Constructing a Mobius transformation that acts on any two points of the upper half complex plane:

Question

I would like to construct a Mobius transformation that sends any two points $z_1$ and $z_2$ from the upper half of the complex plane to i and to $iR^+$, i.e., given any two points $z_1$ and $z_2$, and a Mobius transform A, then A($z_1$) = i, while A($z_2$) goes to some place on the vertical line $iR^+$.

This transform A belongs in the group of matrices PSL(2,R), the projective special linear group of 2x2 real matrices, with determinant = 1.

Any thoughts on how I can construct this mapping (and have it be a member of PSL(2,R))?

Thanks,

Answer 1

1. Map to the unit disk so that $z_1$ goes to center.
2. Rotate the disk so that the image of $z_2$ is real.
3. Map back to the upper half plane with $z\mapsto i\dfrac{1+z}{1-z}$.
4. The composition of the above maps achieves the goal.
5. Any Möbius transformation of the upper halfplane onto itself is represented by a $PSL(2,\mathbb{R})$ matrix; Wikipedia.