Distance in hyperbolic geometry

In summary, when dealing with the Beltrami-Poincare half plane model, the distance between two points on a horizontal line can be calculated by finding the semicircle that passes through the two points. However, in some cases, such as when the points are (-9,12) and (9,12), there may be no solution to the equations needed to calculate the distance. This means that the distance on a horizontal line may be undefined in the Beltrami-Poincare model for hyperbolic geometry.
  • #1
gravenewworld
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When you are dealing with the Beltrami-Poincare half plane model, and you have an h-line that is horiztonal, how can you calculate the distance of two points on the horizontal line? For example, say you have the points (-9, 12) and (9,12). Then to calculate the distance you need a semicircle through those two points. A semicircle has the general form of x^2+y^2+ax=b so you have 9^2+12^2-9a=b and 9^2+12^2+9a=b so it is obvious then that there is no solution to the equations. So is the distance on a horizontal line undefined in Beltrami-Poincare model for hyperbolic geometry?
 
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  • #2
You mean a=0 and b=225 isn't a solution? When did that happen?
 
  • #3
yeah what the hell was i thinking :biggrin: Arithmetic has always been my weakest subject.
 

What is hyperbolic geometry?

Hyperbolic geometry is a non-Euclidean geometry that describes the properties of geometric figures in a curved space. It is characterized by the hyperbolic postulate, which states that through a given point, there can be drawn an infinite number of lines parallel to a given line.

How is distance measured in hyperbolic geometry?

In hyperbolic geometry, distance is measured using a metric called the hyperbolic distance. Unlike in Euclidean geometry, where distance is calculated as the straight-line distance between two points, hyperbolic distance is calculated based on the angle between two lines in a hyperbolic space.

What is the difference between distance in hyperbolic geometry and Euclidean geometry?

The main difference between distance in hyperbolic geometry and Euclidean geometry is the way it is measured. In Euclidean geometry, distance is measured as the straight-line distance, while in hyperbolic geometry, distance is measured using the hyperbolic distance, which takes into account the curvature of space.

Can distance in hyperbolic geometry be visualized?

It can be difficult to visualize distance in hyperbolic geometry, as it is not based on the traditional concept of distance. However, there are visual representations, such as the Poincaré disk model or the hyperboloid model, that can help understand and visualize distance in hyperbolic geometry.

What are some real-world applications of hyperbolic geometry?

Hyperbolic geometry has a variety of real-world applications, including in the design of curved structures such as bridges and buildings, in computer graphics and animation, and in the study of non-linear systems in physics and economics. It also has applications in navigation and mapping, as well as in the study of black holes in astrophysics.

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