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Home , Hyperbolic geometry, Reflection (mathematics)

We determine the motions in the hyperbolic that preserve distances

In regular Euclidean , there are four kinds of actions that preserve distances. That is, there are three things we can do that keep distances between objects the same. These actions are called isometries and one way of thinking about geometry is the study of what kinds of actions preserve various geomtrical quantities. In , the three kinds of isometries are , , reflection, and glide reflection. We can prove that these are the only kinds of isometries as follows. Given a 4ABC if we perform an to the plane it will turn into another triangle 4A0B0C0. There is a translation that moves A to A0, then a rotation that will rotate B onto B0 while leaving A0 alone, and finally, if needed, a reflection that will put C on top of C0. Once these points are fixed, all other of the plane are fixed. So every isometry can be made up from the three types. We didn’t even need glide reflections, because they can be made from the other types. But we’ll keep them around for convenience. Combining any two of the four types gives another of the four types, as listed below:

• Two translations compose to another translation.

• Translation followed by reflection or vice versa is a reflection (in a different axis) or glide reflection (if axis is to translation).

• Translation followed by rotation or vice versa is a rotation (in a different center).

• Translation followed by glide reflection or vice versa is a reflection, unless the transla- tion and glide are in the same direction in which case you get a glide reflection.

• Two rotations compose to a rotation, unless they are rotations by the same in opposite directions, in which case you get a translation.

• A rotation composed with a reflection or with a glide reflection is a reflection, or possibly a glide reflection.

• Two reflections composed yield a rotation, unless they are in parallel lines of , in which case you get a translation.

• A reflection composed with a glide reflection or vice versa is a rotation, or a translation if the reflection is parallel to the glide.

• Two glide reflections composed yield a rotation, or a translation if the glides are par- allel.

It turns out that we can also create any isometry from a composition of at most three reflections. For the isometry is determined by its effect on a triangle, so use each reflection to move one of the triangle to its target location. Now, what are the isometries of ? We’ll start with an easy one.

1 Theorem Horizontal translation is an isometry. PR · QS Proof: Remember the distance formula | ln(y /y )| for vertically aligned points, and | ln( )| 2 1 QR · PS for non-vertically aligned points P and Q, where R and S are the endpoints of the semicircle through P and Q on the x-axis. Since none of these Euclidean distances change as we move purely hoizontally, hyperbolic distances between points do not change.

Here’s an odd surprise—an isometry of hyperbolic geometry that is definitely not one for Euclidean:

Theorem Dilation (x, y) 7→ (kx, ky) for some positive constant k is an isometry of hyper- bolic geometry. Proof: in the distance formulas, all distances get multiplied by k. But then all the k’s cancel out!

Dilation is really just translation along the y-axis! Since the y-axis has an unusual distance on it, this is what will cause things to move parallel to the y-axis and in equal distances with each application. Combining with horizontal translations, we can now get a translation in any direction.

Theorem Inversions in are isometries. Proof: Inversion in the y-axis sends (x, y) to (−x, y). Other vertical lines behave similarly. Semicircular geodesics have inversions. That the vertical reflection is an isometry should be obvious. For the circle inversion, keep in mind all the similar involved in the inversion process and note that all the factors will cancel out in the hyperbolic distance formula. See the diagram:

In the diagram we are inverting in the solid circle; call its center O. Our points are P and Q, which invert to P 0 and Q0. Points R and S are the points where the semicircle through P and Q meet the x-axis, and similarly for R0 and S0. We wish to show that P 0R0 · Q0S0 PR · QS | ln( )| = | ln( )|. But from the similar trianlges that come from inver- Q0R0 · P 0S0 QR · PS

2 sion, P S/P 0S0 = OS/OP 0, QR/Q0R0 = OR/OQ0, P R/P 0R0 = OR/OP 0, and QS/Q0S0 = P 0R0 · Q0S0 PR(OP 0/OR)QS(OQ0/OS) PR · QS OS/OQ0. So = = as desired. Q0R0 · P 0S0 QR(OQ0/OR)PS(OP 0/OS) QR · PS

How do these isometries combine? As in Euclidean, everything is a combination of reflections. Here are the combinations: • One reflection/inversion is an inversion.

• Two reflections through geodesics that meet at a point P are a rotation around P . Points move around the hyperbolic centered at P (which are Euclidean circles that are not centered at P !) in fixed angular motions—where the have vertex at P and consist of hyperbolic sides!

• Two reflections through geodesics that do not meet gives a translation. Given any two such geodesics, there is a unique to both of them. Points on this geodesic are translated along it. Points not on this geodesic move in hypercycles parallel to this geodesic.

• Two reflections through geodeiscs that meet only at the x-axis result in a horolation which is like a rotation around the point at infinity. Points move in that are to the x-axis at the common point of the geodesics.

• Combinations of any two of the above give another of the above with one exception. If you reflect in three geodesics, two of which are perpendicular to the third, you get a glide reflection. A cool way to look at all this is with complex numbers. If we are looking at the entire complex plane, we can easily do isometries by simple operations on complex numbers! Translation is simply adding a fixed complex number: z 7→ z + b where b is a complex constant. Reflection in the x-axis is accomplished by z 7→ z, the complex conjugate of z. Dilation from the origin is simply multiplying by a positive real number; if you want to dilate around some other point you can first translate that point to the origin, then dilate, then translate back. Rotation about the origin is accomplished by multiplying by cis(θ). If you want to rotate about some other point, you can do the translate-rotate-translate-back trick. Inversion through the is accomplished by first taking z 7→ 1/z and then taking the conjugate. Other centers can be accommodated via the translation trick, and other radii by simply dilating. So all of our operation can be accomplished by combinations of multiplying, adding, reciprocating, and taking conjugates. The most general thing you can do is then to send az + b a complex number z to , where a, b, c, and d are complex constants. This is called cz + d a linear fractional transform (because it is a fraction of two linear functions) or a M¨obius transformation. In a M¨obiustransformation, the combination ad − bc cannot equal zero, otherwise az + b b adz + bd b adz + bd b = = = is a boring constant. Because of this, it is often cz + d d bcz + bd d adz + bd d

3 √ convenient to divide each of the constants by ad − bd. (Remember that we are working with complex numbers, so this root always exists!) That way, ad − bc = 1. This is called normalizing. Now these transformations are a little bit too general. For any three complex numbers, there is such a M¨obiustransformation that will take them to any other three complex num- bers. For hyperbolic geometry, we need to make sure that the x-axis stays in place, and that the points in the upper half-plane stay there. That means we will need to find restrictions on the the coefficients a, b, c, and d that will cause this to happen. One thing we can do is plug in z = 0. Since this is real, it must go to a real number. So b/d must be real. Similarly, by plugging in a very large real number for z, we find that a/c must also be real. So we could say b = k1d and a = k2c. Substituting these and plugging in k c + k d (k − k )c z = 1 now tells us that 2 1 = k + 2 1 must be real. So c/(c + d) must be real. c + d 2 c + d Snce we can multiply all of a, b, c, and d by any constant without changing the , we can assume that c is real. But then a = k2c is real, and d and finally b must all be real. So to make the x-axis go to the x-axis, we can choose all the coefficients to be real. Further analysis will require ad − bc > 0√ if we want the upper half-plane to get sent to the upper-half plane. Thus, we can divide by ad − bc and everything stays real. We have proved:

Theorem a M¨obiustransformation is an isometry of the upper half-plane hyperbolid plane if and only it has all real coefficients and ad − bc = 1. It might be easier to deal with the M¨obiustransformation in terms of a . Make the  a b  az + b matrix correspond to the M¨obiustransformation . Amazingly, combining c d cz + d az + b transformation works by multiplying matrices! If you first take z and send it to , and cz + d ez + f then send the result of that to you get gz + h

e az+b + f e(az + b) + f(cz + d) (ea + fc)z + (eb + fd) cz+d = = az+b g(az + b) + h(cz + d) (ga + hc)z + (gb + dh) g cz+d + h while the matrix multiplication gives

 e f   a b   ea + fc eb + fd  = g h c d ga + hc gb + hd

So the matrices combine just like the M¨obiustransformations do. Neat! Notice that these are real matrices with one, in case you have studied these to any extent in another class. As long as the transformation has been normalized, it is also easy to tell what it is doing. You really need some advanced theory of matrices to prove this, but we can just state the results. Look at the trace a + d of the matrix (or the transformation). If it is: • equal to 2, you have a translation.

• less than 2, you have a rotation.

4 • more than 2, you have a dilation (which, remember is like translating in the y-direction.

One last thing: the matrices and complex number stuff cannot do reflections or inversions. They must preserve orientation. If there is a little looping arrow going counterclockwise, it will be transformed into another counterclockwise arrow. Reflections and inversions reverse the direction. The operation z 7→ 1/z is a combination of inversion followed by a reflection across the x-axis. A true reflection would involve the complex conjugate.

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