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Consider the disc model of hyperbolic plane $\mathbb{D}^2$ and a line $g$ through the origin $(0,0)\in \mathbb{D}\subset\mathbb{C}$, i.e. a diameter of the circle $\partial \mathbb{D}=S^1$. Let $P\notin g$ be a point not on the line $g$. I want to construct the perpendicular to the line $g$ which contains $P$. How can I Construct (draw) this on a sheet of paper? Is there a compass and straightedge construction?

I'm interested in such kind of constructions in hyperbolic plane, in particular in the upper half plane model and the disc model. Does anyone knows a good book which explains basic constructions in hyperbolic plane like above?

Hopefully someone can help me, best wishes

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2 Answers 2

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In the case of the Klein model many constructions can be done in Euclidean way. For instance if a hyperbolic line $p$ goes through the "origin" then from a point $P$ not on $p$ dropping a Euclidean perpendicular from $P$ to $p$ will be perpendicular to $p$ in the hyperbolic sense as well.

If the line $p$ does not go through the origin then still exists an Euclidean construction as shown in the figure below:

enter image description here

I hope that the construction is clear. (You have to construct the tangents and then join P with the intersection of the tangents.)

However, these are misleading constructions in a certain sense. At least these constructions derail your hyperbolic intuition whose development would be the goal of studying hyperbolic geometry. Let alone that even a hyperbolic circle (being an ellipse in the Euclidean sense in the Klein model) cannot be constructed with Euclidean methods, (Except if the center of the circle is in the center of the Klein model.) Note that the "center" of the Klein model is one of the most misleading concepts given that the hyperbolic plane does not have a center. So watch out when you try to help yourself in understanding hyperbolic concept by means of Euclidean methods.

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For the Poincare disk model:

( https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model )

Point $P'$ is the inversion point of point $P$ with relation to the boundary circle.

( see https://en.wikipedia.org/wiki/Inversive_geometry#Inverse_of_a_point )

Point $M$ is the midpoint of $P$ and $P'$

Line $m$ is the line perpendicular to $PP'$ through $M$

where $m$ and (lengthend) line $g$ meet is the centre of the circle that contains the arc you are looking for.

When line g is just a hyperbolic geodesic,then it boils down to the euclidean problem:

"Given two orthogonal circles $C_1$ and $C_2$ construct a circle $C_3$ through a point $P$ which is orthogonal to both"

and that you can do by:

Draw line $l$ through the points where $C_1$ and $C_2$ intersect.

Point $P'$ is the inversion point of $P$ with relation to one of the circles.

Point $M$ is the midpoint of $P$ and $P'$

Line $m$ is the line perpendicular to $PP'$ through $M$

where $m$ and line $l$ intersect meet is the centre of the circle $C_3$.

(did the last 3 lines sound familiar :)

For more klein disK constructions:

I am updating the webpages of wikipedia on both disk models and will be adding some constructions as well (but feel free to add them yourself as well :)

Enjoy

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