Visualizing the Poincare Disc: Understanding its Limits

[shounakbhatta]

Hello,

I am facing some problem with Poincare disc.

(1) How to visualize a Poincare disc?
(2) The arc which runs at the end cannot be reached and runs till infinity. How does it happen?

 

[HallsofIvy]

The Poincare disc is a model for non-Euclidean geometry in Euclidean geometry. I'm not sure what you mean by "how does this happen". The Poincare disc model includes only the points inside the boundary circle, NOT points on the circle so an arc, representing a line, does not have end points, just as a Euclidean line does not have endpoints. To talk about "infinity" you need to have a "distance" or "metric" defined on the model- that is ds= \frac{|dz|}{1- |z|^2}.

For "visualizing" it, think of the Poincare disc as not flat but curved upward as you move from the center to the edges, the edges at "infinite" height. Since you are looking directly down the "cylinder" what you see appears to be projected on the plane below it.

 

[shounakbhatta]

Hello HallsofIvy,

Thank you very much for this specific answer. it has helped me to clear the concept and also visualize.

Thank you very much.

 

[Apogee]

The Poincare disc is a model for non-Euclidean geometry in Euclidean geometry. I'm not sure what you mean by "how does this happen". The Poincare disc model includes only the points inside the boundary circle, NOT points on the circle so an arc, representing a line, does not have end points, just as a Euclidean line does not have endpoints. To talk about "infinity" you need to have a "distance" or "metric" defined on the model- that is ds= \frac{|dz|}{1- |z|^2}.

For "visualizing" it, think of the Poincare disc as not flat but curved upward as you move from the center to the edges, the edges at "infinite" height. Since you are looking directly down the "cylinder" what you see appears to be projected on the plane below it.

That was a simple and concise explanation! But why is that the metric for the Poincare disc? And are there any practical applications of the Poincare disc model to other fields of mathematics?

 

[Incnis Mrsi]

The Poincare disc model includes only the points inside the boundary circle, NOT points on the circle so an arc, representing a line, does not have end points, just as a Euclidean line does not have endpoints.

It depends on definition of a point. HallsofIvy deems that Poincaré disc excludes ideal points, whereas Ī deem it includes them. They do not belong to Lobachevski’s plane though, like “points at infinity” of projective geometry do not belong to affine/Euclidean space. But hyperbolic triangles with one, two, or three ideal vertices are perfectly well-defined.