Re: Is it permitted in math to go beyond?

Hero wrote:


... So, one can do geometry with the points of the boundary circle and
beyond - an ideal geometry.

Probably it's better to refer to it as a projective geometry.

But i'm still in trouble with these points. Where are these in respect
to the true non-euclidian plane?

Their role is very much like the role of the "points at infinity" in the
real projective plane.

The Poincare disk might be the best way of thinking about these points. The
Poincare disk is the set of all points in the interior of the unit disk:
x^2 + y^2 < 1. Now think about all the circles orthogonal to the unit
circle (orthogonal: they intersect the unit circle and the tangents at the
points of intersection are perpendicular). The portion of those circles
within the Poincare disk are the lines -- often called h-lines. One can
show that the Poincare disk with the h-lines is a model of the hyperbolic
plane.

Now this is all embedded in the Euclidean plane, and one can take advantage
of this embedding in order to deduce a lot of things about the hyperbolic
plane -- for example, much of hyperbolic trigonometry is developed this
way. So, one way to think of the ideal and ultra-ideal points is to
visualize this embedding. The ideal points lie on the unit circle x^2 +
y^2 = 1, while the ultra-ideal points satisfy that x^2 + y^2 > 1.

One would not routinely use a hyperbolic metric on these points, just as one
would not use a Euclidean metric on ideal points in the real projective
plane. They would be infinitely far away -- in the hyperbolic measure.

--charlie

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