Hyperbolic-orthogonal
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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
When c = 1 and the y's and z's are zero, then (x,t) and (x1,t1) 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.
Suppose lines m and n are related so that for every p ∈ m and q ∈ n, p is hyperbolic orthogonal to q. Then the relation of m to n is the same as the relation of a hyperbola H between asymptotes and the conjugate hyperbola K sharing the same asymptotes: the reflection across an asymptote that takes H to K also interchanges m and n.
[edit] References
- Herman Minkowski (1908) "Raum und Zeit".
- Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the American Academy of Arts and Sciences 48:387-507.
- Synthetic Spacetime (excerpt from above)
