Commentary 2.2 Sphere manifold

In section 2.2 on "What is a manifold?" after equation (2.9) Sean Carroll says check the equation for yourself. The equation is

ϕ₁ is the stereographic mapping from coordinates on a sphere to the 'southern plane' as shown below. (Similar to Carroll's Figure 2.16)

The diagram shows a section through the sphere which has radius 1. The section is in the plane of (y¹, y²) and the X³ axis, so it looks oval.  The point (x¹, x², x³) on the sphere is projected onto (y¹, y²) on the southern plane which is at x³=-1. We then drop a perpendicular (dashed line) from (x¹, x², x³) onto the southern plane to (x¹, x²)  and from there to the Y¹ axis. It hits at (x¹, 0). We also drop a perpendicular from  (y¹, y²) to the Y¹ axis, it hits at (y¹, 0). From there we draw a line back up to the North Pole and draw a line straight up from (x¹, 0) to intersect it. The intersection point is inside the sphere. We now have two right angled, similar triangles in Y¹X³ plane. These give us the equation below. The left hand side is from  the sides of the larger triangle......

Read more at Commentary 2.2 sphere manifold.pdf with additional material working up to the 'Northern plane'

This was my first attempt at a 3-D diagram. They improve!