On this page, you can experiment with hyperbolic lines and segements
using CabriJave applets.
The facts that you are asked to observe will be proved later.
Contents


hyperbolic points and lines
The figure shows some points and hlines.
You can move any of the points A, B and P.
The software draws the hlines AB and OP.
Note. If you drag a point outside C, the hline vanishes.
By experimenting, you should convince yourself of
the following facts:
 Through any two points of hyperbolic space,
there is an hline.
 An hline which passes through the centre O
is a euclidean segment, i.e. a diameter of C.
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properties of hyperbolic lines
This figure shows hlines AB, PQ and XY.
Note. The points A, P, X, B, Q and Y lie on C,
so do not belong to the geometry.
You can vary the lines by moving any of these points.
By experimenting, you should convince yourself of
the following facts
 Two distinct hlines meet at most once.
 Two distinct hlines may have one
common boundary point.
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parallels and ultraparallels
For a pair of hlines, we have three possibilities
 The lines may intersect (i.e. share a point of C)
Such lines are said to be intersecting
 The lines may share a common boundary point
Such lines are said to be parallel
 The lines may neither intersect nor share a boundary point.
Such lines are said to be ultraparallel
In euclidean geometry, if lines M and N are parallel to a line L,
then M and N are parallel.
The figure on the right has hlines XY parallel to XZ,
and WZ parallel to XZ.
By moving W, you should convince yourself that
XY and ZW may be intersecting,
parallel or ultraparallel!
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hyperbolic triangles
The figure shows a hyperbolic triangle PQR
You can move any of the vertices, P, Q and R,
but if any point is dragged outside the circle C,
the associated hyperbolic segments vanish.
By experimenting, you should convince yourself of
the following facts
 When a vertex is close to O,
the associated segments are almost straight
Java bug: the segments vanish if a vertex is at O.
 When a vertex is close to the boundary C,
the angle betwen the segments is close to zero.
You should look at cases where O is inside, on
and outside the triangle, and observe the shapes.
In particular, note that
 When O is inside the triangle, the angles are less
that those of the euclidean triangle PQR
so sum to less than π.
We shall see that this is true in general.
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