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Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic n-space. The stabilizer of any particular line is isomorphic to the product of the orthogonal groups O( n ) and O(1), where O( n ) acts on the tangent space of a ...
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 timeline. This dependence on a certain timeline is determined ...
Orthoptic (geometry) In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Parabola. Orthoptic of the parabola (its directrix) Ellipse. Orthoptic of the ellipse (its director circle) Minimum bounding box of the ellipse ( circumscribed by the orthoptic circle) Major and minor ...
In geometry, two conic sections are called confocal if they have the same foci. Because ellipses and hyperbolas have two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally (at right angles).
The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation. The hyperbola xy = 1 is rectangular with semi-major axis , analogous to the circular angle equaling the area of a circular sector in a circle ...
In mathematics, a hyperbola is a ... the intersection points of orthogonal tangents lie ... the circle that is centered at the hyperbola's center and that passes ...
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.
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...