Review Sheet There will be an exam on Thursday, February 14. The exam will cover topics up through material from projective geometry through Day 3 of the DIY Hyperbolic geometry packet. Below are some practice problems. Students are expected to study outside of class, but some class time will be set aside on Tuesday and Wednesday for students to ask questions about topics related to the exam. We will do some practice problems in class on several days leading up to the exam. Formulas: Students are expected to know the formulas and definitions relevant to the class. A brief summary of topics covered: Finite models of the projective plane and coordinates Infinite models of the projective plane including: the three dimensional model, the Euclidean plane with a line at infinity, the unit sphere Homogeneous coordinates, graphing lines and planes Converting from Euclidean plane with a line at infinity (sitting at z = 1) to homogeneous coordinates Projections from lines to lines geometrically and algebraically. Fractional linear transformations. The cross ratio Models of hyperbolic space and their geodesics: the homogeneous model approximated by triangles fitting seven to a point, the Poincare disc, the upper half plane Mobius transformations: in particular, the analysis of the map f(z) = (z i)/(z + i) taking the upper half plane to the Poincare disc The hyperbolic distance formula for the upper half plane Areas of spherical and hyperbolic triangles
Projective Plane 1. Do all points on the projective plane look like all other points? Why or why not? 2. In the model of the Euclidean plane with a line at infinity, we say that parallel lines meet at infinity. When we embed this model into the three dimensional model, with the Euclidean plane sitting at z = 1, at what point do parallel lines meet? 3. Why do we draw the Euclidean plane at z = 1 usually? Where else could the Euclidean plane sit inside the three dimensional model of projective space? Homogeneous Coordinates 4. Write four different names for the point [3 : 4 : 1]. 5. What is the Euclidean coordinates for the point corresponding to the projective point [5 : 1 : 3]? 6. What is the equation for the projective line whose image on the Euclidean plane is given by y = 4x + 2? 7. What is the intersection in homogeneous coordinates of the lines 3x + 2y z = 0 and 6x 4y 5z = 0? Is this a point at infinity or on the Euclidean plane? 8. What is the intersection in homogeneous coordinates of the lines x + 2y 3z = 0 and 3x 5z 2y = 1? Is this a point at infinity or on the Euclidean plane? 9. Suppose we took the plane at z = 1 and looked at the projective line y + 2x = 1. What is its point at infinity? What is the equation for the 3D plane defining this line? Now, we want to look at where this projective line intersects the 3D plane z 4x = 2 instead. Where does the point (1, 1) go? What about the point ( 2, 5)? What projective points are fixed under projection from z = 1 to z 4x = 2? 10. Draw the projective line 3z 2y = 0 in the three dimensional model. What is its image in the Euclidean plane at z = 1? 11. What points at infinity should the projective curve y = x 3 have? 12. In the spherical model of the projective plane, draw the curve x 2 + y 2 4z 2 = 0. 13. Try redoing any of the problems from the worksheet on homogeneous coordinates (the typed ones with problems (a) through (g).)
Projections 14. Construct two projections which give the function f(x) = 2x + 5. For each step, tell whether the projection is between parallel or perpendicular lines, and whether it is from a point at infinity or a finite point. 15. Prove that the composition of any number of projections is a fractional linear transformation. 16. Give a picture for the projection f(x) = 1/x. Where do the points x = 0 and x = go under this projection? 17. Is every fractional linear transformation a projection? What is the relationship between fractional linear transformations and projections? 18. Is it possible for a projection to fix just one point? That is, is it possible to have a fractional linear transformation f(x) with only one x satisfying f(x) = x? What about exactly two? What about exactly zero, three, and four? The Cross Ratio 19. Show that the cross ratio is preserved under projections. 20. Consider four points of the form x, x + kx, x + k 2 x, x + k 3 x where x and k are real numbers. For what values of k can this be the projection of four equally spaced points? 21. What is the relationship between [p, q, ; r, s] and [p, r; s, q]? 22. Try reproving one or two of the theorems from the cross ratio reading without looking at the proof. 23. Suppose f is a function which preserves the cross ratio, and has values f(0) = 1, f(1) = 2, and f(3) = 1. What is the value of f(5)? Write an equation for f. Hyperbolic Geometry Models 24. Do all points in the hyperbolic plane look like all other points? Why or why not? 25. In the Poincare disc, what are the geodesics? Practice finding a geodesic between two points. 26. In the Poincare disc, draw a line l and a point not on that line P. Draw several straight lines through P which do not intersect l.
27. Show that the combinatorial circumference of a disc of radius r in the hyperbolic plane is bounded above by 5 r (7/5) and below by 2 r (7/2). 28. In the upper half plane model, what are the geodesics? Practice finding a geodesic between two points. 29. Find two parallel lines which diverge as you go towards infinity in both directions. 30. Find two parallel lines which converge on one side as you go to infinity. Mobius Transformations 31. Show that Mobius transformations preserve cross ratio. 32. Draw the images of the vertical lines from real points 2, 1, 0, 1, and 2 under the map f(z) = (z i)/(z + i) taking the half plane to the disc. 33. Show that the map g(z) = (z 1/2)/(1 (1/2)z) takes the disk to the disk. Describe where the boundary goes under g and where 0 goes. 34. Consider the map h(z) = 1/z. Describe what this does to the unit circle, z C such that z < 1. What does it do to the boundary of the unit circle z C such that z = 1? Hyperbolic Distance Formula 35. What is the ratio between the hyperbolic distance between 2i 0.1 and 2i + 0.1? What about between 3i 0.1 and 3i + 0.1? 36. What is the hyperbolic distance between the points 2 + i and 4 + i in the upper half plane? Do you expect this to be bigger or smaller than the distance between 2 + 2i and 4 + 2i? What about compared to the distance between 1 + i and 3 + i? 37. Show that d(z, w) = d(w, z) for any two points z, w in the upper half plane. 38. Draw the intersection of circle on the upper half plane model centered at 2i with radius 1 and the imaginary axis. Areas of Spherical and Hyperbolic Triangles 39. Prove that you cannot make a rectangle in the projective plane.
40. Suppose I cut a sphere into six slices (six lunes), like a clementine, using lines of longitude that are equally spaced. Then I slice it at the equator. This makes twelve equally sized triangles. What are their angle measurements and area? 41. Draw a triangle with one, two, or three angles equal to 0 degrees on the Poincare disc. 42. What is the area of a pentagon with five right angles? 43. Why does the concept of similarity not mean much in the hyperbolic plane?