Mathematics 138
Problems due Monday, May 9, 2005
For some of the problems, you will need a calculator that does hyperbolic functions.  The built-in calculator in Windows will do this in scientific mode, but it is probably easier to use Mathematica.
1. (This is Ryan, problem 2, p.179, restated without Ryan's special cross product.)
Consider the situation where two spacelike lines r and s, with poles x and h respectively, intersect in a lightlike point.
a. Draw a diagram depicting this state of affairs in the Klein model.
b. Show that <x, Bx><h, Bh> = <x, Bh>2.
c. Strictly speaking, r and s do not intersect in hyperbolic geometry: they are parallel.  But show that if you use the formula for angle in hyperbolic geometry for this case, you conclude that parallel lines intersect at an angle of zero.

2a. Ryan, problem 5 on page 179.  In our notation, part i. should call for "a multiple of Bx x Bh," i.e. a vector that represents where the polars of x and h intersect.
b. Draw a diagram in the Klein model showing x, h, their polar lines r and s, and the third element z of the orthonormal basis along with its polar line t.

3. Let s be the hyperbolic line that is represented in the Klein model by the horizontal diameter y = 0, and let P be the point represented by the timelike vector (0.6, 0, 1) (i.e. by the point
x = 0.6, y = 0 in the Klein model).
a. Find a timelike unit vector e3 that represents P and a spacelike unit vector e1 that represents the pole of s.
b. Find a spacelike unit vector e2, representing a point Q of the (projective) line s, such that e1, e2, and e3 are an orthonormal basis.
c. Let a(t) = (cosh t) e3 + (sinh t) e2. Find the points represented by a(t) for the three cases
tanh t = -3/4, tanh t = - 5/12, tanh t = 0, and tanh t = 5/12.  In each case rescale the vector so that the third component is 1, so that the first two components show how to plot the point in the Klein model. 
d. Draw a diagram in the Klein model showing the line s, the points P and Q, and the other three points from part c.  Also indicate what points are the limit of a(t) as t -> -¥ and as t -> +¥. 
4. Suppose that on a spacelike line s you choose a point P represented by a spacelike unit vector e2 and a point Q represented by a timelike unit vector e3, such that <e2, Be3> = 0.  Show that the parameterization a(t) = (cosh t) e2 + (sinh t) e3 generates all the spacelike points of the projective line on which P and Q lie, but none of the timelike points that are true hyperbolic points.  Draw a diagram in the Klein model showing P, Q, and how the points on the line are traced out as t ranges from -¥ to +¥. (This line is a line of doubly hyperbolic geometry, which would make a good term project.)

5. In the Klein model, each point on the hyperbolic line y = 0 is represented by a timelike vector (tanh t, 0, 1).  Let J and I be the lightlike points represented by (-1, 0, 1) and (1, 0, 1) respectively, and let A and B be arbitrary points on the line represented by (tanh tA, 0, 1) and (tanh tB, 0, 1) respectively.  Show that d(A, B) equals half the natural logarithm of CR( A, B, I, J), where, as you may recall, the cross ratio CR( A, B, I, J) is defined in terms of Euclidean segment lengths as
CR( A, B, I, J) = . Hint: write everything in terms of exponentials of tAand tB.

6. N. I. Lobachevsky has gone to hyperbolic heaven and taken up residence in New Kazan, located at (0, 3/5,1).  He learns that Janos Bolyai is completing some interesting work in New Budapest, at (12/13,0,1).  With considerable difficulty, Lobachevsky finds a former KGB agent and sends him by air to steal Bolyai's work.
a. Find unit vectors that represent New Kazan and New Budapest.
b. What is the distance from New Kazan to New Budapest (along a hyperbolic line, of course)? Confirm that the Pythagorean theorem holds for the right triangle whose vertices are New Kazan, New Budapest, and the the offices of the Hyperbolic Review at  (0,0,1).
c. How far east of south is the airplane heading immediately after takeoff from New Kazan?  How far north of west does the plane head after taking off on the return flight from New Budapest?
d. What would be the heading for a plane that took off from New Kazan on a line parallel to the positive x axis?
e. The instant his plane returns, Lobachevsky attaches his own name to Bolyai's article and sends it to the editorial offices of the Hyperbolic Review, located at (0,0,1).  At the same instant, Bolyai dispatches the article from New Budapest, using a plane that flies twice as fast as Lobachevsky's.  Who publishes first?

7. A hyperbolic triangle has the following three vertices:
P represented by (4/13, -3/13, 1)
Q represented by (0, 12/13,1)
R represented by (-12/17, -9/17,1)
Note: these are appropriate for the Klein model, but they all convert neatly to unit vectors.
a. Find the lengths p, q, r of the three sides of the triangle and the three angles P,Q,R.
b. Confirm that the Law of Sines holds.
c. Confirm that the Law of Cosines for cos P holds.
It may be a good idea to use Mathematica to do the arithmetic, and you can confirm your answers by plotting the points in CK.

8. A hyperbolic landowner has seven children.  He decides to leave each of them an equilateral triangle of land so that the triangles fit neatly together to form a hyperbolic heptagon.  What is the angle between the sides of one of these triangles, and how long is each side?
9. Let AX be a hyperbolic translation along the x axis through hyperbolic distance s and let AY be a hyperbolic translation along the y axis through the same hyperbolic distance s.
a. Show that AYAX can be represented as the product of a reflection, a half-turn about the origin, and another reflection.
b. Show that AYAX can be represented as a half-turn about a point on the x axis followed by a half-turn about a point on the y axis, and that AXAY is the product of the same half-turns in the opposite order.
c. Show that half-turns do not commute in hyperbolic geometry and that therefore translations do not commute either. Remember that orthogonal lines are represented in the Klein model by orthogonal Euclidean segments if and only if one segment is a diameter.
10. Find the midpoint of the hyperbolic line segment joining the point represented in the Klein model by (4/5,0,1) to the point represented in the Klein model by (0,3/5,1), and find the pole of the perpendicular bisector. Represent your answers also by vectors whose third component is 1, and draw a Klein model diagram to show them.
You can get the answers quickly by using formulas in the notes, but first you must represent the points by unit vectors!