hyperbola - Definition. Was ist hyperbola
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Was (wer) ist hyperbola - definition

TYPE OF SMOOTH CURVE, LYING IN A PLANE
Hyperbolas; Hiperbola; Rectangular hyperbola; Conjugate hyperbola; Hyperbolic arc; Hyperbolae; Hyperbola (mathematics); Equilateral hyperbola; Right hyperbola
  • Hyperbolas as declination lines on a sundial
  • Hyperbola (red): two views of a cone and two Dandelin spheres ''d''<sub>1</sub>, ''d''<sub>2</sub>
  • ''a'' {{=}} ''b'' {{=}} 1}} giving the [[unit hyperbola]] in blue and its conjugate hyperbola in green, sharing the same red asymptotes.
  • Hyperbola as an affine image of the unit hyperbola
  • Hyperbola as affine image of ''y'' = 1/''x''
  • Hyperbola (red): features
  • Hyperbola: definition with circular directrix
  • Hyperbola: definition by the distances of points to two fixed points (foci)
  • Rotating the coordinate system in order to describe a rectangular hyperbola as graph of a function
  • Hyperbola: construction of a directrix
  • Hyperbola: definition with directrix property
  • Hyperbola: directrix property
  • Hyperbola: semi-axes ''a'',''b'', linear eccentricity ''c'', semi latus rectum ''p''
  • Point construction: asymptotes and ''P''<sub>1</sub> are given → ''P''<sub>2</sub>
  • Hyperbola: pole-polar relation
  • Hyperbola: Polar coordinates with pole = focus
  • Hyperbola: Polar coordinates with pole = center
  • Hyperbola: the midpoints of parallel chords lie on a line.
  • Hyperbola: inscribed angle theorem
  • Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes.
  • Hyperbola: Steiner generation
  • Hyperbola: tangent-asymptotes-triangle
  • Tangent construction: asymptotes and ''P'' given → tangent
  • Hyperbola: the tangent bisects the lines through the foci
  • Three rectangular hyperbolas <math>y=A/x</math> with the coordinate axes as asymptotes<br />
red: ''A'' = 1; magenta: ''A'' = 4; blue: ''A'' = 9
  • Hyperbola: 3 properties
  • Hyperbola: Pin and string construction
  • alt=The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half; the boundary curve of the slice on the cone is the hyperbola. A double cone consists of two cones stacked point-to-point and sharing the same axis of rotation; it may be generated by rotating a line about an axis that passes through a point of the line.
  • Trisecting an angle (AOB) using a hyperbola of eccentricity 2 (yellow curve)
  • Hyperbola ''y'' = 1/''x'': Steiner generation
  • Pencil of conics with a common vertex and common semi latus rectum
  • Hyperbola with its orthoptic (magenta)
  • [[Central projection]] of circles on a sphere: The center ''O'' of projection is inside the sphere, the image plane is red. <br />
As images of the circles one gets a circle (magenta), ellipses, hyperbolas and lines. The special case of a parabola does not appear in this example.<br />
(If center ''O'' were ''on'' the sphere, all images of the circles would be circles or lines; see [[stereographic projection]]).

Hyperbola         
·noun A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base than the side of the cone makes. It is a plane curve such that the difference of the distances from any point of it to two fixed points, called foci, is equal to a given distance. ·see Focus. If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. Both curves are regarded as branches of the same hyperbola. ·see ·Illust. of Conic section, and Focus.
hyperbola         
[h??'p?:b?l?]
¦ noun (plural hyperbolas or hyperbolae -li:) Mathematics a symmetrical open curve (or pair of curves) formed by the intersection of a cone (or pair of oppositely directed cones) with a plane at a smaller angle with its axis than the side of the cone.
Origin
C17: mod. L., from Gk huperbole 'excess' (from huper 'above' + ballein 'to throw').
rectangular hyperbola         
¦ noun a hyperbola with rectangular asymptotes.

Wikipedia

Hyperbola

In mathematics, a hyperbola ( (listen); pl. hyperbolas or hyperbolae (listen); adj. hyperbolic (listen)) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

Hyperbolas arise in many ways:

  • as the curve representing the reciprocal function y ( x ) = 1 / x {\displaystyle y(x)=1/x} in the Cartesian plane,
  • as the path followed by the shadow of the tip of a sundial,
  • as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or, more generally, any spacecraft (or celestial object) exceeding the escape velocity of the nearest planet or other gravitational body,
  • as the scattering trajectory of a subatomic particle (acted on by repulsive instead of attractive forces but the principle is the same),
  • in radio navigation, when the difference between distances to two points, but not the distances themselves, can be determined,

and so on.

Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve y ( x ) = 1 / x {\displaystyle y(x)=1/x} the asymptotes are the two coordinate axes.

Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).