Recently, I've been thinking about projective geometry and the way it's transformed into specific cases of elliptic, Euclidean and hyperbolic geometry, and I realized that there might be more interesting stuff there.

In this model, points in projective plane are lines in R3 (three-dimensional Euclidean space) that pass through the origin. Lines, analogically, are planes that pass through the origin. We can get Euclidean geometry by studying the intersections of these lines and planes with fixed plane z = 1. We can also study elliptic geometry by using sphere x^2 + y^2 + z^2 = 1, or hyperbolic geometry by using two-sheet hyperboloid x^2 + y^2 - z^2 = -1.

But all three of these are actually just specific cases of more general equation: ax^2 + by^2 + z^2 = 1. For a=b=1, we get the sphere, for a=b=0, we get a pair of planes, and for a=b=-1, we get the hyperboloid.

(Getting two planes in Euclidean case instead of one is actually correct: in each case, the line passing through origin cuts the projection surface in two points, but this pair of points is treated like a single point. In the elliptic case this matters and differentiates it from spherical geometry, but in Euclidean and hyperbolic case, the projection surface is made up by two disjoint surfaces in the first place and so one of them can be safely omitted.)

But what happens if a and b have other values? Well, for starters, the resulting planes will be anisotropical. Hyperbolic geometry, unlike Euclidean one, has preferred way to measure lengths, but it doesn't really have a preferred way to state directions. These geometries, on the other hand, would come with absolute system of directions.

Secondly, in the three standard geometries, all straight lines are equal. Each straight line can be moved to any other straight line by some isometry; here, it is no longer the case, again because the absolute system of directions.

Now, what combinations of a and b are possible?

Well, one option is that you put both as positive numbers or both as negative numbers, just not the same. I haven't looked at these cases closely, but they would probably look like squashed versions of elliptic or hyperbolic geometry. Absolute length would differ based on the direction you'd measure it in. The point is that these planes would differ from elliptic or hyperbolic plane metrically, but not topologically.

But let's have a look at three basic hybrid cases:

1) a=1, b=0 (Elliptic/Euclidean or EP ["P" for "parabolic"])

Here, the projection surface is a cylinder x^2 + z^2 = 1. Antipodal points of this cylinder are identical.

2) a=1, b=-1 (Elliptic/hyperbolic or EH)

Here, the projection surface is a one-sheet hyperboloid x^2 - y^2 + z^2 = 1. Antipodal points are identical. (Here, once again, we could have various other cases by setting a and b to other pairs of positive/negative number.)

3) a=0, b=-1 (Euclidean/hyperbolic or PH)

Here, the projection surface is a hyperbolic cylinder x^2 - z^2 = -1. Since this has two sheets, we don't need to concern ourselves with antipodal points.

When I looked at these, I realized something. These cases fill in some unused options for structure of geometric infinity.

For any projection surface, you can separate the lines through origin (i.e. points of projective plane) into three categories:

REAL points (line intersects the projection surface)

IDEAL points (line doesn't intersect the projection surface but is asymptotic to it, i.e. comes arbitrarily close to it)

ULTRAIDEAL points (line doesn't intersect the projection surface and has some minimum distance from it).

In elliptic geometry, all points are real. Euclidean geometry has infinitely many ideal points that lie on a single ideal line. And all other points are real -- there are no ultraideal points in Euclidean geometry. Hyperbolic geometry has also infinitely many ideal points, but they don't lie on a line: they lie on a conic. Furthermore, real points of hyperbolic plane lie inside that conic; points outside the conic are ultraideal.

These three hybrid geometries, however, have other arrangements.

First, apart from three kinds of points, there are also three kinds of lines. Elliptic lines contain no ideal points; they are closed and finite. Parabolic lines contain a single ideal point, while hyperbolic lines contain two ideal points. Of course, elliptic geometry contains only elliptic lines, Euclidean geometry only parabolic lines, and hyperbolic geometry only hyperbolic lines.

EP geometry has a single ideal point. It has elliptic and parabolic lines. Each pair of lines intersects, except for pairs of parabolic lines which are parallel. This geometry has a weird system of conics: there are two kinds of ellipses (depending on whether the ideal point is outside or inside the ellipse). One has finite inside, which is normal, but the other has finite OUTSIDE. Since the part of the plane outside of the ellipse has no ideal points, it must be finite.

PH geometry has two ideal lines that intersect in a special ideal point. Lines that pass through this intersection are parabolic while all other lines are hyperbolic.

EH geometry is sort of dual of hyperbolic geometry -- its ideal points form a conic like in hyperbolic geometry, but the real points are the ones outside the conic. This means that there are no ideal or ultraideal lines -- all lines must contain some real points. And all three types of lines, elliptic, parabolic and hyperbolic, exist in this geometry. A typical point has two parabolic lines through it, dividing the pencil of lines through the point into elliptic and hyperbolic ones.

What I'm interested in is:

What is the metric of these geometries? The metric must satisfy the condition that straight lines, as projected onto the surfaces, are geodetics.

Do they have absolute positions? Do properties of a point depend on where exactly it is?

Do hyperbolic projections like Klein and Poincare work on these as well?