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This property provides a possibility to construct points of a hyperbola if the asymptotes and one point are given. This property of a hyperbola is an affine version of the 4-point-degeneration of Pascal's theorem. [13]
If P 0 is taken to be the point (1, 1), P 1 the point (x 1, 1/x 1), and P 2 the point (x 2, 1/x 2), then the parallel condition requires that Q be the point (x 1 x 2, 1/x 1 1/x 2). It thus makes sense to define the hyperbolic angle from P 0 to an arbitrary point on the curve as a logarithmic function of the point's value of x. [1] [2]
The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At = the asymptotes are at right angles. With > the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion ...
Hyperbolic coordinates plotted on the Euclidean plane: all points on the same blue ray share the same coordinate value u, and all points on the same red hyperbola share the same coordinate value v. In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane
The center of the Kiepert hyperbola lies on the nine-point circle. The center is the midpoint of the line segment joining the isogonic centers of triangle A B C {\displaystyle ABC} which are the triangle centers X(13) and X(14) in the Encyclopedia of Triangle Centers.
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]
The Poincaré coordinates of a point are the Cartesian coordinates of the point when the point is mapped in the Poincaré disk model of the hyperbolic plane, [1] the x-axis is mapped to the segment (−1,0) − (1,0) and the origin is mapped to the centre of the boundary circle. The Poincaré coordinates, in terms of the Beltrami coordinates, are:
Lines through a given point P and asymptotic to line R. Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry: For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not ...
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