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Since both the transverse axis and the conjugate axis are axes of symmetry, the symmetry group of a hyperbola is the Klein four-group. The rectangular hyperbolas xy = constant admit group actions by squeeze mappings which have the hyperbolas as invariant sets.
A ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
In this context the unit hyperbola is a calibration hyperbola [3] [4] Commonly in relativity study the hyperbola with vertical axis is taken as primary: The arrow of time goes from the bottom to top of the figure — a convention adopted by Richard Feynman in his famous diagrams. Space is represented by planes perpendicular to the time axis.
There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point (the origin) on a chosen directed line (the x-axis) and after that many choices exist. The Lobachevsky coordinates x and y are found by dropping a perpendicular onto the x-axis. x will be the label of the foot of the perpendicular.
Since hyperbolas in Q correspond to lines parallel to the boundary of HP, they are horocycles in the metric geometry of Q. If one only considers the Euclidean topology of the plane and the topology inherited by Q , then the lines bounding Q seem close to Q .
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.
As approaches from above, the limit of the pencil of confocal hyperbolas degenerates to the relative complement of that line segment with respect to the x-axis; that is, to the two rays with endpoints at the foci pointed outward along the x-axis (an infinitely flat hyperbola). These two limit curves have the two foci in common.
Like other rectangular hyperbolas, the orthocenter of any three points on the curve lies on the hyperbola. So, the orthocenter of the triangle A B C {\displaystyle ABC} lies on the curve. The line O I {\displaystyle OI} is tangent to this hyperbola at I {\displaystyle I} .