## Hyperbolic conics

Higher-dimensional geometry (previously "Polyshapes").

### Hyperbolic conics

Here's a question that ocurred to me today: How many distinct conics exist in hyperbolic plane?

When you intersect the circle of Poincaré projection with a line or a circle, you'll get four categories of curves; circles, horocycles, pseudocycles and straight lines (which are a special case of pseudocycles).

Analogically, I guess that general hyperbolic conics would be made by intersecting the horizon circle with an Euclidean ellipse, parabola or hyperbola and that the only important thing is the number of intersection points with the horizon.

That would mean:

0. Ellipse inside the horizon -- ellipse analogue
1. Ellipse touching the horizon from inside -- horoellipse?
2. Ellipse, parabola or hyperbola intersecting the horizon in two points -- parabola analogue
3. Ellipse, parabola or hyperbola intersecting the horizon in two points and touching it from inside in a third -- not sure how to call this
4. Ellipse, parabola or hyperbola intersecting the horizon in four points -- hyperbola analogue

What would be the properties of these curves? Curve 2 would have a single "asymptote", line joining its two horizon points. So would the curve 3. Curve 4, on the other hand, would have 6 distinct asymptotes joining all pairs of the four horizon points. How could the foci of these curves be defined and what properties would they have?

EDIT: Just realized there's one extra type: Ellipse touching the horizon from the inside in two points.
Marek14
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### Re: Hyperbolic conics

OK, so I asked around and read some articles. It looks like the "hyperbolic conics look like Euclidean ones in projection" hypothesis holds only in Klein projection, not in Poincaré.

This went better after I realized the duality of points and lines -- every point outside the horizon has a dual line that intersects it and every line that doesn't intersect horizon has a dual point inside it.

So, the conics are classified by type and multiplicity of their intersections with the horizon. There are always four intersections, but some of them can be multiple or imaginary.

Four distinct imaginary intersections -- ellipse.
Two of double imaginary intersections -- circle.
Two distinct real intersections, two imaginary intersections -- semihyperbola.
One double real intersection, two imaginary intersections -- elliptic parabola.
Four distinct real intersections -- hyperbola; this is further subdivided into convex hyperbola where the region between branches lies "outside" (you can construct tangents from the points in this region), and concave hyperbola which allows construction of tangents from regions outside the branches.
One double real intersection and two other real intersections -- hyperbolic parabola; this is further subdivided into convex hyperbolic parabola with one real branch and an ideal point outside this branch, 1-branch concave hyperbolic parabola with one real branch and an ideal point inside this branch and 2-branch concave hyperbolic parabola with two branches meeting at an ideal point.
Two double real intersections -- equidistant conic, representing both equidistants with a given distance from a line.
One triple real intersection and one other real intersection -- osculation parabola.
One quadruple real intersection -- horocycle.

The original (as far as I know) article, by William E. Story from 1883, goes into discussing huge amount of special points and lines for the general conic, but this can be simplified. Each special point is dual to one special line, and only one of these is inside the horizon. I'll look in this further.
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### Re: Hyperbolic conics

I'm not sure exactly how to approach this, because i don't necessarily think the 'conics' and 'cone sections' are the same thing there.

The ellipse that is tangential on a the horizon is most likely a parabola, it would necessarily be bitangential on one of the inscribed horocycles (ie a circle tangential on the same point and one other.

I still struggle to make of an ellipse bi-tangential twice on the large circle. For the diametric case, the two arcs get further apart, like )(, but still retain their parabolic nature. In other words, it's a kind of parabola with a line, rather than a point, as a focus. Such is mootly.

You can draw a non-tangential ellipse to a circle, such that there is, for example, no line within it. This, and what the diametric case from the previous example might give, suggests to me that the poincare projection is not the go, but the beltrami klein might be.

Still, it could help if someone found a piccie of what hyperbolae and parabolae and ellipses might look like under inversion.
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### Re: Hyperbolic conics

wendy wrote:I'm not sure exactly how to approach this, because i don't necessarily think the 'conics' and 'cone sections' are the same thing there.

The ellipse that is tangential on a the horizon is most likely a parabola, it would necessarily be bitangential on one of the inscribed horocycles (ie a circle tangential on the same point and one other.

I still struggle to make of an ellipse bi-tangential twice on the large circle. For the diametric case, the two arcs get further apart, like )(, but still retain their parabolic nature. In other words, it's a kind of parabola with a line, rather than a point, as a focus. Such is mootly.

You can draw a non-tangential ellipse to a circle, such that there is, for example, no line within it. This, and what the diametric case from the previous example might give, suggests to me that the poincare projection is not the go, but the beltrami klein might be.

Still, it could help if someone found a piccie of what hyperbolae and parabolae and ellipses might look like under inversion.

I was directed to the pictures of the conics in the book "Geometry of Lie Groups" by Rosenfeld. The doubly-tangent ellipse is simply a representation of the two equidistants to a line in Klein projection. I played a bit with the Cinderella program and found that a circle with center outside of horizon displays as such equidistant conic. Apparently, equal distance (complex) from an ultra-ideal point equals equal distance from its polar line.
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### Re: Hyperbolic conics

Well, the foci of the conics look pretty fun.

So, each hyperbolic conic has four "absolute tangents"; these are Euclidean lines tangent to both the horizon (or absolute) and to the conic itself. These absolute tangents can be real or imaginary.
The four absolute tangents intersect in six points (three pairs) which are the foci of the conic. If a focus lies outside the horizon, then we can consider its polar line instead, the directrix.

I think that the foci have special reflective properties. In Euclidean ellipse, a ray sent from one focus will reflect into the other. In hyperbola, it will reflect, as if it was coming from the other focus. In both cases, a tangent to the curve at a point is an axis of some angle between straight lines containing that point and first, resp. second focus.

If this holds in hyperbolic plane, then a conic would reflect rays in various ways.

Double real focus: circle
Any ray from the center will be reflected back to the center.

Two real foci: ellipse or convex hyperbola
These would work just like the Euclidean case, ellipse sending all rays from one focus to the other and convex hyperbola sending rays from one focus as if they were coming from the other.

One real and one ideal focus: elliptic parabola and convex hyperbolic parabola.
In elliptic parabola, rays from the real focus will be reflected towards the ideal focus, along one of convergent parallels.
In convex hyperbolic parabola, rays from the real focus will be reflected as if they were coming from the ideal focus.

One real focus and one directrix: semihyperbola
Rays from the focus will be reflected perpendicular to the directrix. The directrix will be perpendicular to the axis of semihyperbola, but it might have different positions. If it passes through the semihyperbola, some rays reflected from the focus will converge before passing the directrix and starting to diverge. But it might also lie entirely outside the semihyperbola, in which case the rays from focus will always diverge after reflection.

One double ideal focus: horocycle
Any ray from the focus will rebound back into it.

Two ideal foci: equidistant
Any ray from one ideal focus will be reflected to the other one. This suggest that an equidistant-shaped mirror would be very useful for distance optical communication as it would guide any ray towards its destination with only one rebound! So an equidistant conic has 2 independent focus systems.

Ideal focus and directrix: 2-branched concave hyperbolic parabola
Rays from the ideal focus will reflect perpendicular to the directrix.

One double directrix: equidistant
Apart from "two ideal foci", equidistant can be also considered to have a double directrix (the line it's equidistant from). Any line perpendicular to it will be also perpendicular to the equidistant, so it will reflect back.

Two directrices: here, ray perpendicular to one directrix will be reflected in a direction perpendicular to the other.
The directrices can intersect (concave hyperbola), be convergent parallels (1-branched concave hyperbolic parabola), or be ultraparallels (concave hyperbola). In case of concave hyperbola, there seem to be three focus pairs, one with intersecting directrices and two with ultraparallel ones.

The only case not explored yet is the osculation parabola. We can imagine this as a limit case of 2-branched concave hyperbolic parabola, where we keep position of one branch and move the other branch to infinity. This conic has an ideal focus and a directrix like 2-branched concave hyperbolic parabola, but it's special: the directrix actually passes through the ideal focus.
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### Re: Hyperbolic conics

Now what I'm interested in is "inside" definitions of these conics.

For example, Euclidean ellipse can be defined as set of points with given sum of distances from two points. A hyperbola is a set of points with given difference of distances from two points (let's say square of the difference to eliminate signs). A parabola is traditionally defined as a set of points with the same distance from a point and from a line -- this can be rephrased as that the difference of these distances is zero.

Are there similar definitions in hyperbolic plane? At least for horocycle, nothing comes to mind (as a horocycle is a completely scale-free object: any two are congruent).
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### Re: Hyperbolic conics

Here's another question:

I know that the circumference of a circle is 2*pi*sinh(r). This is analogical to spherical geometry, where circumference of a circle is 2*pi*sin(r) (if circumference of the whole sphere is 2*pi).

The area is similarly found by integration from 0 to r, so it's 2*pi*(cosh(r) - 1) for hyperbolic plane and 2*pi*(cos(x) - 1) for spherical plane.

Can I use this goniometric/hyperbolic symmetry for length of the equidistant?

Obviously, equidistant is infinite, but I'm interested in this: given a line segment of length a, what is length of part of an equidistant belonging to it at distance b?

On sphere, equidistant to a straight line at distance b is a circle with radius pi/2 - b. Its circumference is 2*pi*sin(pi/2 - b) = 2*pi*cos(b). As length of straight line is 2*pi, length of equidistant belonging to a segment of length a is a*cos(b). Area between the segment and the equidistant would be a * sinh(b).

So can I consider length of part of equidistant, similarly, to be a*cosh(b)?

This would mean that if I compare length of arc of constant-curvature curve with length of "parallel" curve at some distance x, their ratio should converge to e^x for large circles and large equidistants, and presumably should be exactly equal to e^x for horocycles.
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### Re: Hyperbolic conics

Are there cartesian equations for these hyperbolic conics? Or would they be in a different coordinate system?
in search of combinatorial objects of finite extent
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### Re: Hyperbolic conics

ICN5D wrote:Are there cartesian equations for these hyperbolic conics? Or would they be in a different coordinate system?

Well, in Klein projection, their equations are the same as equations of Euclidean conics -- you just take the part that is inside the unit circle.
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### Re: Hyperbolic conics

Spent some more time researching conics and quadrics in various articles. The Poincare disk images can be obtained by replacing x with "2 x/(1 + x^2 + y^2)" and similarly for other coordinates and/or dimensions.

So, for example, a center conic in Klein projection is a * x^2 + b * y^2 = 1, where various types of conics are obtained by putting different values for a and b. Poincare-projection equation of the same conic is 4 * (a * x^2 + b * y^2) / (1 + x^2 + y^2)^2

a > b > 1: Ellipse
a = b > 1: Circle
a > 1, b = 1: Equidistant
a > 1, 0 < b < 1: Concave hyperbola
a > 1, b = 0: Two ultra-parallel lines
a > 1, b < 0: Convex hyperbola
(Other combinations contain no real points.)

Putting 0 on right side gives you two concurrent lines / point / imaginary circle/ellipse.

For parabolas, the master equation is a * x^2 + (b + 1)*y^2 - 2 * y = b - 1, with Poincare version 4 * (a * x^2 + (b + 1) * y^2) / (1 + x^2 + y^2)^2 - 4 * y / (1 + x^2 + y^2) = b - 1. The real conics are:

b > a > 0: One-branch concave hyperbolic parabola
a = b > 0: Horocycle
a > b > 0: Elliptic parabola
a = 0, b > 0: Line (plus second line that touches the Absolute)
a < 0, b > 0: Convex hyperbolic parabola
a < 0, b = 0: Two convergent parallel lines
0 > b > a: Two-branch concave hyperbolic parabola

Then there's the semihyperbola equation a * x^2 + 2 * b * y^2 - 2 * y = 0. This requires abs(b) < 1 to yield semihyperbola.

Finally, the equation of osculating parabola is a bit complicated (1 - x^2 - y^2) + 2 * a * y * (x + 1) = 0.

All of these equations except the last one have an obvious generalization into higher dimensions, yielding:
Ellipsoids/hyperboloids with center and three planes of symmetry, which cut them in ellipses/hyperbolas. Cones also have a center, but they pass through it and planes of symmetry cut them in lower-dimensional cones.
Paraboloids with two planes of symmetry which cut them in parabolas.
Semihyperboloids with two planes of symmetry which cut them in semihyperbolas.
And probably some strange beast without axes of symmetry that features osculating parabola somewhere.

For example: a generic equation of paraboloid is a * x^2 + b * y^2 + (c + 1)*z^2 - 2 * z = c - 1. If you put a = 2, b = 1, c = 1, you'll get a horocycle for x = 0 and elliptic parabola for y = 0. If you track this shape along the z coordinate (cutting it by planes perpendicular to z axis), you'll start with a point that changes into an ellipse. The short axis of the ellipse will grow to approach a constant (corresponding to how elliptic parabola grows wider and wider but never surpasses certain limit width), while the long axis will grow beyond any limits (corresponding to how line can cut a horocycle in a pair of arbitrarily distant points). In the limit, the cuts approach an equidistant curve.
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