Search results
Results from the WOW.Com Content Network
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 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}}.}
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.
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).
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 ...
Cassini ovals and their orthogonal trajectories (hyperbolas) Orthogonal trajectories of a given pencil of curves are curves which intersect all given curves orthogonally. For example the orthogonal trajectories of a pencil of confocal ellipses are the confocal hyperbolas with the same foci. For Cassini ovals one has:
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.