Hyperbolic rotation

A hyperbolic rotation is a transformation of the hyperbolic plane which has only one fix point. A hyperbolic rotation is the composition of two reflections which cut in the fix point.

We can find the hyperbolic rotation of a point from the tools just selecting two points of a hyperbolic line. To construct the tool for obtaining the hyperbolic reflection of a point, we first plot the hyperbolic line and then we use the inversion tool to plot the image of the point, that's to say, the hyperbolic reflection of the point.

To create the tool we mark the fix point P and the rotation angle. Then, we follow these steps:

    (1) Construct the hyperbolic circle with center P and point, the point we want to transform, Q. The image point must lie in this hyperbolic circle.
    (2)As the Half-Plane Model is conformal with the Euclidean plane we can make the Euclidean rotation with the same center.
To obtain the point we have to rotate:
    (3) Plot the hyperbolic line which contains the center and the point which we want to transform.
    (4) Plot the tangent line at P.
    (5) Plot the perpendicular euclidean line to the tangent line at P.
    (6) Consider the intersection between the lines plotted at steps (5) and (6).

    (7) Rotate (euclidean rotation) the intersection point at (6) and with the fixed angle and center the point P.
    (8) Plot the euclidean line which contains P and the rotated point. This line is the tangent to the hyperbolic line which contains the image for the hyperbolic rotated point Q.
    (9) Plot the euclidean circle with center at the boundary line and passing through P which has as tangent the line in (8).
    (10) Consider the intersection between the circle just constructed and the hyperbolic circle plotted at (1). This is the hyperbolic rotation of Q.

The macro we have created allows also to rotate segments, triangles and circles. To construct this other tools we just transform each of the defining points and then we construct again the object from the images points.

Note that the rotation of the hyperbolic circles with center the fix point are invariant. From this fact, we obtain the name hyperbolic rotation.

 In the next figure it is plotted the rotation of a circle, which is not invariant.


Hyperbolic geometry
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