Search results
Results from the WOW.Com Content Network
The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers. The model generalizes directly to an additional dimension: a hyperbolic 3-space three-dimensional hyperbolic geometry relates to Minkowski 4-space.
It is the logarithm dist(p 1, p 2) = log ((s + d) 2 /h 1 h 2) Distance between two points can alternately be computed using ratios of Euclidean distances to the ideal points at the ends of the hyperbolic line.
Choose a line (the x-axis) in the hyperbolic plane (with a standardized curvature of -1) and label the points on it by their distance from an origin (x=0) point on the x-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates x and y by dropping a perpendicular onto the x-axis.
Fixing a point yields a natural distance on : two points represented by rays , originating at are at distance ( ()). When X {\displaystyle X} is the unit disk, i.e. the Poincaré disk model for the hyperbolic plane, the hyperbolic metric on the disk is
Then n-dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the square root of the scalar square. If the signature (+, −, −) is chosen, scalar square between distinct points on the hyperboloid will be negative, so various definitions of basic terms must be adjusted, which can be inconvenient.
Specializing to the case where one of the points is the origin and the Euclidean distance between the points is r, the hyperbolic distance is: (+) = where is the inverse hyperbolic function of the hyperbolic tangent. If the two points lie on the same radius and point ′ = (′,) lies between the origin and point = (,), their hyperbolic ...
The vertical bars indicate Euclidean distances between the points in the model, where ln is the natural logarithm and the factor of one half is needed to give the model the standard curvature of −1. When one of the points is the origin and Euclidean distance between the points is r then the hyperbolic distance is:
The hyperbolic distance between an ideal point and any other point or ideal point is infinite. The centres of horocycles and horoballs are ideal points; two horocycles are concentric when they have the same centre.