Hyperbola

hyperbolicrectangular hyperbolahyperbolaseccentricity exceeding 1.0equilateral hyperbolahyperbolic archyperbolic curvehyperbolic segmenthyperbolic shapenegative curve
In mathematics, a hyperbola (plural hyperbolas or hyperbolae) 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.wikipedia
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Ellipse

ellipticalellipticeccentricity
(The other conic sections are the parabola and the ellipse.
Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded.

Conic section

conicconic sectionsconics
The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix.
3. Hyperbola]]

Hyperbolic function

hyperbolic tangenthyperbolichyperbolic cosine
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). Just as the trigonometric functions are defined in terms of the unit circle, so also the hyperbolic functions are defined in terms of the unit hyperbola, as shown in this diagram.
form the right half of the equilateral hyperbola.

Paraboloid

hyperbolic paraboloidelliptic paraboloidhyperbolic paraboloids
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).
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).

Focus (geometry)

focifocusfocal points
Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix.
For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola.

Eccentricity (mathematics)

eccentricityeccentriceccentricities
Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix.
The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound.

Smoothness

smoothsmooth functionsmooth map
In mathematics, a hyperbola (plural hyperbolas or hyperbolae) 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.
An ellipse tends to a circle as the eccentricity approaches zero, or to a parabola as it approaches one; and a hyperbola tends to a parabola as the eccentricity drops toward one; it can also tend to intersecting lines.

Hyperbolic navigation

hyperbolicradio navigation
Plotting all of the potential locations of the receiver for the measured delay localizes the receiver to a hyperbolic line on a chart.

Gravity assist

gravitational slingshotgravitational assistslingshot
This oversimplified example is impossible to refine without additional details regarding the orbit, but if the spaceship travels in a path which forms a hyperbola, it can leave the planet in the opposite direction without firing its engine.

Apollonius of Perga

ApolloniusApollonius of PergeApollonian
The term hyperbola is believed to have been coined by Apollonius of Perga (c.
His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.

Sundial

sundialssun dialsun clock
Hyperbolas may be seen in many sundials.
The corresponding light-spot or shadow-tip, if it falls onto a flat surface, will trace out a conic section, such as a hyperbola, ellipse or (at the North or South Poles) a circle.

Parabola

parabolicparabolic curveparabolic arc
(The other conic sections are the parabola and the ellipse.
Remark: Steiner's generation is also available for ellipses and hyperbolas.

Mathematical object

mathematical objectsobjectsgeometric object
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).

Orbit

orbitsorbital motionplanetary motion
An open orbit will have a parabolic shape if it has velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a hyperbola when its velocity is greater than the escape velocity.

Unit hyperbola

hyperbolaasymptotes to the unit hyperbolaone hyperbola
Just as the trigonometric functions are defined in terms of the unit circle, so also the hyperbolic functions are defined in terms of the unit hyperbola, as shown in this diagram.
The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale.

Plane curve

complex plane curvecurvecurve lying in a plane
In mathematics, a hyperbola (plural hyperbolas or hyperbolae) 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.

Menaechmus

Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones.
Menaechmus (Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher Plato and for his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube using the parabola and hyperbola.

Pascal's theorem

hexagrammum mysticumGeneration of Conic SectionsHexagrammum Mysticum Theorem
This property of a hyperbola is an affine version of the 3-point-degeneration of Pascal's theorem.
In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem) states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon.

Similarity (geometry)

similarsimilaritysimilar triangles
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.

Asymptote

asymptoticasymptoticallyasymptotes
Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms.
The hyperbola

Hyperbolic geometry

hyperbolic planehyperbolichyperbolic surface
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).

Matrix (mathematics)

matrixmatricesmatrix theory
An affine transformation of the Euclidean plane has the form, where A is a regular matrix (its determinant is not 0) and \vec f_0 is an arbitrary vector.

Multilateration

time difference of arrivalTDOAUnilateration
A hyperbola is the basis for solving multilateration problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points.
When these possible locations are plotted, they form a hyperbolic curve.

Orthoptic (geometry)

Isopticorthoptic
:This description of the tangents of a hyperbola is an essential tool for the determination of the orthoptic of a hyperbola.
# The orthoptic of a hyperbola

Degenerate conic

degeneratedegenerate casedegenerate cases
A special case of a hyperbola—the degenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero:
This degenerate conic occurs as the limit case a=1, b=0 in the pencil of hyperbolas of equations The limiting case a=0, b=1 is an example of a degenerate conic consisting of twice the line at infinity.