Geometry Explorer Help
Exploring in a Hyperbolic World:
Using Geometry Explorer
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Windows, Tools, etc.:
Working within Hyperbolic Geometry with Geometry Explorer is
essentially
no different than working within Euclidean Geometry. Almost all
of
the tools work in both environments, with a couple of notable
exceptions
which are described below. Even recordings made in one
environment
can be played back in the other environment. Thus, for example, the Koch
snowflake example can be recorded in a Euclidean canvas and played
back on a hyperbolic canvas, or vice-versa. This flexibility was
deliberately
designed to give the user the maximum opportunity to explore and
contrast
different geometric "universes" and thus to develop an intuition for
how
it "feels' to live in one geometry versus another.
Differences Between Euclidean and Hyperbolic Canvases:
There are four main differences between the two environments:
- In the Euclidean canvas the parallel button in the tool
panel
will
result in the unique parallel being created, once a linear object
(segment,
ray, or line) and a point have been selected. In Hyperbolic
geometry, using this tool (with the same selection of a linear object
and
a point) will result in the creation of the two unique ultraparallels
to
the given linear object and point. Thus, recordings made using
parallels
in either environment will not be playable in the other environment.
- In the Euclidean canvas, circles and arcs can be defined using
three
points.
This construction depends on the Euclidean parallel postulate, and thus
is not available in Hyperbolic Geometry.
- There is no graph option available in Hyperbolic Geometry.
- Transformations cannot be defined in terms of measurements in
Hyperbolic Geometry. (At least not yet!)