Poincare disk model. Circle question.

[greenmath]

Homework Statement

Use the Poincare (disk) model to show that in the hyperbolic plane, there exists two points A, B lying on the same side S of a line l such that no circle through A and B lies entirely within S.

Homework Equations

The hint was to use this proposition:
A P-circle is a Euclidean circle in the disk, and conversly, but the P-center differes from the Euclidean center except when the center is the origin.

Basicallly in the disk circles still look like they normally would but the center might be different from what it would be in a eudclidean sense.

The Attempt at a Solution

I'm quite lost as to how to do this actually, I realize you should be able to do it by contradiction and just show that any circle you draw would have to either intersect the line l or it would have to touch the boundary of the disk. But it seems like you should be able to draw a circle quite easily that would go through two points, be inside another circle and not cross a line.

The book is eucliden and non-euclidean geometries development and history by marvin jay greenberg. Problem p-20.

 

[mighty2000]

Dear fellow forum member,

Thank you for your question. I will use the Poincare disk model to show that in the hyperbolic plane, there exists two points A, B lying on the same side S of a line l such that no circle through A and B lies entirely within S.

First, let's define the Poincare disk model. In this model, the hyperbolic plane is represented by a disk, with the boundary of the disk representing the line at infinity. Lines in the hyperbolic plane are represented by arcs of circles that intersect the boundary of the disk perpendicularly. Points in the hyperbolic plane are represented by points within the disk.

Now, let's consider a line l in the hyperbolic plane and two points A and B lying on the same side S of l. Since l is represented by an arc of a circle that intersects the boundary of the disk perpendicularly, we can draw a line segment AB connecting A and B that is perpendicular to l. This line segment represents a hyperbolic line in the Poincare disk model.

Next, let's consider a circle through A and B. In the Poincare disk model, this circle is represented by a Euclidean circle that intersects the boundary of the disk perpendicularly at points A and B. However, according to the proposition given in the homework, the center of this P-circle may be different from the Euclidean center, unless the center is at the origin.

Now, let's draw a circle with center C that intersects the line segment AB at points A and B. Since C is not necessarily at the origin, this circle will intersect the line l at some point D. This means that the circle does not lie entirely within the same side S of the line l as points A and B, as it intersects the line l at point D.

Therefore, we have shown that in the hyperbolic plane, there exists two points A and B lying on the same side S of a line l such that no circle through A and B lies entirely within S. This is because any circle through A and B will intersect the line l at some point, as shown in our example.

I hope this explanation helps you understand the concept better. If you have any further questions, please don't hesitate to ask. Good luck with your studies!