Let N 001 G S2 and let Po be the plane z 0 in R3 Denote th

Let N = (0,0,1) G S^2, and let Po be the plane z = 0 in R^3. Denote the stereographic projection map by T: S^2 \\ {N} rightarrow P_0. In class, we found that T(x, y, z) = (x/1 - z, y/1 - z, 0), (x, y, z) S^2\\ {N}. Note: we can identify the plane Po with R^2 by mapping (x, y,0) G P_0 (x, y) R^2. The stereographic projection map can also be thought of as a map T: S^2 \\ {N} rightarrow R^2, given by Let P be the plane x = 2y in R^3. Let C be the great circle formed by the intersection of S^2 with P. Find a parametric expression for the line of intersection between P and PQ. Using the identification Po R^2 described above, give the equation of this intersection line in R^2. Let (2b,b, c) C be an arbitrary point in C. Find the image of this point under stereo- graphic projection to R2, and show that the image satisfies the equation of the line that you stated in part (a).

Solution

(a) The line of intersection is the set of points common to the planes

                                         x=2y and z=0.

The parametric equation of this line is given by

                                x/2=y/1=z/0 = c, where c is a parameter.

Or                                x=2c,y=c and z =0

(b) Under T , (2b,b,c) maps to

                                            (2b/(1-c), b/(1-c),0) in P₀ (This is the image under stereographic projection)

and to                                  (2b/(1-c), b/(1-c)) in R² under the identication (x,y,0) with (x,y)

and this point clearly satisfies the equation

                                                 x =2y