# Find the image of the following circles on the sphere under stereographic projection

Find the image of the following circles on the sphere under stereographic projection from the north pole onto the equatorial plane:

1. $C = S^2 ∩ \{(x_1, x_2, x_3) \;|\; x_1 = x_2\}$
2. $C = S^2 ∩ \{(x_1, x_2, x_3) \;|\; x_3 = \frac12\}$
3. $C = S^2 ∩ \{(x_1, x_2, x_3) \;|\; x_3 = −\frac12\}$
4. $C = S^2 ∩ \{(x_1, x_2, x_3) \;|\; x_1 + x_2 − x_3 = 0\}$

Can anyone do one or two and explain to me?
thank you

• What are the coordinates of the north pole in your coordinate system? – MvG Mar 18 '14 at 14:26
• The question did not say it but lets just assume N = (0, 0, 1) – user134607 Mar 18 '14 at 14:35

My general approach would be as follows: identify the point on the circle closest to the north pole, and the point farthest away from it. Project these two points, e.g. by formulating lines through them and intersecting these with the equatorial plane. Or for $N=(0,0,1)$ you can use the following computation:
The idea behind this formula is that $1-x_3$ describes the third component of the vector pointing from the north pole to the point $(x_1,x_2,x_3)$ on the sphere. You want to scale things in such a way that this length becomes one, which means you divide by this expression.