On the plane or on spheres, rotations and reflections are both intrinsic in the sense that they are experienced by a 2-dimensional bug as rotations and reflections. These intrinsic rotations and reflections are also extrinsic in the sense that they can also be viewed as isometries of 3-space. (For example, the reflection of a sphere through a great circle can also be viewed as a reflection of 3-space through the plane containing the great circle.) Thus rotations and reflections are particularly easy to see on planes and spheres. In addition, on the plane and sphere all rotations and reflections are global in the sense that they take the whole plane to itself or whole sphere to itself. (For example, any intrinsic rotation about a point on a sphere is always a rotation of the whole sphere.) On cylinders and cones, intrinsic rotations and reflections exist locally because cones and cylinders are locally isometric with the plane. However
a. Which intrinsic rotations and reflections on which cones and cylinders are also extrinsic? Which are global?
Be sure to look at the cone points. The answers to the two questions are not exactly the same.
Now, we can see from our physical hyperbolic planes that geodesics exist joining every pair of points and that these geodesics each have reflection-in-themselves symmetry. (If you did not see this in Problem 5.1c, then go back and explore some more with your physical model. In Chapter 16 we will prove rigorously that this is in fact true by using the upper half plane model.) In Chapter 16 we will show that these reflections are global reflections of the whole hyperbolic space. However, note that there do not exist extrinsic reflections of the hyperbolic plane (in Euclidean 3-space). However, given all this, it is not clear that there exist intrinsic rotations, nor is it necessarily clear what exactly intrinsic rotations are.
b. Let l and m be two geodesics on the hyperbolic plane which intersect at the point P. Look at the composition of the reflection Rl through l with the reflection Rm through m. Show that this composition RmRl deserves to be called a rotation about P. What is the angle of the rotation?
Figure 5.13 Composition of two reflections is a rotation
Let A be a point on m and B be a point on l, and let Q be an arbitrary point (not on m or l). Investigate where A, B, and Q are sent by Rl and then by RmRl . See Figure 5.13.
We will study symmetries and isometries in more detail in Chapter 11. In that chapter we will show that every isometry (on the plane, spheres, and hyperbolic planes) is a composition of one, two, or three reflections.
c. Show that Problem 3.1 (VAT) holds on cylinders, cones (including the cone points), and hyperbolic planes.
If you check your proof(s) of 3.1 and modify them (if necessary) to only involve symmetries, then you will be able to see that they hold also on the other surfaces.
d. Define "rotation about P through an angle q " without mentioning reflections in your definition.
What does a rotation do to a point not at P?
e. A popular high school textbook series defines a rotation as the composition of two reflections. Is this a good definition? Why? or Why not?