This problem shows that great circles on the sphere correspond to
circles and lines (``generalized circles'') in the Euclidean plane.
We're going to define a map from the sphere x^{2} +
y^{2} + z^{2} = 1 to the Euclidean plane as follows:
draw a line from the north pole (0,0,1) through the point (x,y,z) on
the sphere as in the figure.
This will intersect the xyplane in a point
(x_{0},y_{0},0) (provided the point (x,y,z)
is not the north pole!). (See the pdf file for the pictures.)
It might also help to view this picture from the side of the
xyplane, as seen below. (See the pdf file for the pictures.)

Use this second figure and some similar triangles to show that

1z

=

√ (x_{0}^{2} + y_{0}^{2}).



Use similar triangles to show that

x_{0}
x

= 
y_{0}
y

= 
1
1z

. 

Hint: Look at the projection of this situation into the
three planes: the xyplane, the xzplane, and the yzplane.

From part (a), conclude that
x_{0}^{2} + y_{0}^{2} = 
1z^{2}
(1z)^{2}

= 1 + 
2z
1z

. 

A great circle on the sphere x^{2} + y^{2} + z^{2} = 1 is the
intersection of the sphere with a plane Ax+By+Cz=0 (this is the
equation of a plane through the origin  the center of the
sphere). We begin by assuming that this great circle does
not pass through the north pole (0,0,1), which means that
C ¹ 0. We can assume that C=1 (just divide through by C and
rename the resulting variables). For the next parts, therefore,
assume that Ax + By + z = 0.
 Using the fact that Ax + By + z = 0, show that the equation
x_{0}^{2} + y_{0}^{2} = 1 + [ 2z/(1z)] from part (c) simplifies to
x_{0}^{2} + 2 A x_{0} + y_{0}^{2} + 2 B y_{0} = 1. 


The equation from the previous part is a circle. Find the
center and radius of this circle. (Your answer will involve A
and B.)
Now assume that the great circle on the sphere passes through the
north pole (0,0,1). In terms of the equation Ax+By+Cz=0, this
means that C=0, so Ax+By=0. For this last part, therefore,
assume that Ax + By = 0.
 Using the fact that Ax + By = 0, show that the equation from
part (b) simplifies to A x_{0} + B y_{0} = 0. This is a line.

At the beginning of this problem, I called circles and lines
``generalized circles'' on the plane. We've shown that great
circles on the sphere correspond to circles in the plane and
certain lines in the plane. What lines are these? That is,
what lines in the plane correspond to great circles on the
sphere?