How many different types of conics exist in hyperbolic plane?
Euclidean geometry has three, of course. But when I was trying to find out results for the hyperbolic plane, the best thing I found freely accessible was this: http://www.mathnet.ru/links/0116e85b4ef8e4fdf19cbe340a1eb771/ivm1975.pdf
My Russian is a bit rusty, though. I can see that this guy found 9 distinct types, but I have no clue how they look or what properties they have.
Can anyone help me, please?
Best Answer
Defining Conic Sections
The first question, of course, is how to even define conic sections on the hyperbolic plane. The usual method is to use the hyperboloid model, which identifies the hyperbolic plane with the hyperboloid $$ x^2 + y^2 - z^2 = -1,\qquad z\geq 1. $$ In this model, isometries of the hyperbolic plane correspond to linear transformations of $\mathbb{R}^3$ that map this hyperboloid to itself.
Using this model, define a conic section in $\mathbb{H}^2$ to be any intersection of the hyperboloid with a set of the form $$ \{\textbf{x}\in\mathbb{R}^3 \mid \textbf{x}^{T}S\textbf{x} = 0\}\tag*{(*)} $$ where $S$ is a nondegenerate $3\times 3$ symmetric matrix. That is, a conic section is any subset of the hyperboloid $x^2+y^2-z^2 = -1$ defined by an equation of the form $$ Ax^2 + Bxy + Cy^2 + Dxz + Eyz + Fz^2 = 0 $$ where $A,B,C,D,E,F\in\mathbb{R}$.
Note that the sets of the form $(*)$ given above are precisely the elliptic doubles cones with a vertex at the origin. (This assumes $S$ is not positive or negative definite, in which case the given set consists only of the origin.) This justifies the name “conic section”. Note also that these conic sections are invariant under isometries of $\mathbb{H}^2$ (since isometries correspond to linear maps), which seems like a good property for conic sections to have.
The Klein Model
We can understand these conic sections better by switching to the Klein model of the hyperbolic plane. In this model, $\mathbb{H}^2$ is the interior of a unit disk, and hyperbolic lines are straight chords of the disk. (This is different from the more common Poincaré disk model, in which hyperbolic lines are circular arcs.) The Klein model can be thought of as the disk $x^2+y^2=1$ in the $z=1$ plane of $\mathbb{R}^3$, which projects onto the hyperboloid from the origin.
Now, here's a few things to know about the Klein model:
It's not a conformal model. That is, the angle at which two chords appear to intersect is different from the actual angle of intersection in the hyperbolic plane.
The isometries of the Klein model are precisely the projective transformations of the plane that map the unit disk to itself.
The reason that the Klein model is helpful is that hyperbolic conic sections in the Klein model are precisely Euclidean conic sections that intersect the unit disk. This has to do with the fact that the Klein disk really lies on the plane $z=1$ in $\mathbb{R}^3$, and the intersection of this plane with the elliptic cones defined above gives conic sections on the plane.
Therefore, understanding conic sections on the hyperbolic plane is precisely the same thing as understanding them on the Euclidean plane up to projective transformations that preserve the unit disk.
The Classification
It appears that the classification of hyperbolic conic sections goes back to an 1882 paper by William E. Story:
Story, William E. “On non-Euclidean properties of conics”. American Journal of Mathematics 5, no. 1 (1882): 358–381.
Story classifies conic sections according to the number and multiplicities of intersections between the conic section and the boundary circle. This results in eight types of conic sections
An ellipse is an ellipse contained entirely within the interior of the unit disk.
A hyperbola is a conic section that intersects the unit circle at four different points. (Such a conic section may be a portion of an ellipse, parabola, or hyperbola in the Euclidean plane, though which of these three types it is may change under hyperbolic isometries.)
A semi-hyperbola is a conic section that intersects the unit circle transversely at two different points. (Again, this may be an ellipse, parabola, or hyperbola in the Euclidean plane.)
An elliptic parabola is an ellipse or circle in the disk that intersects the unit circle at one point of tangency.
A hyperbolic parabola is a conic section that intersects the unit circle three times, with one being a point of tangency.
A semi-circular parabola is a conic section that has the unit circle as one of its osculating circles (i.e. they have a point of third-order contact) and also intersects the unit circle at one additional point.
A horocycle (called a “circular parabola” by Story) is an ellipse in the unit disk that has a fourth-order contact with the unit circle. That is, it is an ellipse that has the unit circle as an osculating circle at one of the endpoints of its minor axis.
A circle is a circle contained entirely in the interior of the unit disk, and an equidistance conic is an ellipse that is tangent to the unit disk at two points. (Story refers to both of these cases simply as “circles”.)
Story mentions that some of these eight types can be further subdivided, and later authors often increased the number of types to 11 or 12. For example, a classification into 12 types can be found on pg. 142 in the following book:
Liebmann, Heinrich. Nichteuklidische geometrie. Vol. 49. GJ Göschen, 1905.
I don't read German, so I can't confirm this, but this classification is reproduced in English on pg. 257 of the following book, which also has some nice figures on the following page:
Rosenfeld, Boris, and Bill Wiebe. Geometry of Lie groups. Vol. 393. Springer Science & Business Media, 2013.
In particular, Rosenfeld and Wiebe (presumably following Liebmann) make the following distinctions:
They further classify Story's “circles” into true circles and equidistant conics.
They call a hyperbola “concave” if it has four common tangent lines with the unit circle in the Euclidean plane, and “convex”
They classify hyperbolic parabolas into three types depending on the number of branches and the number of common tangent lines in the Euclidean plane.
This results in a total of 12 types. This classification has the advantage that it is symmetric with respect to projective duality, since it uses both the intersection points and the common tangent lines.