Download Math 11 Adv Distance on a sphere – another application of

Math 11 Adv
Distance on a sphere – another application of trigonometry
The idea of “the shortest distance between two points” is different on a sphere – the
shortest distance is found as the length of an arc on a great circle (a portion of a
circumference).
Lines of longitudes are great circles. However (with the exception of the Equator) lines of
latitude are not. Thus, traveling along a line of longitude is traveling on a great circle, while
traveling along a line of latitude between two points may seem like the most direct path,
but it is not.
Our geographic coordinate system is great for figuring distances that are parallel to lines of
latitude or longitude. But if the paths traveled don’t align with this system, the calculation
becomes more complicated. In this image, angle  represents the latitude and angle  represents longitude. Latitude is measured
from the equator, longitude from the primer meridian (in Greenwich, England).
Questions 1-4 can be done with basic trigonometry and circle geometry. Assume the earth’s radius is 6370km.
1. St Martin French/Dutch island in the Caribbean, and it sits at 18N, 63W. The shortest route from St Martin to Halifax (45N, 63W)
is directly along the 63W line of longitude.
a) What distance must be traveled to make this trip?
b) Lines of longitude are relatively easy to use for calculating shortest distance. Why?
2. Casablanca, Morocco (33.5N, 7.6W) and Damascus, Syria (33.5N, 35.3E) are at the same latitude. If you traveled along this
latitude from one to the other, what distance would you travel?
3. Dublin, Ireland and Karachev, Russia are at the same longitudes as the two cities in 2), but they are at 53N latitude instead. How
far do you have to travel along this latitude from one to the other?
4. Rebecca and Jono, who seem to know their sailing, are in a two-person vessel in the middle of the Atlantic (21N, 5W). They sail
their boat due west for 400km. Where do they end up? (Just coordinates)
Read this, then do that.
Imagine two points at the same latitude, like Halifax and Belgrade, Serbia. The great circle connecting these two points is not easy to
visualize, and does not align with our coordinate system. In this scenario, to figure out shortest distance, we need the Law of
Cosines for Spherical Geometry. This is different than the Law of Cosines for Plane Geometry. It says:
cos( A)  sin 1  sin 2   cos 1  cos 2  cos   
Where
1 , 2
are the latitudes of locations 1 and 2 (north is positive latitude, south is negative)
 is the difference in longitudes between locations 1 and 2 (east is positive longitude, west is negative)
A is the angle formed by the arc on the great circle between locations 1 and 2.
If we can determine angle A, then we can determine the shortest distance between two locations – it represents the portion of the
entire circumference of the earth.
5. New York (41N, 74W or -74o) and Paris (49N, 2E or +2o)… ahh, New York and Paris!
a) what is the angle formed from the arc of the great circle between these two cities?
b) what is the shortest distance between them?
6. Point A sits at 15o South latitude, while point B sits at 23o N latitude. The angle formed by the great circle arc between these two
locations is 41o. How far apart are they in an east/west direction (longitudinally)?
Latitude, Longitude, arcs, great circles, rhumb lines, spherical cosine law… aren’t you glad you have Google Maps and a GPS on your
phone?