If we now
imagine that
we are parallel transporting the stick around the edges of a triangle
ABC,
starting and ending at A, then the direction that the stick makes after
returning to A will be identical to the direction that the stick
originally
made with the path at A.
Translation in Hyperbolic
Geometry is not quite so simple. Let us look at what happens as
we
translate a segment around a triangle. Start with triangle ABC and edge
AD. Translate AD from A to B by multiple-selecting A and then B and
then
choosing "Vector" from the "Mark" menu in the Transform Panel. When the
dialog box pops up choose "Rectangular" for a simple translation and
then
hit "Okay". We have now defined a hyperbolic translation from A to B.
Now
select segment AD and hit the now-active translate button in the
Transform
Panel. Segment AD will be parallel transported to point B
resulting
in the segment BE, as shown.
Next, define a new
translation
vector from point B to point C and translate segment BE, resulting in
segment
CF, as shown.
Finally, define a translation
vector
from C to A and translate CF, resulting in segment AG, as shown.
Clearly, the original segment
AD and
the parallel transport around the triangle, segment AG, are not in the
same direction, as was the case in Euclidean Geometry. To see how
much the segment has changed let's measure the angle DAG, as this
measures
the net rotation of AD to AG. (Note: In Euclidean Geometry, the
combination
of two translations is again a translation. The example we just
constructed
shows that the combination of translations in Hyperbolic Geometry need
not be a translation, and in fact is a rotation in this case.) As
a comparison, let's also compute the defect of triangle ABC.
That is interesting, the defect
and
the net angle change appear complimentary. Let's use the
Calculator
to add these two measurements together. Then, we will evaluate the sum
and add this new measure back to the canvas. (If you need help on
using the Calculator check the help section on that topic here.)
Does this relationship persist? Let's move point A around and check...
The relationship does hold. In fact, this relationship is connected with a very important idea in geometry -- that of holonomy.
We define the Holonomy of triangle ABC to be the smallest angle measured counterclockwise from the original position of a segment AD to the final position of AD (which in this example is segment AG) as AD undergoes parallel transport counterclockwise around the triangle.
There are several conclusions we can tentatively draw from this example.