# Maths stuff

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

1. ### DoraemonsaurusGuest

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

2. ### William ElliotGuest

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

William Elliot, Nov 23, 2003

3. ### George JonesGuest

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