Show Poincare Disk is incidence geometry

In summary, the conversation is about showing that the Poincare disk satisfies the incidence axiom that any line contains at least two points. The Poincare disk has two kinds of lines - straight lines going through the origin, and lines described by the equation (x-a)^2 + (y-b)^2 = a^2 + b^2 - 1 with x^2 + y^2 < 1. The attempt at a solution involves finding (x,y) and (x',y') coordinates on the line, but the person is unsure if they are headed in the right direction. They clarify that a "line" in the Poincare disk model can be a Euclidean straight line or an arc of a Euclidean
  • #1
murmillo
118
0

Homework Statement


I have to show that the Poincare disk satisfies the incidence axiom that any line contains at least two points.


Homework Equations


There are two kinds of lines on the Poincare disk. I've found 2 points for the first kind, which are straight lines going through the origin. The second kind are lines
L a,b = {(x,y) | (x-a)2 + (y-b)2 = a2 + b2 - 1, with x2 + y2 < 1}

The Attempt at a Solution


Given a and b, I need to find (x, y) and (x',y') that lie on the line.
I've shown that x^2+y^2+1 = 2ax+2by, but I don't know if I'm headed in the right direction.
 
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  • #2
I'm confused as to what you are doing here. In the "Poincare' disk model for hyperbolic geometry", a "line" is either a Euclidean straight line throug the center of the disk or an arc of a Euclidean circle contained in the disk. In either case, it follows immediately from Euclidean geometry that they contain an infinite number of points. Are you required to actually calculate coordinates of two points in the "line"? If so, what coordinate system are you using?
 
  • #3
Yes, I have to show that there exist at least two points on the line. I need to find coordinates in terms of a and b. I'm using the normal coordinate system for points in R2.
 

1. What is the Poincare Disk model?

The Poincare Disk model is a geometric representation of hyperbolic geometry developed by French mathematician Henri Poincare. It is a two-dimensional disk with the properties of the hyperbolic plane, including non-Euclidean angles and distances. The model is useful for visualizing and studying hyperbolic geometry, particularly in the field of non-Euclidean geometry.

2. How is the Poincare Disk model related to incidence geometry?

The Poincare Disk model is a specific example of an incidence geometry, where points, lines, and circles are the fundamental objects. It is a model that follows the same axioms and rules as incidence geometry, making it a useful tool for studying and understanding the principles of incidence geometry.

3. What are the basic principles of the Poincare Disk model?

The Poincare Disk model follows several basic principles, including the fact that all geodesics (straight lines) are arcs of circles perpendicular to the boundary of the disk. Additionally, all angles in the Poincare Disk are hyperbolic angles, which follow different rules than Euclidean angles. Finally, the distance between two points is measured along the shortest path, which is a hyperbolic line, rather than a straight line.

4. How does the Poincare Disk model differ from the Euclidean model?

The Poincare Disk model differs from the Euclidean model in several ways. In Euclidean geometry, the angles of a triangle add up to 180 degrees, but in the Poincare Disk model, the angles of a hyperbolic triangle add up to less than 180 degrees. The Poincare Disk also has different rules for parallel lines and the distance between points, which are both dependent on the curvature of the hyperbolic plane.

5. What are some real-life applications of the Poincare Disk model?

The Poincare Disk model has various real-life applications, particularly in the fields of physics and computer science. It is used to model the behavior of light in gravitational fields and to study the geometry of spacetime. Additionally, the Poincare Disk model is used in computer graphics and animation to create visually appealing and realistic 3D images. It is also used in game development and virtual reality technology.

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