Hyperbolic geometry - relations between lines, curves, and hyperbolas

In summary, the individual is seeking help understanding the relationship between Euclidean and hyperbolic geometry, particularly in terms of transformations and models. They are also curious about how a specific function would be represented in non-Euclidean coordinates. They apologize for any errors in their understanding and are eager to learn from experts on the subject.
  • #1
Reuel
3
0
Hi.

I studied calculus a while back but am far from a math god. I have been reading around online about hyperbolic geometry in my spare time and had a simple question about the topic.

If a straight line in Euclidean geometry is a hyperbola in the hyperbolic plane (do I have that right?) then what is the "transformation" from one to the other? For example, the line y=x in the Cartesian coordinate system would be what in the hyperbolic plane? That is, what hyperbola corresponds to y=x? Can the two be related?

The ultimate reason I am interested in knowing specifically how to go from one to the other is because I am curious as to how the hyperbolic rational expression of the form

[itex]f(x)=\frac{ax}{b+cx}[/itex]​

would be expressed in non-Euclidean terms and what straight line in Euclidean geometry would lead to such a hyperbola in non-Euclidean geometry.

If any of that is nonsense, I apologize. I don't know much about the subject but am willing to learn.

Thank you for your help.
 
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  • #2
I am not sure what you are talking about. A "straight line in Euclidean geometry" isn't in the "hyperbolic plane" and so does not correspond to any thing there. You probably are talking about a "model" for the hyperbolic plane in the Euclidean plane such as the "Klein model", the "disc model" or the "half plane" model. But what a straight line represents in such a model depends upon which model you are referring to and, depending on the model, exactly which straight line. For example, any Euclidean straight line in the Klein model represents a hyperbolic straight line while only a straight line perpendicular to the boundary in the half plane model represents a straight line in hyperbolic geometry
 
  • #3
Yeah. I'm sure my original post had plenty of flaws. I guess all I meant was, how would the function I mentioned be represented in non-Euclidean coordinates? You mentioned several "models" and I do not know enough about them to know which would be best suited for such. Hence seeking help from experts on such subjects.
 

FAQ: Hyperbolic geometry - relations between lines, curves, and hyperbolas

What is hyperbolic geometry?

Hyperbolic geometry is a type of non-Euclidean geometry that describes the properties and relationships between lines, curves, and figures in a curved space. It is based on the concept of a hyperbola, a type of curve that is formed by intersecting a cone with a plane at a specific angle.

How is hyperbolic geometry different from Euclidean geometry?

In Euclidean geometry, the sum of the angles in a triangle is always 180 degrees, and parallel lines never intersect. In hyperbolic geometry, the sum of the angles in a triangle is always less than 180 degrees, and parallel lines can intersect at multiple points. Hyperbolic geometry also follows different rules for distance and area calculations compared to Euclidean geometry.

What are some real-world applications of hyperbolic geometry?

Hyperbolic geometry has many practical applications, including in the fields of architecture, astronomy, and computer science. For example, hyperbolic geometry is used in the design of space telescopes and satellite dishes, as well as in the development of algorithms for computer graphics and virtual reality.

How is hyperbolic geometry used to model curved space?

Hyperbolic geometry is used to describe the properties of negatively curved spaces, such as saddle-shaped surfaces. This type of geometry provides a more accurate model for the geometry of the universe, as it accounts for the effects of gravity and the curvature of space-time predicted by Einstein's theory of general relativity.

What are some famous theorems and results in hyperbolic geometry?

One of the most well-known theorems in hyperbolic geometry is the Gauss-Bonnet theorem, which relates the curvature of a surface to its topology. Other important results include the Poincaré disc theorem, which states that any two polygonal regions in the hyperbolic plane are equivalent, and the Beltrami-Klein model, which provides a visual representation of hyperbolic geometry on a flat surface.

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