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Consequence: for any pair of points , of a chord there exists a skew reflection with an axis (set of fixed points) passing through the center of the hyperbola, which exchanges the points , and leaves the hyperbola (as a whole) fixed.
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
More generally, for any collection of points P i, weights w i, and constant C, one can define a circle as the locus of points X such that (,) =.. The director circle of an ellipse is a special case of this more general construction with two points P 1 and P 2 at the foci of the ellipse, weights w 1 = w 2 = 1, and C equal to the square of the major axis of the ellipse.
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).
The unit hyperbola is blue, its conjugate is green, and the asymptotes are red. In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation = In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length
the x-coordinate of a point is the signed distance of its projection onto the line (the foot of the perpendicular segment to the line from that point) to the origin; the y-coordinate is the signed distance from the point to the line, with the sign according to whether the point is on the positive or negative side of the oriented line.
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 .
The centre of a parabola is the contact point of the figurative straight. The centre of a hyperbola lies without the curve, since the figurative straight crosses the curve. The tangents from the centre to the hyperbola are called 'asymptotes'. Their contact points are the two points at infinity on the curve.