Tools to work with the Half-Plane model

From the tools of the following list you will be able to draw straight lines, circumferences, triangles,... all hyperbolic in a similar way as you would make in the Euclidean case.

All of the following tools allow us to plot the objects in any position. For example, if you want to construct a hyperbolic line from two given points, the tool will plot either the euclidean circle or the euclidean straight line perpendicular to the boundary line.

This fact, was not true in an all version of these tools. In those version, it just was possible to plot, for example, the hyperbolic lines when the two defining points did not lie in the same euclidean perpendicular line. As we believe that to study and understand geometry it is important to construct the objects in these more simple positions, we did this new version of the tools.

Clicking on the tools in the list below you will obtain how we have constructed the object in the general case, when the initial points do not lie in the same perpendicular line. Anyway, you can find some of the general characteristics about the construction of the tools in any case.

The macros can be downloaded from Half-Plane_Model2.gsp. To use them you need The Geometer Sketchpad software.

- Hyperbolic line,

- Hyperbolic ray,

- Hyperbolic segment,

- Parallel lines ,

- Length of segments,

- Measure of angles,

- Midpoint of a segment,

- Perpendicular bisector,

- Perpendicular from an exterior point,

- Perpendicular from a point on line,

- Angle bisector,

- Angle of parallelism (an important concept of Hyperbolic Geometry),

- Circumference given the center and the radius,

- Circumference given the center and a point,

- Circumference that passes through 3 points (not always exist),

- Hyperbolic arc of a circumference,

- Hyperbolic equidistant,

- Horocycle and

- Hyperbolic triangle.

Up to now, we have created the tools that allow us to construct objects in the Hyperbolic Geometry in the Half-Plane model but in no moment we have talked about how to move objects so that we obtain an equivalent object using the applications that preserve the distances, that is to say, the isometries.

In the same file,
Half-Plane_Model2.gsp you can find the tool that allows to transport segments. Given a segment we can transport it to another position in a unique way if we fix the initial point and the direction.

Transport of segments.

The isometries of the hyperbolic plane are composition of inversions with respect to hyperbolic lines. Or, thinking with Euclidean terms, the isometries are the inversions with respect to the circumferences with center in the boundary line and arbitrary radius. You can also study the isometries of the Half-Plane Model.  We have also created the tools Hyperbolic_Isometries.gsp to work with the transformations. With them you can transform points, segments, triangles and circles. These tools are explained at Isometries of the hyperbolic plane.

Hyperbolic geometry
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