Hyperbola

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 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.

[1] In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others.

), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).

Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones.

[3] The names of the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment.

from the center and parallel to the minor axis are called directrices of the hyperbola (see diagram).

of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:

The inverse statement is also true and can be used to define a hyperbola (in a manner similar to the definition of a parabola): For any point

The process of "reciprocation in a circle C" consists of replacing every line and point in a geometrical figure with their corresponding pole and polar, respectively.

For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is

Two hyperbolas are geometrically similar to each other – meaning that they have the same shape, so that one can be transformed into the other by rigid left and right movements, rotation, taking a mirror image, and scaling (magnification) – if and only if they have the same eccentricity.

Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2f.

The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram).

(see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas: For two points

If the chord degenerates into a tangent, then the touching point divides the line segment between the asymptotes in two halves.

Other orthogonal two-dimensional coordinate systems involving hyperbolas may be obtained by other conformal mappings.

Besides providing a uniform description of circles, ellipses, parabolas, and hyperbolas, conic sections can also be understood as a natural model of the geometry of perspective in the case where the scene being viewed consists of circles, or more generally an ellipse.

One sees a hyperbola whenever catching sight of a portion of a circle cut by one's lens plane.

On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light.

The intersection of this cone with the horizontal plane of the ground forms a conic section.

In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the declination line).

The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon.

The collection of such hyperbolas for a whole year at a given location was called a pelekinon by the Greeks, since it resembles a double-bladed axe.

Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from a LORAN or GPS transmitters.

Conversely, a homing beacon or any transmitter can be located by comparing the arrival times of its signals at two separate receiving stations; such techniques may be used to track objects and people.

appears as one solution to the Korteweg–de Vries equation which describes the motion of a soliton wave in a canal.

As shown first by Apollonius of Perga, a hyperbola can be used to trisect any angle, a well studied problem of geometry.

Hyperbolas appear as plane sections of the following quadrics: Brozinsky, Michael K. (1984), "Reflection Property of the Ellipse and the Hyperbola", College Mathematics Journal, 15 (2): 140–42, doi:10.1080/00494925.1984.11972763 (inactive 2024-12-16), JSTOR 2686519{{citation}}: CS1 maint: DOI inactive as of December 2024 (link)

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.
A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone . The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.
Hyperbola (red): features
Hyperbola: definition by the distances of points to two fixed points (foci)
Hyperbola: definition with circular directrix
Rotating the coordinate system in order to describe a rectangular hyperbola as graph of a function
Three rectangular hyperbolas with the coordinate axes as asymptotes
red: A = 1; magenta: A = 4; blue: A = 9
Hyperbola: directrix property
Hyperbola: definition with directrix property
Pencil of conics with a common vertex and common semi latus rectum
Hyperbola: construction of a directrix
Hyperbola (red): two views of a cone and two Dandelin spheres d 1 , d 2
Hyperbola: Pin and string construction
Hyperbola: Steiner generation
Hyperbola y = 1/ x : Steiner generation
Hyperbola: inscribed angle theorem
Hyperbola as an affine image of the unit hyperbola
Hyperbola as affine image of y = 1/ x
Tangent construction: asymptotes and P given → tangent
Point construction: asymptotes and P 1 are given → P 2
Hyperbola: tangent-asymptotes-triangle
Hyperbola: semi-axes a , b , linear eccentricity c , semi latus rectum p
Hyperbola: 3 properties
Here a = b = 1 giving the unit hyperbola in blue and its conjugate hyperbola in green, sharing the same red asymptotes.
Hyperbola: Polar coordinates with pole = focus
Hyperbola: Polar coordinates with pole = center
Animated plot of Hyperbola by using
A ray through the unit hyperbola at the point , where is twice the area between the ray, the hyperbola, and the -axis. For points on the hyperbola below the -axis, the area is considered negative.
Hyperbola: the tangent bisects the lines through the foci
Hyperbola: the midpoints of parallel chords lie on a line.
Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes.
Hyperbola with its orthoptic (magenta)
Hyperbola: pole-polar relation
Sinusoidal spirals ( r n = –1 n cos( nθ ), θ = π /2 ) in polar coordinates and their equivalents in rectangular coordinates :
n = −2 : Equilateral hyperbola
n = −1 : Line
n = −1/2 : Parabola
n = 1/2 : Cardioid
n = 1 : Circle
n = 2 : Lemniscate of Bernoulli
Central projection of circles on a sphere: The center O of projection is inside the sphere, the image plane is red.
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.
(If center O were on the sphere, all images of the circles would be circles or lines; see stereographic projection ).
Hyperbolas as declination lines on a sundial
The contact zone of a level supersonic aircraft's shockwave on flat ground (yellow) is a part of a hyperbola as the ground intersects the cone parallel to its axis.
Trisecting an angle (AOB) using a hyperbola of eccentricity 2 (yellow curve)