DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§2. An Introduction to Hyperbolic Geometry 91PαrβQrRFigure 2.6(dividing the angle at P into n angles of 2π/n each), we obtain a regular n-gon with the propertythat ∑ ι j =2π, asshown (approximately?) in Figure 2.7 for the case n =8. The point is thatFigure 2.7because the interior angles add up to 2π, when we identify edges as in Figure 2.5, we will obtain asmooth 2-holed torus with constant curvature K = −1. The analogous construction works for theg-holed torus, constructing a regular 4g-gon whose interior angles sum to 2π.EXERCISES 3.21. Find the geodesic joining P and Q in H and calculate d(P, Q).*a. P =(0, 1), Q =(1, 2)b. P =(−1, 8), Q =(2, 7)c. P =(0, 9), Q =(4, 7)2. Suppose there is a geodesic perpendicular to two geodesics in H. What can you prove aboutthe latter two?3. Prove the angle-angle-angle congruence theorem for hyperbolic triangles: If ∠A ∼ = ∠A ′ , ∠B ∼ =∠B ′ , and ∠C ∼ = ∠C ′ , then △ABC ∼ = △A ′ B ′ C ′ . (Hint: Use an isometry to move A ′ to A, B ′along the geodesic from A to B, and C ′ along the geodesic from A to C.)