# curves in Poincare half space (3 dimensional hyperbolic geometry)

Okay maybe I am going a bit ahead of my self

The Poincare half plane still has many mysteries for me

But still I was puzzeling about the 3 dimensional variant of it.

So lets assume an hyperbolic 3 dimensional space.

In this space there are 4 kinds of regular surfaces:

• hyperbolic planes
• hyperbolic Spheres
• Horospheres and
• Hyperspheres (or what is the three dimensional variant of the hypercycle , the set of points equidistant of an hyperbolic plane)

Then lets try to put then into my poincare half- space model $(x,y,z) | z \gt 0$ that represents hyperbolic space. and call the $z=0$ plane the boundary plane

Then if i am not mistaken I think:

1) Hyperbolic planes are:

1a Euclidean planes orthagonal to the boundary plane

or

1b Euclidean halfspheres whose centre is in the boundary plane

2) Hyperbolic spheres are:

2a Euclidean spheres that are fully inside the half space , (Not touching the boundary plane and the hyperbolic centre differs from the euclidean centre)planes orthagonal to the boundary plane

3) Horospheres are:

3a Euclidean planes parallel to the boundary plane

or

3b Euclidean spheres that are tangent to the boundary plane

4 Hyperspheres are

4a Euclidean planes not orthagonal or parallel to the boundary plane

4b Euclidean spheres, that are not fully in the halfplane and whose center is not on the boundary plane.

But then curves and it gets complicated.

5) an hyperbolic line I can imagine is an intersection of two hyperbolic planes so they can be:

5a a line orthagonal to the boundary plane

5b a half circle who's centre in on the boundary plane and who's plane is orthagonal to the boundary plane .

6) an hyperbolic circle is a circle who is completely inside the half space

But then some hyperbolic circles can be constructed in different ways - it can be an intersection of two horospheres - the intersection of a hyperbolic plane and a circle

and here I lose track of it all

But then it gets complicated and i lose track, lets start at the other end what euclidean curves do we have:

7 ) euclidean lines:

7a) an euclidean line normal to the boundary plane --> a hyperbolic line

7b) a line parallel to the boundary plane (this could be the intersection of an horosphere and an hyperplane, or an horosphere and an hyperbolic plane) --> ????

7c) a line orthogonal but not normal to the boundary plane (intersection of a hyperplane, and a hyperbolic plane ) -->> ????

7d) a line in no way orthogonal to the boundary plane (intersection of two hyperplanes) --> ????

8 ) For euclidean circles we have the following options:

• a they are inside the halfplane,
• b there centre is on the boundary plane,
• c they touch the boundary plane ,
• d or they cut the boundary plane.

also there are two orientations:

• 1 they have a radius normal to the boundary plane
• 2 they do not have a radius normal to the boundary plane.

That all combuined give the following options:

8a) euclidean circles completely in the halfspace -> hyperbolic circles

8b) euclidean circles which centre is on the boundary plane:

8b1) euclidean circles which centre is on the boundary plane and that have a radius that is normal to the boundary plane -> hyperbolic line

8b2) euclidean circles which centre is on the boundary plane and that don't have a radius that is normal to the boundary plane -> ????

8c) euclidean circles which touch the boundary plane:

8c1) euclidean circles which touch the boundary plane and that have a radius that is normal to the boundary plane -> horocycle

8c1) euclidean circles which touch the boundary plane and that do not have a radius that is normal to the boundary plane -> ????

8d) euclidean circles which cut the boundary plane:

8d1) euclidean circles which cut the boundary plane and that have a radius that is normal to the boundary plane -> hypercycle

8d1) euclidean circles which cut the boundary plane and that don't have a radius that is normal to the boundary plane -> ????

What are the names of the unnamed curves? (if they have a name)

and are there more special curves or surfaces in hyperbolic space?

What are all possible permutations and how to distinghuis an horocycle from an circle and a hypercycle and so on or are there maybe even more special curves in 3 dimensional hyperbolic geometry)?

• Everything up to 6) sounds fine. Keep in mind, just because a hyperbolic circle "can be constructed" in different ways, that's irrelevant to the fact that a hyperbolic circle is a circle which is completely inside the half space. For that matter, a point in the Euclidean plane "can be constructed" in different ways, as the intersection of many different pairs of lines. – Lee Mosher Mar 28 '15 at 13:42
• Thanks I was wondering if the model as I described it really did model hyperbolic space and even is there only one kind of hyperbolic space? (for the last see en.wikipedia.org/wiki/… , no I am not claiming to understand it) ) I have expanded the question, hope you can take another look. – Willemien Mar 29 '15 at 13:28
• You have not described a model of hyperbolic 3-space, you have only listed some properties. For description of a model see en.wikipedia.org/wiki/Hyperbolic_space. – Lee Mosher Mar 29 '15 at 15:05
• that wikipedia article is not very clear (i was just trying myself to split the section "the poincare models" in two sections (one for the poincare ball and another for the poincare halfspace model , but my knowledge is not (yet) advanced enough to do it – Willemien Mar 29 '15 at 15:59
• I did add sections to en.wikipedia.org/wiki/Poincaré_half-plane_model for the 3 and higher dimensional cases but they are still very stub class – Willemien Mar 29 '15 at 16:01