What transformation of the complex plane lifts to the rotation of the unit sphere by 180 degrees about the line whose equations are x=y and z=0 in R^3? (three dimensional plane of all reals)(I think that's what that means R^3?)

Thank you!
Jared

To answer this question, we need to understand the concept of stereographic projection. Stereographic projection is a way to map points on a sphere to a plane using a specific projection point called the "north pole." In this case, we will consider the unit sphere centered at the origin in three-dimensional space (R^3).

1. Setting up the Unit Sphere:
The equation x^2 + y^2 + z^2 = 1 represents the unit sphere centered at the origin in R^3. The north pole of the unit sphere is defined as (0, 0, 1).

2. Stereographic Projection:
To perform the stereographic projection, we need to define a projection point on the unit sphere. In this case, we will use the south pole, which is the point (0, 0, -1).

3. Mapping the Plane:
We will map the points on the unit sphere except for the south pole to a plane located below the unit sphere. The idea is to draw a line from the projection point (south pole) through each point on the unit sphere, and wherever this line intersects the plane, that will be the projected point.

4. Finding the Line of Rotation:
The line of rotation corresponds to the intersection of the unit sphere with the plane z = 0. In this case, the line is given by the equations x = y and z = 0.

5. Mapping the Points on the Line:
To find the transformation that lifts to the rotation of the unit sphere by 180 degrees about the given line, we need to consider the equation of the line as an input to our stereographic projection.

Therefore, any point on the line (x, y, z) that lies on the unit sphere can be mapped to the plane using the stereographic projection with the south pole as the projection point.

To find the image point (u, v) on the plane corresponding to the point (x, y, z) on the unit sphere, use the following formulas:

u = (x / (1 - z)) and v = (y / (1 - z))

So, the transformation of the complex plane that lifts to the rotation of the unit sphere by 180 degrees about the line x = y and z = 0 in R^3 can be obtained by using the above formulas to map each complex number to its projection on the plane.