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
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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