Hyperbolic Geometry: Parameterization of Curves for Hyperbolic Distance

In summary, the conversation discusses the points P and Q, lying on a half circle with radius one and centered at (1,0). The task is to compute the hyperbolic distance between P and Q, using both the definition and properties of hyperbolic distance and the coordinates of the images of P and Q through standard inversion. The student is struggling with understanding the correct parameterization for the problem and asks if there is a quick trick for parameterizing difficult curves in the hyperbolic plane.
  • #1
Phoenixtears
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Homework Statement



Consider the points P = (1/2, √3/2) and Q = (1,1). They lie on the half circle of radius one centered at (1,0).

a) Use the deifnition and properites of the hyperbolic distance (and length) to compute dH(P,Q).

b) Compute the coordinates of the images of Pa nd Q through the standard inversion and use that to evaluate again dH(P,Q).


Homework Equations



Our professor gave us a few trig identites to use, but other than that we need the equation for Hyperbolic distance:

∫ [itex]\sqrt{}x'(t)2+y'(t)2[/itex] [itex]\frac{}{}y(t)[/itex]
From a to b where a < t < b

The Attempt at a Solution



Ignoring the question almost entirely, the part that I can't figure out is how to parameterize this correctly. My professor gave us a solution that gave one option for a parameterization, but I'm not sure how he got there.

His solution:
x(t) = cost +1
y(t)= sint

I believe that more than one parameterization would work, but how do you come up with the one that makes things simplest?

Thank you so much!

Phoenix
 
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  • #2
Okay, I feel a little silly for not understanding his parameterization, but I do understand it now. My question then moves to if there is a quick trick to parameterizing potentially more difficult curves in the hyperbolic plane?
 

What is hyperbolic geometry?

Hyperbolic geometry is a non-Euclidean geometry that explores the properties of shapes and spaces in a curved plane. It differs from Euclidean geometry in that it follows the rules of hyperbolic trigonometry rather than Euclidean geometry's rules of trigonometry. It has been studied extensively in mathematics and has applications in various fields such as physics, computer science, and art.

What is the parameterization of curves for hyperbolic distance?

The parameterization of curves for hyperbolic distance is a method used to measure the distance between points in a hyperbolic space. It involves finding a mathematical formula or set of equations that describes the curve and using it to calculate the distance between points on the curve. This technique is crucial in understanding the properties of hyperbolic geometry and its applications.

How is hyperbolic distance different from Euclidean distance?

In Euclidean geometry, the distance between two points is the length of the straight line connecting them. However, in hyperbolic geometry, the distance between two points is measured along the shortest path on a curved surface, which can result in a different value compared to Euclidean distance. This is due to the curvature of the hyperbolic space, which affects the concept of distance in this geometry.

What are some real-world applications of hyperbolic geometry?

Hyperbolic geometry has been applied in various fields, including physics, computer science, and art. In physics, it has been used to study black holes and other curved spacetimes. In computer science, it has been used in image and video compression algorithms. In art, it has been used to create visually stunning works, such as the hyperbolic crochet coral reef project.

What are some famous theorems in hyperbolic geometry?

The most well-known theorems in hyperbolic geometry are the Poincaré disk model, the Poincaré half-plane model, and the hyperboloid model. These models describe the hyperbolic space and allow for easier visualization and understanding of its properties. Other notable theorems include the Gauss-Bonnet theorem, which relates the curvature of a surface to its topology, and the hyperbolic triangle inequality, which states that the sum of the angles in a hyperbolic triangle is less than 180 degrees.

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