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Concept
Hyperbolic orthogonality
Summary
In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the relativity of simultaneity.
Two lines are hyperbolic orthogonal when they are reflections of each other over the asymptote of a given hyperbola.
Two particular hyperbolas are frequently used in the plane:
The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote A, a pair of lines (a, b) are hyperbolic orthogonal if there is a pair (c, d) such that , and c is the reflection of d across A.
Similar to the perpendularity of a circle radius to the tangent, a radius to a hyperbola is hyperbolic orthogonal to a tangent to the hyperbola.
A bilinear form is used to describe orthogonality in analytic geometry, with two elements orthogonal when their bilinear form vanishes. In the plane of complex numbers , the bilinear form is , while in the plane of hyperbolic numbers the bilinear form is
The vectors z1 and z2 in the complex number plane, and w1 and w2 in the hyperbolic number plane are said to be respectively Euclidean orthogonal or hyperbolic orthogonal if their respective inner products [bilinear forms] are zero.
The bilinear form may be computed as the real part of the complex product of one number with the conjugate of the other. Then
entails perpendicularity in the complex plane, while
implies the ws are hyperbolic orthogonal.
The notion of hyperbolic orthogonality arose in analytic geometry in consideration of conjugate diameters of ellipses and hyperbolas. If g and g′ represent the slopes of the conjugate diameters, then in the case of an ellipse and in the case of a hyperbola.
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