I'm a complex analysis student reading about quasiconformal maps, and I have been wondering something I have been unable to answer myself. Suppose a and b are points in the unit disc U, is there a quasiconformal homeomorphism of the sphere which: 1) Takes a to b and 2) Is the identity outside U? It seems like it should be true, but I can neither come up with a construction nor can I prove it impossible. The best idea I've had is to prove it locally -- that is, for each a in U there is an r such that if |a - b| < r, there is a qc-map with the properties listed taking a to b. Then I would proceed by connectedness.