Mathematics 138
Problems due Wednesday, April 20, 2005

For some of these problems, you will need a calculator with trig functions.  Even better, you can use Mathematica or a similar software package, which will do cross products automatically.  Printed Mathematica output is fine to hand in. This problem set does not do full justice to the topic, but it is as long as seems reasonable for a single week.
In problems 1 and 3, latitudes are measured from the equator, as is conventional in
geography.  Keep this in mind when converting "coordinates" into vectors. 
1. Quito, Ecuador (point Q) is located on the equator, 1.38 radians west of the prime meridian. London (point L) is located on the prime meridian, 0.91 radians north of the equator.  So Quito, London, and the point P where the prime meridian intersects the equator form a spherical right triangle.
In coordinates, P = , Q = , L = 
a. What is the distance p in radians from Quito to London?
b. If a plane flies the great circle route from Quito to London, how many radians north of east
 should it be heading right after takeoff?
c. When the plane returns from London to Quito, how many radians south of west should it be heading right after takeoff?
d. Find a parametric representation in the form (cos t)Q + (sin t)R for the flight path from
Quito to London. Check that when t = p, the plane is at point L.
e. If the right angle of a right triangle is at point C, prove that 
tan A = tan a / sin b, and tan B = tan b / sin a, 
where a is the length of the side opposite point A and b is the length of the side opposite point B.  Use this to check your answers to parts b and c.
2. Ryan, problem 6 on page 121.

3. An ecumenical group has acquired sites in Rome (latitude 42 N, longitude 12 E), Jerusalem (latitude 32 N, longitude 35 E) and Mecca (latitude 21 N, longitude 40 E).  Their
sites define a spherical triangle with vertices R, J, and M.
a. Find a unit vector for each vertex of the triangle.  The x axis is through 0N, 0E, the y axis is through 0N, 90E, and the z axis is through the north pole.
b. Determine the distances m = d(R,J), j = d(R,M), and r = d(J,M).  Express your answers in radians.
c. In Rome the group draws two arrows on the floor, labeled "This way to Jerusalem" and "This way to Mecca." Of course these point along the great circle routes from Rome to Jerusalem and Mecca.  What is the angle JRM between the arrows?
d. Using the Law of Sines, answer the same questions for arrows placed in Jerusalem and Mecca.
(The fact that the ratio in the Law of Sines is very close to 1 is a numerical fluke. Be careful about Jerusalem, since there are two angles between 0 and p with a given value of the sine function.)
e. Sensing trouble, the U.S. State Department issues a travel advisory for the interior of the spherical triangle RJM.  What fraction of the Earth's surface does this cover?

4. Prove the formula
cos a = (cos A + cos B cos C)/( sin B sin C)
on page 109 of Ryan, and check that your results from problem 3 agree with it.
The shortest proof, using cross products, is almost dual to the one given in lecture for cos A in terms of lengths of sides, but you need to be careful about the distinction between interior angles like A and exterior angles like p-A in order to get the correct + sign in the numerator. The discussion of dual triangles in the lecture notes may be useful.
5. A wealthy landowner in the asteroid belt decides to give property to his five sons. Each receives a plot in the form of an equilateral spherical triangle with one vertex at the north pole, and the five plots share common boundaries heading down from the north pole.
a. What are the angle and side (both in radians) for one of these plots?
b. Show that each plot occupies 1/20 of the surface of the asteroid.
c. The landowner who owns the southern half of the asteroid gives land to her five daughters in the same way, aligning the midpoint of the northern boundary of each lot with a southern vertex of one of the northern-hemisphere lots.  The five sons and five daughters pair up and marry.  Each couple has two children. Ten new triangular lots for the children are created by joining each vertex in the north to the two nearest vertices in the south. Show that each new lot is congruent to the original lots (so that the vertices define a regular dodecahedron on the asteroid and the centers of the lots define a regular icosahedron).
6 a. Find the 3x3 matrix that represents a reflection in a line (great circle) whose pole is the unit vector .
b. Line m has pole , while line n has pole . Find a matrix for the product T =Wn Wm
Determine the pole of this rotation. (You know that +1 is an eigenvalue for the pole.)
7. By following the same approach as on p. 30 of Ryan, prove that the product WaWbWg of
successive reflections in three nonconcurrent lines of S2 is a nontrivial glide reflection. Sketch a diagram showing how to find the axis of the glide reflection.
8. Ryan, problem 30 on page 122.
9. Ryan, problem 41 on page 123.

10. The following verse from I Kings, Chapter 7, is frequently cited to show that in ancient Hebrew culture the value of p was approximated by 3. Some claim that Hebrew letters can be used to represent numbers and that the value 3.141 can be deduced from the text, but my expert looked at the Hebrew text and told me "It says 30 cubits and no other reading is possible."
"23. And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about."
a. If the "molten sea" were on the surface of a sphere, centered for convenience at the North Pole, and a cubit were angle measure, then this could be accurate spherical geometry: you would have a circle centered at the North Pole for which the circumference (a parallel of latitude) is three times the diameter (measured along a meridian up to the Noth Pole and back down to the other side). For what radius, measured in degrees, is this true? What would this imply about the number of degrees in a cubit? (In fact, this reading is untenable, since a cubit is a unit of linear measure, from the elbow to the tip of the longest finger.)
b. Given the spherical interpretation of the text, does the area of the "molten sea" equal 3 r2? To get the area of a spherical cap you can evaluate an integral, or, since this is a long problem set, you can just look up the answer in a calculus text or mathematical handbook.