| 1. | Make sure that you have a working copy of Geometer's Sketchpad version 4. The software can be purchased from Key Curriculum Press. You can download a free demonstration version of Geometer's Sketchpad from the Key Press site. The demo version is similar to the full version, but you cannot save, print, or export your work. |
| 2. |
Download the desired tools into your preferred directory. I recommend
creating a separate directory (perhaps "hyptools" or something like
that) to hold the different Sketchpad files.
|
The Geometer's Sketchpad version 4 file klein.gsp contains a template of the Klein disk and custom tools for hyperbolic constructions in this model. The tools include those for the 10 "standard" constructions: constructing lines and segments, measuring lengths and angles, constructing angle bisectors, raising and dropping perpendiculars, constructing perpendicular bisectors, and constructing circles by either center and point or by center and two points defining a radius segment. (See the hyperbolic tools page for a more careful description of these constructions.) There are several additional construction tools included in this package. These include tools to
One particular difficulty with the Beltrami-Klein model is that unlike the Poincaré models, Klein circles are not Euclidean circles - they are ellipses. As such, in Geometer's Sketchpad, they must be drawn as loci. This creates a practical problem: Sketchpad does not know how to intersect loci. Also in the Klein package, then are additional tools which
In creating these last three tools, it proved useful to use the natural isomorphism between the Klein disk and the Poincaré disk (described in Greenberg's Euclidean and Non-Euclidean Geometry, 3rd ed, p. 236). To find the intersection of, say, two Klein circles, one can use the isomorphism to map the Klein circles (i.e. ellipses) to the Poincaré disk (where their images are true Euclidean circles), use Sketchpad to find their intersections in the Poincaré disk, then map these points back to the Klein disk. For this reason, the Klein package also includes a tool to
The Geometer's Sketchpad version 4 file poinhalf.gsp. contains a template of the Poincaré half-plane and custom tools for hyperbolic constructions in this model. The tools include those for the 10 "standard" constructions: constructing lines and segments, measuring lengths and angles, constructing angle bisectors, raising and dropping perpendiculars, constructing perpendicular bisectors, and constructing circles by either center and point or by center and two points defining a radius segment. ( See the hyperbolic tools page for a more careful description of these constructions.) There are two additional construction tools included in this package, which
construct the midpoint of a Poincaré segment.
construct the reflection of a given point about a Poincaré line.
Most of the tools for hyperbolic constructions in the Poincaré disk are included with the Geometer's Sketchpad version 4 sample files in a standard distribution. The Poincaré disk file is usually found at
"Samples\Sketches\Investigations\Poincare Disk.gsp"
in the Sketchpad default directory.
This Sketchpad document contains a template of the Poincaré disk as well as the tools for nine of the ten "standard" hyperbolic constructions. However, it does not include the construction to "Raise a perpendicular" through a point on a line (construction #6 on the hyperbolic tools page).
The Geometer's Sketchpad version 4 file pdsktool.gsp contains two tools, which
To incorporate all of the tools into a single
file, open both the "Poincare Disk.gsp" sample file and the file
"pdsktool.gsp" simultaneously. Then, with the "Poincare
Disk.gsp" as your active sketch, use the custom tool button on the bottom
left of the Sketchpad desktop to copy the tools from "pdsktool.gsp"
to the "Poincare Disk.gsp" file. Don't forget to save your
sketch under a new name, else you will overwrite the sample sketch!
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