unit hyperbola requires the conjugate hyperbola y 2 − x 2 = 1 {\displaystyle y^{2}-x^{2}=1} to complement it in the plane. This pair of hyperbolas share...

In geometry, a conjugate hyperbola to a given hyperbola shares the same asymptotes but lies in the opposite two sectors of the plane compared to the original...

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution...

diameters of a hyperbola are conjugate when each bisects all chords parallel to the other. In this case both the hyperbola and its conjugate are sources...

through (a, 0) and (−a, 0). The case a = 1 is called the unit hyperbola. The conjugate hyperbola is given by { z : ‖ z ‖ 2 = − a 2 } {\displaystyle \left\{z:\lVert...

a hyperbola is perpendicular to the conjugate axis and to each directrix. The product of the perpendicular distances from a point P on a hyperbola or...

ellipse, parabola, or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is...

hyperbola with asymptote A, its reflection in A produces the conjugate hyperbola. Any diameter of the original hyperbola is reflected to a conjugate diameter...

surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse...

over 200. Osgood and Graustein used the rectangular hyperbola, its conjugate hyperbola, and conjugate diameters to rationalize tie rods at 15 degree radial...

\over e^{2}-1}.} The transverse axis of a hyperbola coincides with the major axis. In a hyperbola, a conjugate axis or minor axis of length 2 b {\displaystyle...

Lorentz transformation". This diagram included the unit hyperbola, its conjugate, and a pair of conjugate diameters. Since the 1960s a version of this more...

unit hyperbola x2 − y2 = 1, and its conjugate x2 − y2 = −1 can be used to form a polar decomposition based on the branch of the unit hyperbola through...

equation represents a hyperbola; if we also have A + C = 0 {\displaystyle A+C=0} , the equation represents a rectangular hyperbola. A quadric, or quadric...

notions preceded physical interpretation. For instance, conjugate diameters of conjugate hyperbolas are related as space and time. The principle of relativity...

duration lies on a hyperbola. Since the hyperbola is traced by a hyperbolic angle, and velocity determines a point on the hyperbola, there is a hyperbolic...

point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines). All these degenerate...

{\displaystyle \eta } -coordinates such that the wanted conic section appears as a hyperbola and the points U , V {\displaystyle U,V} become the points at infinity...

relativistic wave equations relativistic mass split-complex number unit hyperbola world line black holes no-hair theorem Hawking radiation Hawking temperature...

Edinburgh, and included a sheet of nine figures, two of which display conjugate hyperbolas. Having been stung in the Great Vector Debate over the non-associativity...

sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian...

{\displaystyle OI} is tangent to this hyperbola at I {\displaystyle I} . The hyperbola is the isogonal conjugate of O I {\displaystyle OI} , the line joining...

Alvis–Curtis duality Artin–Verdier duality Beta-dual space Coherent duality Conjugate hyperbola De Groot dual Dual abelian variety Dual basis in a field extension...

terminology stems from the use of conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal. Complemented...

ellipse. The midpoint of a segment connecting a hyperbola's vertices is the center of the hyperbola. The perpendicular bisector of a side of a triangle...

runs from S1 to P behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from P to F2 runs and...

point of rotational symmetries. Similarly the centre of an ellipse or a hyperbola is where the axes intersect. Several special points of a triangle are...

{\displaystyle \det A_{33}=AC-{\frac {B^{2}}{4}}} : Q {\displaystyle Q} is a hyperbola if and only if det A 33 < 0 {\displaystyle \det A_{33}<0} , Q {\displaystyle...

counter-sphere. One example is the unit hyperbola that forms a quasi-sphere of the hyperbolic plane, and its conjugate hyperbola, which is its counter-sphere. The...

The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane)....