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In hyperbolic geometry, ideal triangles are all congruent to each other.

Convex Ideal quadrilaterals ( a quadrilateral where all 4 points are ideal points) all have the same area $(2\pi)$

But I think for the rest they are not all congruent for example the angle the diagonals make can be different.

But that made me think of we have two ideal quadrilaterals that have the same angle between the diagonals are they congruent ? Or is there more to concider?

Willemien
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1 Answers1

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I'd suggest thinking about this in the context of the Poincaré disk model. An isometry of the hyperbolic plane is uniquely determined by mapping three ideal points to their images. That's because that isometry is a Möbius transformation, which is uniquely determined by three points and their images. (Actually if the order of ideal points changes, then you's have to compose this with an inversion in the unit circle, because the Möbius transformation would exchange inside and outside.)

So you can map any three points to any three points in an isometric way. Conversely, if you fix three corners, the position of the fourth uniquely determines the shape of the quadrilateral.

So what parameters can you use to describe that shape? The angle between the diagonals does seem like an intuitive choice. Coming from a background of projective geometry, I'd probably pick the cross ratio of the four ideal vertices. You can compute this e.g. as $\lambda=\frac{(a-c)(b-d)}{(a-d)(b-c)}$ with $a,b,c,d\in\mathbb C$ having absolute value $1$. Both approaches have benefits and drawbacks.

One thing worth considering is whether you allow for self-intersecting quadrilaterals. If two edges may intersect, then the diagonals would not intersect. Extending them to infinite lines, one could see them intersect in the Beltrami-Klein model, forming an imaginary angle of intersection. So you may want to forbid self intersection, or allow for complex angles. Cross ratios cover the self-intersecting case using real numbers (and $\infty$ for the special case of two certain points coinciding). To forbid them you could restrict the value of the cross ratio to $1<\lambda<\infty$ for $(a,b,c,d)$ cyclic in that order, or to $0<\lambda<1$ for $(a,b,d,c)$ cyclic in that order.

MvG
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  • Thanks added convex to the question :) I choose the angles between the diagonals because that is model independent (and my dynamic geometry program does not understand complex numbers :( ) did lead to another question on cross ratio:) – Willemien Jul 23 '17 at 09:33
  • @Willemien: Actually the cross ratio is model independent as well. The Beltrami-Klein model agrees with the Poincaré Disk model on the unit circle, and the Poincaré Half Space can be obtained from the Disk via a Möbius transformation which preserves cross ratios as well. As a wider picture, Cayley-Klein metrics express all angles as cross ratios, with Laguerre's formula for the Euclidean case. But that's the cross ratio not of four points but of four lines, two of them complex tangents to the unit circle from the point of intersection. This is where the complex angle for non-convex comes from. – MvG Jul 25 '17 at 20:32