This document contains mathematical calculations and coordinates for the north and south poles of a rotated sphere, including the x and y coordinates of the north pole (-sin(45)*sin(45), cos(45)), south pole (0.5, -cos(45)), and radial circles (sin(theta), cos(theta)). It also lists the y-coordinates for points on an upright sphere and calculates the lengths of the upright and rotated sphere axes.

1. South Pole of the Rotated Sphere, and coordinates of the centres of the haloes on
the upright sphere.
Sharon Liu.
2023/08/09.
North pole x: -sin(45)*sin(45).
North pole y: cos(45).
The south pole, on the xy projection, is the mirror image, of the north pole.
South pole x: 0.5 .
South pole y: -cos(45).
Radial circle x: sin(theta).
Radial circle y: cos(theta).
r_sphere = 1.

2. Upright sphere: x coordinates are 0.
DU45, y: cos(45).
DE, y: 0.
DD45, y: cos(135).
Next steps.
Length of the upright axis, is 2.
Length of the rotated sphere axis, is sqrt(1^2 + (2*cos(45))^2). (Used pythagoras.)