Standard 2D geometries, elliptic, Euclidean and hyperbolic, can be all derived from the same basic idea: start with projective geometry formed by lines and planes through origin in $R^3$ and then put some quadric in its way. $x^2 + y^2 + z^2 = 1$ for elliptic, $z^2 = 1$ for Euclidean, and $x^2 + y^2 - z^2 = 1$ for hyperbolic. This can be rewritten as $1x^2 + 1y^2 + z^2 = 1$, $0x^2 + 0y^2 + z^2 = 1$, and $(-1)x^2 + (-1)y^2 + z^2 = 1$.

I tried to vary the parameters for the projection surface and found some possibilities for hybrid geometries that are no longer isotropic because they contain non-isomorphic lines.

Cylinder $x^2 + z^2 = 1$ leads to a hybrid of elliptic and Euclidean geometry, hyperbolic cylinder $x^2 - z^2 = 1$ to a hybrid of Euclidean and hyperbolic geometry and one-sheet hyperboloid $x^2 - y^2 + z^2 = 1$ to a hybrid of elliptic and hyperbolic geometry.

If we consider elliptic geometry to contain no ideal points, Euclidean geometry to contain an ideal line (line at infinity), and hyperbolic geometry to contain an ideal conic (the absolute), then these three hybrids have a single ideal point, two ideal lines that intersect, and an ideal conic, respectively. The difference between the last one and hyperbolic geometry is that hyperbolic geometry has real points inside the conic and ultra-ideal points outside of it, while the elliptic/hyperbolic hybrid is reversed: its real points are outside the conic.

This is as far as I got -- there should be a way to impose metric on these geometries so that straight lines would be shortest distances between points. I'd be interested in knowing whether someone has studied this further?


2 Answers 2


I would place your question strongly in the realm of projective geometry. There you'd more likely embed the plane at some offset to the origin (conventionally at $z=1$), and instead have your cone have its apex at the origin. This gives you homogeneous coordinates, and also homogeneous equations for the conics. I believe the question you are asking is equivalent to one in that setup.

Using a projective formalism, and following Perspectives on Projecive Geometry and lectures by it's author J. Richter-Gebert, I'd summarize what you describe under Cayley–Klein geometries. Your definition is not capturing the complete picture. To measure angles, you need not only the primal conic, as a set of incident points, but also the dual conic, as a set of tangent lines. For non-degenerate conics, one can simply compute one from the other. But for conics that factor into a single component with multiplicity two, as in the case of $z^2=0$, you need to explicitly state the dual conic. For Euclidean geometry, that conic consists of the ideal circle points. In standard coordinates (homogenized using $(x,y)\mapsto[x:y:1]$) those ideal circle points would be $[1:\pm i:0]$. So they are two points on the line at infinity with complex coordinates. They are incident with every Euclidean circle.

The general recipe for distance measurements in any of these is the following: given two points, draw the line through them. Interact that line with the fundamental conic; these intersections might be complex. Compute the cross ratio of these for points then take the natural logarithm. Apply a cosmetic factor to match conventions, in particular to get real values in common cases. For angles you do the dual procedure: find the point of intersection, then draw tangents to the fundamental conic and compute the cross ratio of these four lines.

Chapter 20.6 of Perspectives classifies geometries based on the signature of the fundamental conic, given as a primal-dual pair.

  1. $(+,+,+),(+,+,+)$ is a purely complex nondegenerate conic leading to elliptic geometry.
  2. $(+,+,-),(+,+,-)$ is a real nondegenerate conic leading to hyperbolic geometry.
  3. $(+,+,0),(+,0,0)$ is a pair of complex lines intersecting in a real point of multiplicity two. This is dual Euclidean geometry, i.e. Euclidean geometry but with the roles of lines and points swapped.
  4. $(+,-,0),(+,0,0)$ is a pair of real lines intersecting in a point of multiplicity two. You can consider this the dual of case 6.
  5. $(+,0,0),(+,+,0)$ is a double line with two designated complex conjugate points, leading to Euclidean geometry.
  6. $(+,0,0),(+,-,0)$ is a double line with two designated real points. This is sometimes called pseudo-Euclidean geometry or Minkowsky geometry. It is also relevant as the geometry of special relativity, since angles add as relativistic speeds do.
  7. $(+,0,+),(+,0,0)$ is a double line with a designated double point. Perspectives calls this Galilean geometry.

For the case 2, it is important to note that you only get traditional hyperbolic geometry if you restrict yourself to points on the inside. Inside in this sense is the set of real points through which every real line will intersect the fundamental conic in two distinct points. In this sense, every conic is equivalent to every other, but the geometry on the outside is different from that on the inside.

You can also classify based on how many real intersections / tangents you get for the measuring process described above. This describes distance and angle measurement each as elliptic, parabolic or hyperbolic. This leads to $3\times3=9$ cases. But three of them are covered by case 2 above, since you get no real tangents for points on the inside, and depending on where the line joining points on the outside goes you may get real intersections or not. For comparison, Euclidean geometry has elliptic angle measure and parabolic distance.

  • $\begingroup$ Number 3 seems to correspond to my hybrid of elliptic and Euclidean geometry (one ideal point). I have also noticed that it's dual to Euclidean geometry since you can swap points and lines. Number 4, similarly, corresponds to the Euclidean/hyperbolic hybrid. And elliptic/hyperbolic hybrid is Number 2 if we restrict ourselves to the points OUTSIDE the conic. Which leaves two more cases that I haven't thought of before. Hmm... $\endgroup$
    – Marek14
    Feb 5, 2019 at 6:18
  • $\begingroup$ One thing that I realized from looking at the CK geometries a bit is that a distance function can be used to describe distance between two points, but also distance between two lines (which is related to their angle). Does that mean that while the standard geometries put constraints on allowed sum of angles in triangle, some geometries could also constrain the minimum or maximum allowed circumference of a triangle? $\endgroup$
    – Marek14
    Feb 5, 2019 at 13:35
  • $\begingroup$ @Marek14: Distance and Angle are dual. Elliptic geometry is self-dual, and the way I see it right now, the perimeter of a triangle there will be less than $2\pi$, same as the sum of exterior angles. Defining interior vs. exterior is hard in elliptic, unless I'm missing something. Case 2 is self-dual as well, so for lines passing the conic the angle sum will satisfy an inequality similar to hyperbolic triangle sum, if you choose a factor to avoid imaginary lengths. And case 3 definitely leads to fixed circumference for any triangle, with the exact length again depending on choice of factor. $\endgroup$
    – MvG
    Feb 5, 2019 at 22:27
  • $\begingroup$ I have pondered it a bit more, read a bit more, and realized what the additional geometries mean. Technically, there are three kinds of projective lines - line with no improper points, which is elliptic and, I presume, has a finite length whatever the actual metric is, line with one improper point, which is parabolic, and line with two improper points, which is hyperbolic and has a proper section and an ultraideal section. If the projective construction contains no parabolic/Euclidean dimension, or just one, it's alright. $\endgroup$
    – Marek14
    Feb 7, 2019 at 15:20
  • $\begingroup$ But once it contains two or more, a special line or a plane arises as an improper object, and this object can be split into multiple categories as well. So, for example, if I have three Euclidean dimensions, the improper points form a plane. I can then give this plane any previously constructed planar geometry. If I make it hyperbolic by creating a conic within it, the resulting 3D geometry will be 2+1 Minkowskian with a special cone defined at each point, dividing spacelike and timelike intervals. $\endgroup$
    – Marek14
    Feb 7, 2019 at 15:23

All right, so what I've learned is that I essentially approached Cayley-Klein geometries. My approach is layman and pretty combinatoric, but I can eventually replicate the geometric structures. The most useful thing I got is the deeper understanding of dualities between various geometries.

So, in two dimensions, we have:

Elliptic geometry (my symbol EE), which is self-dual and whose absolute set is empty.

Euclidean geometry and dual Euclidean geometry (symbols PP;e and EP). Euclidean geometry has line as an absolute set, with no special points on that line. Dual Euclidean geometry has a point as an absolute set.

Hyperbolic geometry and dual hyperbolic geometry (symbols HH and EH). They have a conic as an absolute set and they differ in whether the real points are inside or outside of the conic. They could be considered self-dual or dual to each other. Dual hyperbolic geometry has apparently unavoidable imaginary distances, which means it's more of a spacetime geometry than planar geometry.

Galilean geometry (symbol PP;p). Absolute set is a line with one "lightlike" point. This is basically a spacetime geometry in a world with infinite speed of light. It's self-dual.

Minkowski geometry and its dual (symbols PP;h and PH). Minkowski geometry has absolute set formed by a line with two lightlike points that divide the line into spacelike and timelike portion; the dual has absolute set formed by two lines and their intersection point.

I tried to extend this to more dimensions. This is what I got for three, format is symbol [equation of the quadric surface in $R^4$] (absolute set) ~ dual geometry:

  • EEE [$x^2 + y^2 + z^2 + w^2 = 1$] (empty set) ~ EEE
  • EEP [$x^2 + y^2 + w^2 = 1$] (point) ~ PPP;ee
  • EEH [$x^2 + y^2 - z^2 + w^2 = 1$] (non-ruled quadric with real points outside) ~ HHH
  • EPP;e [$x^2 + w^2 = 1$] (line) ~ EPP;e
  • EPP;p [$x^2 + w^2 = 1$] (line with one lightlike point) ~ PPP;pp
  • EPP;h [$x^2 + w^2 = 1$] (line with two lightlike points) ~ PPH
  • EPH [$x^2 - z^2 + w^2 = 1$] (cone with real points outside) ~ PPP;eh
  • EHH [$x^2 - y^2 - z^2 + w^2 = 1$] (ruled quadric) ~ EHH
  • PPP;ee [$w^2 = 1$] (plane) ~ EEP
  • PPP;ep [$w^2 = 1$] (plane with one lightlike point) ~ PPP;ep
  • PPP;eh [$w^2 = 1$] (plane with lightlike conic, timelike points outside) ~ EPH
  • PPP;pp [$w^2 = 1$] (plane with one lightlike line) ~ EPP;p
  • PPP;ph [$w^2 = 1$] (plane with two lightlike lines) ~ PPP;ph
  • PPP;hh [$w^2 = 1$] (plane with lightlike conic, timelike points inside) ~ PHH
  • PPH [$-z^2 + w^2 = 1$] (two planes) ~ EPP;h
  • PHH [$-y^2 - z^2 + w^2 = 1$] (cone with real points inside) ~ PPP;hh
  • HHH [$-x^2 - y^2 - z^2 + w^2 = 1$] (non-ruled quadric with real points inside) ~ EEH

EPP has a line as an absolute set, and that line can have an additional structure in three ways. PPP has a plane as an absolute set, and there are six ways to introduce lightlike structures into that plane. PPP;hh, for example, leads to a Minkowski space with two spacelike dimensions and one timeline dimensions; each point will have lightlike lines through it that intersect the ideal plane in its lightlike conic, which naturally forms the light cone.

However, what of PPH geometry? I'm not quite sure whether it can be subdivided or not.


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