Hybrid geometry?

[Marek14]

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?

[wendy]

Projective geometry is really interesting if you call back up the projection.

Suppose you have a sphere S, and a plane M, such that M does not cross S. We now insert a point U representing the 'point at infinity'.

Any circle on S represents an isocurve. (circle, horocycle, bollocycle), depending on whether it intersects the horizon zero, once, or twice. The horizon itself is defined by tangents from S passing through U. When U is on S, the tangent is a plane and you get a 'point at infinity'. When U is above S, then the tangent is a cone, and you get a circle-horizon on S. When U is inside, then there are no tangents, and hence no horizon.

The projective plane is M. Straight lines are planes that pass through U, evidently intersect with M as a line. The horizon in the euclidean space is the tangent at U, which when U is not polar to M (ie U to the centre of S, strikes M perpendicularly), then the result is a projective 'line at infinity'. The hyperbolic infinity is a conic of some kind or another.

Parallelisms can be read as, there is through any point P, a single parallel isocurve that matches two other point-conditions. One 'point-like' condition is that the line is straight. In the space described above, the parallelisms are set by lines. With the exception of the line itself, there is exactly one plane through a line and a point.

If the parallelism is of several straight lines, then every line is straight. This is because the point U is on that line. Note we include in this set, the crossing lines at P, because the line of parallelism is the line through P and U. This evidently is in the line at infinity, and thus every set of lines in the projective plane M, cross where PU crosses M.

Note that parallelisms that do not contain U, contains one straight line. The tangent-plane to S, other than U, contains a series of co-tangential circles, that pass through P. The parallelism that passes through two points P and Q, contain the set of isocurves that pass through P and Q. The parallelism that produced by tangent-planes from P and from Q, produce a set of curves that are as the potentials of dipoles. When P and Q are antipodal, and U is in the centre, the first produces lattitude lines, the second longitude lines. The inner and outer parallelisms of P and Q are sets of orthogonal curves in every instance. This is true when P=Q as well.

The projection runs from U to M, which means that circles on S (being isocurves), project as conics on M.

We next notice, that the astronomers do not know whether the space is curved positively or negatively, but it's near enough to zero. This can be represented by replacing U with a small sphere E, set crossing the surface of the sphere S.

[mr_e_man]

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.

And that's a specific case of a more general equation! Ax²+Bxy+Cy²+Dxz+Eyz+Fz²=1. The expression on the left is a generalized dot product ("symmetric bilinear form") applied to the position vector r=xe₁+ye₂+ze₃. But we can always find a coordinate system that diagonalizes the dot product; we can rewrite r=uf₁+vf₂+wf₃, so that r.r=au²+bv²+cw². Furthermore, the u coordinate can be divided by sqrt(|a|) if a=/=0, and likewise for the other two. So without loss of generality, each of a,b,c is 1 or 0 or -1.

See also this.

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.

Already for the hyperbolic plane, we use a non-Euclidean metric for the surrounding space. So in general, I think we ought to use a metric that makes the surface non-squashed. Note also that the set of planes through the origin is preserved by any linear transformations that un-squash the surface, so the transformation of a geodesic is still a geodesic.

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.

...I assume a "conic" is the intersection of the surface with a pair of parallel planes. Aren't the two types of ellipses essentially the same, only differing in a choice of orientation? Couldn't we consistently orient them so that the point at infinity is outside?

Anyway, the natural "metric" here is r.r=x²+z², which becomes ds²=dθ² in polar coordinates on the cylinder. (It's technically not a metric, because y doesn't appear; it's degenerate.) This is preserved by rotations x'=x cos α - z sin α, z'=x sin α+z cos α, and by shear transformations y'=y+αx.

So an ellipse (a plane cutting the cylinder at an angle) can be sheared into a circle y=0. It's clear that the acceleration d²r/dθ² is always orthogonal to the surface; this is one definition of geodesics.

And a line (a plane cutting the cylinder lengthwise like x=0) has acceleration 0, so it's also a geodesic.

One problem is that a helix, or indeed any curve with dθ=/=0, is a geodesic by this definition. An ellipse and a helix may share the same starting point and tangent line (initial position and velocity). This is because the metric is degenerate; the curves drift apart in the y direction.

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?

The geodesics do not determine the metric. See here, here. But maybe you can find some metric that satisfies the constraints (and is non-degenerate).

If we do use the degenerate metric, then all points are equivalent. On the cylinder, any point can be sheared and rotated to any other point isometrically. The other surfaces are similar.

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

These are special cases of ordinary perspective projection. This doesn't essentially depend on the dot product (unless we want to make the camera rigid). We can choose any focal point f, and any independent set of vectors {v₁,v₂,v₃}, and ray-trace along them to any point in space p=f+xv₁+yv₂+zv₃, then take the projected point to be p'=(x/z, y/z).

The Beltrami-Klein projection has focus at (0,0,0). The Poincare projection has focus at (0,0,-1).

[Marek14]

No, conic is not an intersection with pair of parallel planes. Instead, conic seems to be an intersection of a cone (circular or elliptic, with apex in the origin) with the projection surface. If it was parallel planes, you couldn't encircle a finite part of the cylinder inside a conic and so you couldn't draw, say, a circle around a point.

I managed to extend it to three dimensions -- 10 combinations are possible, in general:

EEE - elliptic space, all points are real, all lines and planes are elliptic.
EEP - there is one ideal point. Lines are elliptic or parabolic, planes are elliptic or EP.
EEH - ideal points form a non-ruled quadric, real points are outside, ultraideal points are inside. Lines can be elliptic, parabolic or hyperbolic, planes can be elliptic, EP or EH.
EPP - ideal points form a line. Lines are elliptic or parabolic (plus the ideal line), planes are EP or Euclidean.
EPH - ideal points form a cone, real points are outside, ultraideal points are inside. Lines can be elliptic, parabolic, hyperbolic, wholly ideal or ultraideal with a single ideal point, planes can be EP, EH, Euclidean or PH.
EHH - ideal points form a ruled quadric, real points are on one side, ultraideal points on the other (inside and outside is isomorphic). Lines can be elliptic, parabolic, hyperbolic, wholly ideal, wholly ultraideal or ultraideal with a single ideal point, planes can be EH, PH or hyperbolic.
PPP - Euclidean space, ideal points form a plane. All lines are parabolic or ideal, all planes are Euclidean (plus the ideal plane).
PPH - ideal points form two planes with two quadrants being real and two being ultraideal. Lines are parabolic or hyperbolic. Planes are Euclidean or PH.
PHH - ideal points form a cone, real points are inside, ultraideal points are outside. Lines are parabolic, hyperbolic, wholly ideal, wholly ultraideal or ultraideal with a single ideal point, planes are PH, hyperbolic, or wholly ultraideal with a single ideal point.
HHH - ideal points form a non-ruled quadric, real points are inside, ultraideal points are outside. All lines and planes are hyperbolic.

[mr_e_man]

No, conic is not an intersection with pair of parallel planes. Instead, conic seems to be an intersection of a cone (circular or elliptic, with apex in the origin) with the projection surface. If it was parallel planes, you couldn't encircle a finite part of the cylinder inside a conic and so you couldn't draw, say, a circle around a point.

The part of the cylinder x²+z²=1 between the parallel planes y=Ax+1 and y=Ax-1 is finite. It can be considered the inside of the curve made of the "two" ellipses. It's actually just one ellipse, because of the projective identification r = -r.

But I shouldn't have made the parallel planes assumption, because it's obviously not general enough for the case of plane geometry z²=1.

[Marek14]

I got some more information. Apparently, these structures are Cayley-Klein geometries. They are classified a bit differently (for example hyperbolic geometry and what I labeled the EH type are basically the same thing, just switching real and ultra-ideal) and there are some more types, which got me a bit of thinking to understand, but I think I got it.

Let's start with a line. There are three possible types of lines: elliptic line which has no ideal points, parabolic (Euclidean) line which has one, and hyperbolic line which has two -- and these two ideal points split it into two parts, one real and one ultra-ideal.

In 2D, we get elliptic geometry with empty set of ideal points, then hyperbolic geometry where ideal points form a quadric, EH plane that is dual to it (outside of the ideal quadric).
Then we get the EP geometry with one ideal point (this is dual to Euclidean geometry, so theorems can be carried over in some way; apparently, the triangles in this geometry have all the same circumference?) and the PH geometry with two straight lines. That one point in EP geometry can be also understood as two imaginary lines intersecting in a real point.

And then we get to Euclidean geometry that has a line as its ideal set (absolute figure), and here's where it gets interesting. Because that line itself can have one of the three structures I mentioned before.

If the line is elliptic, we get normal Euclidean geometry which is isotropic and has no preferred direction. But if the line is hyperbolic, we get pseudo-Euclidean geometry which is in fact a 2D version of Minkowski spacetime with one spatial and one temporal dimension. Through each point in this plane, you can draw two special lines that intersect the ideal line in the two special points. That means that any two points will be in spacelike relation (if the line connecting them intersects the ideal line in its "ultra-ideal" portion), timelike relation (if the line intersects it in "real" portion) or lightlike relation (if the line intersects it in one of the "ideal" points).
And if the line is parabolic, then we get Galilean geometry which is spacetime with infinite speed of light. There is just one preferred direction at every point.

And I think this can be generalized into more dimensions. In 3D, if you have two parabolic dimensions, there will be an ideal line that can be given one of the three structures -- and if you have three parabolic dimensions, you end up with ideal plane that can be given any of the plane geometries.