Circumference of a circle in Poincare Half Plane

In summary, the conversation discusses the use of the Poincare Half Plane to calculate the circumference of a circle. It is noted that vertical lines are geodesics and the arclength formula can be used to determine the distance between two points on a vertical line. The formula for finding the radius of a circle using the Poincare metric is also mentioned. The conversation then goes on to discuss the parametrization of a circle and the use of the arclength formula to find its circumference. However, there is confusion as the result obtained using the Poincare metric is the same as the result using the regular euclidean metric. There is a request for assistance in identifying the mistake in the calculation.
  • #1
demonelite123
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i am trying to figure out how to calculate the circumference of a circle in the Poincare Half Plane. I know that vertical lines are geodesics so using the arclength formula, the distance between 2 points [itex] (x_0, y_0) and (x_1, y_1) [/itex] on a vertical line is [itex] ln(y_1/y_0) [/itex]. Thus, if i have a circle centered at (0, b) on the y-axis with radius r, then the radius of the circle using the Poincare metric will be [itex] \frac{1}{2}ln(\frac{b+r}{b-r}) [/itex].

I want to find the circumference of the circle [itex] x^2 + (y-b)^2 = b^2 - 1 [/itex] so i parametrize [itex] x = \sqrt{b^2-1}cos(\theta), y = b + \sqrt{b^2-1}sin(\theta) [/itex]. I then use the arclength formula and get [itex] C = \int^{2\pi}_0 \frac{\sqrt{b^2-1}}{b+\sqrt{b^2-1}sin(\theta)} d\theta [/itex]. I have the extra factor of [itex] y = b + \sqrt{b^2-1}sin(\theta) [/itex] on the bottom since I am using the Poincare metric. This evaluates to [itex] 2\pi\sqrt{b^2-1} [/itex] which i know shouldn't be right since the circumference of this circle using the regular euclidean metric is also just [itex] 2\pi\sqrt{b^2-1} [/itex]. can someone help me figure out what went wrong here?
 
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  • #2
Why do you think your answer is wrong? Can you say more? Justify it better?
 

FAQ: Circumference of a circle in Poincare Half Plane

1. What is the formula for finding the circumference of a circle in Poincare Half Plane?

The formula for finding the circumference of a circle in Poincare Half Plane is C = 2πr, where C represents the circumference and r represents the radius of the circle.

2. How is the Poincare Half Plane related to the circumference of a circle?

The Poincare Half Plane is a mathematical model used to represent hyperbolic geometry, which is one of the non-Euclidean geometries. It is used to visualize and calculate the properties of circles, including the circumference, in this geometry.

3. Can the circumference of a circle in Poincare Half Plane ever be infinite?

No, the circumference of a circle in Poincare Half Plane is always a finite value. This is because the Poincare Half Plane is a bounded space, meaning that there is a maximum distance that can be traveled in any direction.

4. How does the circumference of a circle in Poincare Half Plane differ from the circumference in Euclidean geometry?

In Euclidean geometry, the circumference of a circle is directly proportional to its radius, meaning that as the radius increases, the circumference also increases. However, in Poincare Half Plane, the circumference increases at a much slower rate as the radius increases.

5. What is the significance of studying the circumference of a circle in Poincare Half Plane?

Studying the circumference of a circle in Poincare Half Plane allows us to better understand non-Euclidean geometries and their properties. This has practical applications in fields such as physics, engineering, and computer science, where non-Euclidean spaces are used to model complex systems and phenomena.

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