Mathematics 138
Problems due Wednesday, April 6, 2005
Euclid is online at http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
Don't try the last three problems until after the April 4 lecture)
1a. Invent a concept of Galilean geometry (call it co-area) that is dual to area and prove that the co-area of a coparallelogram is invariant under any Galilean isometry.  (Hint: if you think of this physically, the units are square meters per second)
b Yaglom, problem 10 on page 66. The hint will become useful after you have done the preceding problem. 
2. State and prove a theorem of Galilean geometry that is dual to the theorem of Pappus.
The simplest proof of Pappus's theorem that I know has the three points on one line represented by vectors v, av, and bav, while the three on the other line are w, bw, and abw.  Try using doublets to construct the dual.

3. Yaglom, problem 14 on page 75.  "Tangential coordinates" are defined at the bottom of page 67; they are the "slope" and "intercept" of a line.  It may be helpful (and is certainly enlightening) to think of these transformations in physical terms.

4. Doublet A is formed by the x -axis (its beginning) and the line x = 3y + 1 (its end). Doublet B is formed by the line x+y = 4 (its beginning) and the line y = 2x (its end).
a. Construct the sum A+B graphically.
b. Construct the sum A+B by expressing each doublet as a linear combination of the basis doublets I and O.
c. Construct the doublet 2A-B.  Show it graphically in addition to specifying its components.
5. The Orient Express backs into a station and then heads out of the station, moving always with constant acceleration so that a graph of its motion is a parabola in the Galilean plane.
Observers note the following triangle of events:
Position y = 10 at time t = -2.
Position y = -6 at time t = 0.
Position y = 0 at time t = 3.
a. By taking the ratio of any angle of the triangle to the length of the opposite side, find the acceleration of the train.
b. Find an equation for y as a function of t.
c. On what diameter do all the midpoints of horizontal chords to this parabola lie?  Where is the tangent horizontal?
d. For what slope (speed) k do the midpoints of all chords lie on the special line 3 units to the right of this? What line of this slope is tangent to the parabola?
e. Find the matrix for a cyclic rotation that maps the parabola to itself, carrying the second event above into the third event (so that the matrix acting on  gives . What is the effect of this matrix on a horizontal chord?

6. Find the 3x3 matrix T(v) that represents a cyclic rotation of Galiliean geometry that carries the
parabola y = a(x - x0)2 + c into itself.  When x0 = 0, this should reduce to the matrix given in the lecture notes. Show that T(v) T(w) = T(v+w).

7a. Invent and prove a theorem of Galilean geometry that is similar to Euclid, Book III, Proposition 35. It will involve two intersecting chords of a parabola(instead of a circle) and the distances will be Galilean, not Euclidean. Euclid talks about areas of rectangles just to mean multiplication; your theorem can mention multiplication instead of areas.

Do not write down the equation of a parabola. Instead, use Yaglom's definition of a parabola in terms of the Galilean angle between lines that pass through the two ends of a segment. You will need a sort of "similar triangles" argument, easily done by using the Law of Sines.

b. Restate your theorem as a theorem of physics involving uniformly accelelerated motion and constant-velocity motion.