The rotation that leaves all but two axies fixed is what i call a great arrow. That is, it's a great circle with an arrow on it. I suppose you could construct all rotations out of great arrows. In four dimensions, one constructs all rotations, including great arrows, out of clifford rotations.
With the "projective geometry", there is a loss of detail in some geometries. Those that maintain an infinity have no point-loss, because the antipodal point is not in real space.
When you project a plane by projective geometry, imagine that you have an observer U, above the ground, and that lines are transferred from the horizontal to the vertical, by passing planes through the line on the vertical, and through the point U.
A point P is transfromed from the horizontal to the vertical, by continuing a line from P to U to its image P'.
What this gives is the actual projective, as used by artists.
Hyperbolic geometry is completely preserved by the projective as well.
And, my Google search for 'toric projection' didn't lead to anything glarkable.
Projections, such as the mercartor and cylinderic, rely on the nature of the Earth's equator and its role in defining longitude and lattitude. In four dimensions, you don't have an equator as such.
Instead, what you have is a double rotation (ie in wx, yz). The nature of physics says that energy is transferred from rotational modes, so you get a rotation where wx, yz are at the same frequency: a clifford rotation. These are representable by quarterions. (The great-arrow rotations are effected by quarterions by using xZx', where x' is some kind of complement of x.
Since in a clifford-rotation, every point goes around the centre, one gets lattitude by the rising of say, the zenith star. Any point that has star X as zenith is on the same line of longitude.
One, can, for a pair of different points, calculate the differences of lattitude, because star X will peak in the sky at say, 60 degrees, then the distance between the lattitudes is 30 degrees (ie 90-60). If one makes a map of distances between lattitudes, one gets a sphere surface (glomochorix), where the angles are doubled. A pair of orthogonal great circles appears as antipodal on the sphere.
Longitude corresponds to the rotation, and is measured out in the time of the day: all points at say 150E is at 10pm at night, for example.
One can then take the product of longitude (circle-surface) and lattitude (sphere-surface) to get a circle-sphere torus. Unwapping this leads to interesting projections.
1. You can preserve the sphere-surface, and unwrap the circle into the radially to the sphere. The rotation is then outwards, and reappearing into the centre.
2. You can unwrap the sphere, and the circle, to give a (sphere-projection) * (circle-projection) prism. Since the circle-projection is a line (but still dividable at different points, cf different breaks in the mercartor projection).
You then can project the sphere in different ways. This is in nature, a stack of rectangles (from the thin line in x to the thin line in y). Rotations follow the diagonals of these lines, but can be skewed to go left-to-right.
The next orientation assumes that the earth is tilted: that is, the zodiac is a line that is not parallel to the earth-rotation, and there is a single pair of great arrows, for which each is also right-parallel to these (ie the line of zodiac is equidistant from these lines).
This presents a second and third axis, which effectively turns your lattitude-sphere into (like the earth), where N pole is icy cold, and S pole is tropics. [as if you doubled the distance from the north pole]. The earth-like longitude becomes the year-seasons. That is, for all points that is on the lattitude sphere at 140 E might be having spring, while 40 W is in autumn.
This gives 3 coordinates, (climate, time of year, time of day), which makes a box-shaped thing, and this is the toric projection.
W