Definitions

hyperbolic function

In mathematics, one of a set of functions related to the hyperbola in the same way the trigonometric functions relate to the circle. They are the hyperbolic sine, cosine, tangent, secant, cotangent, and cosecant (written “sinh,” “cosh,” etc.). The hyperbolic equivalent of the fundamental trigonometric identity is cosh2math.z − sinh2math.z = 1. The hyperbolic sine and cosine, particularly useful for finding special types of integrals, can be defined in terms of exponential functions: sinhmath.x = (math.emath.xmath.emath.x) ÷ 2 and coshmath.x = (math.emath.x + math.emath.x) ÷ 2

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Non-Euclidean geometry, useful in modeling interstellar space, that rejects the parallel postulate, proposing instead that at least two lines through any point not on a given line are parallel to that line. Though many of its theorems are identical to those of Euclidean geometry, others differ. For example, two parallel lines converge in one direction and diverge in the other, and the angles of a triangle add up to less than 180°.

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In mathematics, two points in the Cartesian plane are hyperbolically orthogonal if the slopes of their rays from the origin are reciprocal to one another.

If the points are (x,y) and (u,v), then they are hyperbolically orthogonal if

y/x = u/v.

Using complex numbers z = x + y i and w = u + v i, the points z and w in C are hyperbolically orthogonal if the real part of their complex product is zero, i.e.

xu - yv = 0.

If two hyperbolically-orthogonal points form two angles with the horizontal axis, then they are complementary angles.

Since Hermann Minkowski's foundation for spacetime study in 1908, the concept of points in a spacetime plane being hyperbolic-orthogonal to a timeline (tangent to a World line) has been used to define simultaneity of events relative to the timeline. To see Minkowski's use of the concept, click on the link below and scroll down to the expression

$c^\left\{2\right\} t t_1 - x x_1 - y y_1 - z z_1 = 0$.
When c = 1 and the y's and z's are zero, then (x,t) and $\left(x_1,t_1\right)$ are hyperbolic-orthogonal.

To get away from the dependence on analytic geometry in the above definition, Edwin Bidwell Wilson and Gilbert N. Lewis provided an approach using the ideas of synthetic geometry in 1912: the radius to a point on an hyperbola and the tangent line at that point are hyperbolic-orthogonal. They note (p.415) "in our plane no pair of perpendicular [hyperbolic-orthogonal] lines is better suited to serve as coordinate axes than any other pair", an expression of the Principle of relativity.

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