Maths stuff

Discussion in 'General Math' started by Doraemonsaurus, Nov 23, 2003.

  1. You are sitting at home, playing with your little earth model you got
    for christmas, and see two towns on it.
    One lies at [80 00 00 N 007 45 00 E] and the other at [70 00 00 S 007
    45 00 E].
    Now you are becoming curious. You want two questions answered:

    1. How far away are these two towns from each other?
    2. If your car drives 200 km/h, how long will it take you to drive
    from one town to the other?
     
    Doraemonsaurus, Nov 23, 2003
    #1
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  2. Indeed I am. 80 deg N, 7 deg 45' E is North tip of Norwegian Sea
    70 deg S, 7 deg 45' E is the South end of and entrance of the Ross Sea.
    There are no towns there. At best a research station at North tip of
    Norwegian Sea and at most a research vessel at the entrance of the
    Ross Sea.
    Car? Surely you jest. For 200 km/hr you better charter a piper cub
    plane and be prepared for in flight refueling unless you intend the 200
    km/hr to be average flight time including stop overs. That seems more
    practical.

    As were along the same longitude, the nautical distance to cover is 150
    degrees along the great circle of 7 deg 45' E. Now you need to look up
    the radius of the Earth, pi and the oblateness of the Earth. Because of
    rotation, the N-S axial diameter is shorter than the equatorial diameter.

    Here now you have big problem, but much time during the thousands of miles
    of slow flight time, to read up and learn about arc lengths of ellipses
    and other oblate figures. They are no simple matter, being left to
    advance calculus courses.
     
    William Elliot, Nov 23, 2003
    #2
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  3. Doraemonsaurus

    George Jones Guest

    I assume that, given the lattidues and longitudes, you want to know the
    shortest distance between 2 towns on the surface of the Earth.

    If the approximation that the Earth is spherical is made, it is fairly
    easy to give a general answer to 1.

    Let r_A and r_B be the position vectors of towns A and B with respect
    to the centre of the Earth, and let R be the radius of the Earth. Then
    d, the distance between A and B is given by

    d = R*a,

    where a is the angle (in radians) between vectors r_A and r_B. This is
    just the definition of angle.

    If you want to use degrees for angle a,

    d = R*a*pi/180.

    Now, how is the angle a found?

    r_A dot r_B = R^2 * cos(a),

    so,

    a = arccos{(r_A dot r_B)/R^2}

    What about r_A dot r_B ?

    r_A dot r_B = x_A * x_B + y_A * y_B + z_A * z_B

    = R^2 * [sin(t_A)*cos(p_A)*sin(t_B)*cos(p_B)

    + sin(t_A)*sin(p_A)*sin(t_B)*sin(p_B)

    + cos(t_A)*cos(t_B)]

    Lattitude and longitude relate directly to spherical coordinates
    theta = t and phi = p. Longitude is p (take west to be negative) and
    t = 90 - lattitude (take south to be negative).

    I've chosen the convention for phi and theta usually used by physicists,
    which, I think, is opposite to the convention usually used by
    mathematicians.

    Regards,
    George
     
    George Jones, Nov 23, 2003
    #3
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