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Math 731 Assignment #9 Due: F., October 23, 2015 Diagnostic exercises (DX): 4.8, 4.22, 4.24, 4.26, 4.28, 4.33 Submit the following three problems 4.25, 4.29 Problem. (stereographic projection): Consider the map π from the unit sphere S 2 minus the north pole N to all of the xy-plane given by: π maps a point p on the sphere to the point where the line through N and p intersects the xy-plane. The map π is called stereographic projection, see Example 4.16. (a) Derive the following formulas. If (x, y, z) is a point on the sphere, show that y x , . π(x, y, z) = 1−z 1−z Show that the inverse map π −1 applied to a point (u, v) in the plane satisfies 2u 2v u2 + v 2 − 1 −1 π (u, v) = , , . u2 + v 2 + 1 u2 + v 2 + 1 u2 + v 2 + 1 hint: similar triangles (b) What is the image of the northern hemisphere (minus N ) via π? (c) What is the image of the southern hemisphere via π? (d) Show that the set of fixed points of π, i.e., points where π(x, y, z) = (x, y, z), is the equator of S 2. (e) Show that π is a bijection explicitly by computing π (π −1 (u, v)) and π −1 (π(x, y, z)). Convince yourself that π maps open sets of S 2 − {N } bijectively to open sets of R2 , which means it is a homeomorphism. (you do not need to submit anything regarding the previous sentence)