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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, or relativity of simultaneity. In Minkowski's development the hyperbola of type (B) above is in ...
The radius and tangent are hyperbolic orthogonal at a since p(a) and are reflections of each other in the asymptote y = x of the unit hyperbola. When interpreted as split-complex numbers (where j j = +1 ), the two numbers satisfy j p ( a ) = d p d a . {\displaystyle jp(a)={\tfrac {dp}{da}}.}
Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the
The orthoptic of a hyperbola =, > is the director circle + = (in case of a ≤ b there are no orthogonal tangents, see below), The orthoptic of an astroid x 2 / 3 + y 2 / 3 = 1 {\displaystyle x^{2/3}+y^{2/3}=1} is a quadrifolium with the polar equation r = 1 2 cos ( 2 φ ) , 0 ≤ φ < 2 π {\displaystyle r={\tfrac {1}{\sqrt {2}}}\cos(2 ...
A parabola, being tangent to the line at infinity, would have its center being a point on the line at infinity. Hyperbolas intersect the line at infinity in two distinct points and the polar lines of these points are the asymptotes of the hyperbola and are the tangent lines to the hyperbola at these points of infinity.
Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of circular sector area u and hyperbolic functions depending on hyperbolic sector area u. The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle.
Likewise the limit of an analogous pencil of hyperbolas degenerates to a pair of lines perpendicular to the major axis. Thus a rectangular grid consisting of orthogonal pencils of parallel lines is a kind of net of degenerate confocal conics. Such an orthogonal net is the basis for the Cartesian coordinate system.
For example the orthogonal trajectories of a pencil of confocal ellipses are the confocal hyperbolas with the same foci. For Cassini ovals one has: The orthogonal trajectories of the Cassini curves with foci , are the equilateral hyperbolas containing , with the same center as the Cassini ovals (see picture). Proof: