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.
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hyperbolic points and lines
The figure shows some points and h-lines.
You can move any of the points A, B and P.
The software draws the h-lines AB and OP.
Note. If you drag a point outside C, the h-line vanishes.
By experimenting, you should convince yourself of
the following facts:
- Through any two points of hyperbolic space,
there is an h-line.
- An h-line 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 h-lines 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 h-lines meet at most once.
- Two distinct h-lines may have one
common boundary point.
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parallels and ultraparallels
For a pair of h-lines, 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 h-lines 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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