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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.
For a hyperbola =, > the intersection points of orthogonal tangents lie on the circle + =. This circle is called the orthoptic of the given hyperbola. The tangents may belong to points on different branches of the hyperbola.
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 ...
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}}.}
Considering the pencils of confocal ellipses and hyperbolas (see lead diagram) one gets from the geometrical properties of the normal and tangent at a point (the normal of an ellipse and the tangent of a hyperbola bisect the angle between the lines to the foci). Any ellipse of the pencil intersects any hyperbola orthogonally (see diagram).
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.
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, two circles are said to be orthogonal if their respective tangent lines at the points of intersection are perpendicular (meet at a right angle). A straight line through a circle's center is orthogonal to it, and if straight lines are also considered as a kind of generalized circles , for instance in inversive geometry , then an ...