Math 731 Assignment #9

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
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
u2 + v 2 − 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)