Visualizing the Poincare Disc: Understanding its Limits

In summary: There are practical applications of the Poincare disc model to other fields of mathematics, but I'm not familiar with them.
  • #1
shounakbhatta
288
1
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?
 
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  • #2
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 [itex]ds= \frac{|dz|}{1- |z|^2}[/itex].

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.
 
  • #3
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.
 
  • #4
HallsofIvy said:
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 [itex]ds= \frac{|dz|}{1- |z|^2}[/itex].

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?
 
  • #5
HallsofIvy said:
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.
 

1. What is the Poincare Disc and how is it visualized?

The Poincare Disc is a model of hyperbolic geometry, a type of non-Euclidean geometry. It is visualized as a disk with a specific curvature, where straight lines are represented as arcs of circles that intersect the boundary of the disk at right angles.

2. What are the limits of the Poincare Disc?

The Poincare Disc has infinite area but finite circumference, meaning it does not have a well-defined center or edge. Additionally, the distance between points increases exponentially as they move towards the boundary of the disk.

3. How does the Poincare Disc differ from Euclidean geometry?

The Poincare Disc has different rules for measuring distance, angles, and parallel lines compared to Euclidean geometry. In the Poincare Disc, the sum of angles in a triangle is always less than 180 degrees, and there are no parallel lines.

4. What is the importance of visualizing the Poincare Disc?

Visualizing the Poincare Disc helps us understand and explore the concepts of hyperbolic geometry, which has important applications in fields such as physics, computer graphics, and cosmology. It also challenges our traditional understanding of geometry and expands our mathematical thinking.

5. Are there other models of hyperbolic geometry besides the Poincare Disc?

Yes, there are many other models of hyperbolic geometry, including the Poincare Half-Plane, the Klein Model, and the Hyperboloid Model. Each model has its own unique visualization and properties, but they all share the same non-Euclidean geometry principles.

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