Let N = (0,0,1) G S^2, and let Po be the plane z = 0 in R^3. Denote the stereogr

Question

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).

Explanation / Answer

(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

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