Learn Hyperbolic Geometry: Why & How Does Parallelism Work?

In summary, the conversation discusses the basics of hyperbolic geometry, including the upper half plane, hyperbolic lines, and parallels. The boundary of the hyperbolic plane is also mentioned, with a suggestion to study it using the unit disc and conformal mapping.
  • #1
kexue
196
2
I like to learn some basic hyperbolic geometry!

Starting with the hyperbolic plane, the upper half plane with the hyperbolic lines being all half-lines perpendicular to the x-axis, together with all semi-circles with center on the x-axis.

Why and how are there always infinitely many hyperbolic lines parallel to a given hyperbolic line L and passing through a given point p not lying on L?

Does parallel in hyperbolic geometry just mean two lines do not cut each other?

Second problem.

My book says the boundary of the hyperbolic plane can be thought of as real line [tex]\cup[/tex] infinity, which is supposed to be a circle. I can't see this to be a circle. What does the author mean?

thank you
 
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  • #3
kexue said:
I like to learn some basic hyperbolic geometry!

Starting with the hyperbolic plane, the upper half plane with the hyperbolic lines being all half-lines perpendicular to the x-axis, together with all semi-circles with center on the x-axis.

Why and how are there always infinitely many hyperbolic lines parallel to a given hyperbolic line L and passing through a given point p not lying on L?

Does parallel in hyperbolic geometry just mean two lines do not cut each other?

Second problem.

My book says the boundary of the hyperbolic plane can be thought of as real line [tex]\cup[/tex] infinity, which is supposed to be a circle. I can't see this to be a circle. What does the author mean?

thank you

In my opinion hyperbolic geometry is best studied at first on the unit disc where classical Moebuis tranformations of the complex plane that preserve the disc are isometries. It is immediate from this way of looking at it that there are infinitely many parallels to a line through any point.

For the upper half plane, it is good to map it onto the disc conformally and take the induced metric.
 
  • #4
kexue said:
I like to learn some basic hyperbolic geometry!

Starting with the hyperbolic plane, the upper half plane with the hyperbolic lines being all half-lines perpendicular to the x-axis, together with all semi-circles with center on the x-axis.

Why and how are there always infinitely many hyperbolic lines parallel to a given hyperbolic line L and passing through a given point p not lying on L?

Does parallel in hyperbolic geometry just mean two lines do not cut each other?

Second problem.

My book says the boundary of the hyperbolic plane can be thought of as real line [tex]\cup[/tex] infinity, which is supposed to be a circle. I can't see this to be a circle. What does the author mean?

thank you

On the unit disc the boudary is the unti circle, no problem here. This circle is mapped to the real line plus the point at infinity on the Riemann sphere by a conformal mapping of the unit disc onto the upper half plane. I urge you to learn it this way.
 

1. What is hyperbolic geometry?

Hyperbolic geometry is a non-Euclidean geometry that describes the properties of curved spaces, in contrast to Euclidean geometry which describes the properties of flat spaces. It is based on the concept of a hyperbolic plane, where parallel lines diverge and the angles of a triangle add up to less than 180 degrees.

2. Why is it important to learn hyperbolic geometry?

Hyperbolic geometry has many applications in fields such as physics, mathematics, and computer science. It helps us understand and model curved spaces, and has been used to solve problems in fields such as general relativity and graph theory. Additionally, it challenges our understanding of traditional Euclidean geometry and expands our thinking to consider different types of geometry.

3. How does parallelism work in hyperbolic geometry?

In hyperbolic geometry, parallel lines do not exist as they do in Euclidean geometry. Instead, parallelism is defined as the property that two lines never intersect, no matter how far they are extended. This means that in a hyperbolic plane, there can be multiple lines parallel to a given line through a point, and the distance between these lines will increase as they diverge.

4. What are some key concepts in hyperbolic geometry?

Some key concepts in hyperbolic geometry include the Poincaré disk model, which represents the hyperbolic plane as a disk with the center representing infinity, and the hyperbolic metric, which is a measure of distance in hyperbolic space. Other important concepts include hyperbolic trigonometry, which differs from traditional trigonometry in Euclidean geometry, and the concept of negative curvature, which characterizes hyperbolic spaces.

5. How can one learn hyperbolic geometry?

One can learn hyperbolic geometry through various resources such as textbooks, online courses, and lectures. It is important to have a strong foundation in Euclidean geometry and basic algebra before delving into hyperbolic geometry. Practice and visualization are also key in understanding the concepts and properties of hyperbolic geometry. Additionally, collaborating with others and discussing ideas can also aid in the learning process.

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