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Information about Hyperbola

Not to be confused with hyperbole.


In mathematics, a hyperbola (Greek ὑπερβολή literally 'overshooting' or 'excess') is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone.

It may also be defined as the locus of points where the difference in the distance to two fixed points (called the foci) is constant. That fixed difference in distance is two times a where a is the distance from the center of the hyperbola to the vertex of the nearest branch of the hyperbola. a is also known as the semi-major axis of the hyperbola. The foci lie on the transverse axis and their midpoint is called the center.

For a simple geometric proof that the two characterizations above are equivalent to each other, see Dandelin spheres.

Algebraically, a hyperbola is a curve in the Cartesian plane defined by an equation of the form
such that , where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the hyperbola, exists.

The graph of two variables varying inversely on the Cartesian coordinate plane is a hyperbola.

Definitions

The first two were listed above:
  • The intersection between a right circular conical surface and a plane which cuts through both halves of the cone.
  • The locus of points where the difference in the distance to two fixed points (called the foci) is constant.
  • The locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant larger than 1. This constant is the eccentricity of the hyperbola.
A hyperbola comprises two disconnected curves called its arms or branches which separate the foci. At large distances from the foci the hyperbola begins to approximate two lines, known as asymptotes. The asymptotes cross at the center of the hyperbola and have slope for an East-West opening hyperbola or for a North-South opening hyperbola.

A hyperbola has the property that a ray originating at one of the foci is reflected in such a way as to appear to have originated at the other focus. Also, if rays are directed towards one of the foci from the exterior of the hyperbola, they will be reflected towards the other foci.

Enlarge picture
Conjugate unit rectangular hyperbolas
A special case of the hyperbola is the equilateral or rectangular hyperbola, in which the asymptotes intersect at right angles. The rectangular hyperbola with the coordinate axes as its asymptotes is given by the equation xy=c, where c is a constant.

Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.

If on the hyperbola equation one switches x and y, the conjugate hyperbola is obtained. A hyperbola and its conjugate have the same asymptotes.

Equations

Cartesian

East-west opening hyperbola centered at (h,k):
North-south opening hyperbola centered at (h,k):
The major axis runs through the center of the hyperbola and intersects both arms of the hyperbola at the vertices (bend points) of the arms. The foci lie on the extension of the major axis of the hyperbola.

The minor axis runs through the center of the hyperbola and is perpendicular to the major axis.

In both formulas a is the semi-major axis (half the distance between the two arms of the hyperbola measured along the major axis), and b is the semi-minor axis.

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the length of the sides tangent to the hyperbola are 2b in length while the sides that run parallel to the line between the foci (the major axis) are 2a in length. Note that b may be larger than a.

If one calculates the distance from any point on the hyperbola to each focus, the absolute value of the difference of those two distances is always 2a.

The eccentricity is given by


The foci for an east-west opening hyperbola are given by
where c is given by
and for a north-south opening hyperbola are given by
again with


For rectangular hyperbolas with the coordinate axes parallel to their asymptotes:




The simplest example of these are the hyperbolas
.

Polar

East-west opening hyperbola:
North-south opening hyperbola:
Northeast-southwest opening hyperbola:
Northwest-southeast opening hyperbola:


In all formulas the center is at the pole, and a is the semi-major axis and semi-minor axis.

Parametric

East-west opening hyperbola:


North-south opening hyperbola:


In all formulas (h,k) is the center of the hyperbola, a is the semi-major axis, and b is the semi-minor axis.

See also

External links

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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their
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In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix
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plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space.
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locus (Latin for "place", plural loci) is a collection of points which share a property. The term 'locus' is usually used of a condition which defines a continuous figure or figures, that is, a curve.
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Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over").
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In geometry, the foci (singular focus) are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola.
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Dandelin spheres characterized thus:

Each Dandelin sphere touches, but does not cross, both the plane and the cone.


This concept is named in honor of Germinal Pierre Dandelin.
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Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
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locus (Latin for "place", plural loci) is a collection of points which share a property. The term 'locus' is usually used of a condition which defines a continuous figure or figures, that is, a curve.
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line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
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In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed.

Unspecified constants

The most widely mentioned sort of constant
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eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

In particular,
  • The eccentricity of a circle is zero.

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In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle.
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An asymptote is a straight line or curve A to which another curve B (the one being studied) approaches closer and closer as one moves along it. As one moves along B, the distance between it and the asymptote A
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reflection (also spelled reflexion) is a map that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q.
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angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
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trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
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trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
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parametric equations bear slight similarity to functions: they allow one to use arbitrary values, called parameters, in place of independent variables in equations, which in turn provide values for dependent variables.
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ellipse (from the Greek ἔλλειψις, literally absence) is the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant.
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hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", etc.
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hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", etc.
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semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape.
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In geometry, the semi-minor axis (also semiminor axis) is a line segment associated with most conic sections (that is, with ellipses and hyperbolas). One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis.
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eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

In particular,
  • The eccentricity of a circle is zero.

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parabola (from the Greek: παραβολή) (IPA pronunciation: /pəˈrab(ə)lə/
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