Distance in hyperbolic geometry

  • Context: Graduate 
  • Thread starter Thread starter gravenewworld
  • Start date Start date
  • Tags Tags
    Geometry Hyperbolic
Click For Summary
SUMMARY

The discussion focuses on calculating the distance between two points on a horizontal line in the Beltrami-Poincare half-plane model of hyperbolic geometry. Specifically, it examines the points (-9, 12) and (9, 12), highlighting the need for a semicircle to determine the distance. The equations derived from the semicircle's general form, x² + y² + ax = b, reveal that there is no solution for the given points, indicating that the distance on a horizontal line in this model is undefined. The conversation also touches on the challenges of arithmetic in understanding these concepts.

PREREQUISITES
  • Understanding of hyperbolic geometry principles
  • Familiarity with the Beltrami-Poincare half-plane model
  • Knowledge of semicircles and their equations
  • Basic arithmetic and algebra skills
NEXT STEPS
  • Research the properties of the Beltrami-Poincare half-plane model
  • Study the concept of distance in hyperbolic geometry
  • Learn about semicircles and their applications in geometric calculations
  • Explore the implications of undefined distances in hyperbolic models
USEFUL FOR

Mathematicians, geometry enthusiasts, students studying hyperbolic geometry, and anyone interested in the applications of the Beltrami-Poincare model.

gravenewworld
Messages
1,129
Reaction score
27
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?
 
Physics news on Phys.org
You mean a=0 and b=225 isn't a solution? When did that happen?
 
yeah what the hell was i thinking :biggrin: Arithmetic has always been my weakest subject.
 

Similar threads

High School Axes on a hyperbolic plane
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Hyperbolic geometry - relations between lines, curves, and hyperbolas
  • · Replies 2 ·
Replies
2
Views
4K
Graduate Coordinates in hyperbolic geometry?
  • · Replies 6 ·
Replies
6
Views
12K
Undergrad A way to comprehend the geometry of a hypercube
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Computing arc length in Poincare disk model of hyperbolic space
  • · Replies 4 ·
Replies
4
Views
5K
Undergrad Chiral symmetries in E[SUP]^n[/SUP]
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Understanding the Hyperbolic Distance Formula: Deriving Log QA.PB/QB.PA
  • · Replies 2 ·
Replies
2
Views
10K
Graduate Distance formula in hyperbolic metric
  • · Replies 2 ·
Replies
2
Views
5K
Graduate Learn Hyperbolic Geometry: Why & How Does Parallelism Work?
  • · Replies 3 ·
Replies
3
Views
5K
Suggestions for HS geometry book (proofs)
  • · Replies 14 ·
Replies
14
Views
5K