# Hyperbolic Plane

## Hyperbolic Plane

Problem 5.1 What Is Straight in a Hyperbolic Plane?a. On a hyperbolic plane, consider the curves that run radially across each annular strip. Argue that these curves are intrinsically straight. Also, show that any two of them are asymptotic, in the sense that they converge toward each other but do not intersect.b. Find other geodesics on your physical hyperbolic surface. Use the properties of straightness (such as symmetries) you talked about in Problems 1.1, 2.1, and 4.1.c. What properties do you notice for geodesics on a hyperbolic plane? How are they the same as geodesics on the plane or spheres, and how are they different from geodesics on the plane and spheres?Hint for 5.1 a)Some of you seem to be having trouble visualizing 5.1a. This is because on the hyperbolic soccer ball model that most of you made, you can’t see the annuli. You can see the annuli on a crocheted hyperbolic plane, but so far only one person has turned in a crocheted plane. This YouTube video records the construction of an annular hyperbolic plane, which should help you visualize the asymptotic lines on the hyperbolic plane.Problem 7.1 and Problem 8.2 Due date: Feb 28th7.1The Area of a Triangle on a Spherea. The two sides of each interior angle of a triangle A on a sphere determine two congruent lunes with lune angle the same as the interior angle. ‘Show how the three pairs of lunes determined by the three interior angles, a, f, y, cover the sphere with some overlap. (What is the overlap?)Draw this on a physical sphere, as in Figure 7.2b. Find a formula for the area of a lune with lune angle 6 in terms of 6 and the (surface) area of the sphere (of radius p), which you can call Sp. Use radian measure for angles.Hint: What if <9is n? ji/2?c. Find a formula for the area of a triangle on a sphere of radius p.8.2

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