Stereographic projection

Fanny point x E S2 \ N, let t be the unique straight line through (0,0, I) and x. Since
x 7t N, t intersects the (xi, x2)-plane P {r3 = 0). Then f(x) E R2 is defined to be the
point whose coordinates (xi, x2) which agree with the (xl, x2) coordinates of the intersection
point of t with the plane P.
(I) Consider the point x = (4,0, 4). Show that x is a point in S2. Calculate the
linear parametric equation (x(t) = At + B, y(t) = Ct + D, z(t) = Et + F) for the
line t that passes through x and N. Calculate where l intersects the plane P, and
write down 1(x).
(2) For a general point x = (x9, yo, zo) E 52 N, calculate the linear parametric equation
for the line t that passes through x and N (in terms of the constants xo, yo, and
zo). Then calculate where t intersects the plane P, to determine ,f(x) in terms of
ro, yo, zo. This gives a general formula for the function 1.
(3) Looking at your function from problem 2, what is the set of points (x0, yo, zo) ERs
where this function is defined? Show this function is continuous where it is defined.