a disc model for euclidean geometry

Some students find it difficult to accept that a geometry with a disc model
can contain lines of arbitrary length.

One way to appreciate that this can happen is to look at a disc model for
the familiar euclidean geometry.

Let E be the usual model of the euclidean plane, and
let S be the lower half of a sphere (centre O) touching E,
with the "rim" deleted.

For any P on E, let t(P) = Q, the point where OP cuts S.
This defines a bijection t from E to S, so we can regard
the finite surface S as a model for euclidean geometry.

If we now view S from directly above O, we get a model
for the euclidean plane on a disc centred on O.

Of course, the model "distorts" distances and angles on E.

In fact, a line on E through
the point vertically below O
is represented by a
diameter of the disc!

return to the hyperbolic plane