.NNE) would I be pointing and what inclination (eg. 30 degrees down)?

    Very rounded numbers (and compass points) are fine.

    Point A is 45N 60W
    Point B is 60N 100E
     
    Last edited: Nov 27, 2016
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  3. Nov 27, 2016 #2

    Mark44

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    I haven't thought about the direction, but I have given some thought to the inclination of the line joining the two points.
    GCircle.png

    Any two points on a sphere lie on a "great circle," as shown in the figure above. For the inclination of line segment ##\overline{AB}##, the longitude plays no role -- only the latitude is important. Point A's latitude is 45° N, and point B's latitude is 60° N, so rays extending from the center of the earth make angles of 45° and 60°, respectively, with the equator. The triangle shown here is an isosceles triangle whose lower angle is 75°. The angles at A and B are 52.5° each.

    If you draw a line at A that is tangent to the great circle, the angle between this tangent and ##\overline{AB}## will be 37.5°, the angle of declination of the line from point A to point B.

    I leave the other part of your question to others who might be interested.
     
    Last edited: Nov 27, 2016
  4. Nov 27, 2016 #3

    QuantumQuest

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    Some spherical trigonometry would be of great help. If you have the time and patience to read, take a look at the first chapter of this book.
     
  5. Nov 27, 2016 #4

    jedishrfu

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  6. Nov 28, 2016 #5

    mfb

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    @Mark44: If longitude plays no role, what about this situation?
    Point A is 45N 60W
    Point B is 60N 60W
    Clearly the great circle connecting those has 90 degrees inclination. Your approach only works if the longitude is different by 180 degrees.


    If in doubt, calculate the 3D coordinates for both points, everything else follows from that.
     
  7. Nov 28, 2016 #6

    Mark44

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    The approach I described is only for the inclination of the segment joining the two points, not the distance between them, in which the longitude necessarily plays a role. For the scenario you give, the central angle of the isosceles triangle is 15°, so the base angles are each 82.5°. From this, the angle of declination from A to B is -7.5°.

    To find the direction of the segment, I agree that you would need the coordinates (3D rectangular would be easiest, I think) of the two points. From these you could find a vector joining the two points, and from that, find the compass direction from one point to the other.
     
  8. Nov 28, 2016 #7

    mfb

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    That inclination depends on longitude already. Just with the latitude values, you cannot do anything apart from determining the minimal inclination.
     
  9. Nov 28, 2016 #8

    Mark44

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    After thinking about this some more, I realize you're right. Your example of two points at the same longitude didn't really show this, but if the two points are both on the equator (i.e., same latitude), then the angle of inclination comes solely from the longitude. If the two points are at the same longitude, then only the latitudes come into play. If long./lat. are different for both points, then both have to be used.

    My error was in not taking into account that the angles relative to the great circle through the two points can be different from the angles relative to the equatorial plane.
     
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