Wikipedia / Hyperbola
Wikipedia / 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. (The other conic sections are the parabola and the ellipse, with the circle being a special type of 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.
Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflectionss across the tangent line at that point, or as the solution of certain bivariate quadratic equationss such as the reciprocal 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.
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 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).
Etymology and history
The word "hyperbola" derives from the Greek , meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have been coined by Apollonius of Perga () in his definitive work on the conic sections, the Conics. 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. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than the segment or exceed the segment.
Definitions
As locus of points
A hyperbola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:
of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis. It contains the vertices of the foci to the center is called the focal distance or linear eccentricity is the eccentricity.
can be viewed in a different way (see diagram): is called the circular directrix) of the hyperbola.. This property should not be confused with the definition of a hyperbola with help of a directrix (line) below.
Hyperbola with equation y = A/x
If the xy-coordinate system is rotated. (whose semi-axes
Thus, in an xy-coordinate system the graph of a function is a rectangular hyperbola entirely in the first and third quadrants with
the coordinate axes as asymptotes,
as major axis ,
the center and the semi-axis
the vertices
the semi-latus rectum and radius of curvature
the linear eccentricity
the tangent
the semi-axes
as major axis,
the vertices
so that the new center is remain unchanged.
By the directrix property
from the center and parallel to the minor axis are called directrices of the hyperbola (see diagram).
The second case is proven analogously. The inverse statement is also true and can be used to define a hyperbola (in a manner similar to the definition of a parabola):
and any real number is a hyperbola.
yields a parabola an ellipse.)
Proof
produces the equations
This is the equation of an ellipse) or a parabola) or a hyperbola). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
, the x.
Construction of a directrix
are inverse with respect to the circle inversion can be constructed using the theorem of Thales.
(see diagram).
As plane section of a cone
The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see diagram: red curve). In order to prove the defining property of a hyperbola (see above) one uses two Dandelin spheres, which are spheres that touch the cone along circles and are the foci of the hyperbola.
be an arbitrary point of the intersection curve.
The generatrix.
and, hence, are of equal length.
and, hence, are of equal length.
is independent of the hyperbola point have to be on circles simply rotates about apex without changing its length.
Pin and string construction
The definition of a hyperbola by its foci and its circular directrices (see above) can be used for drawing an arc of it with help of pins, a string and a ruler:
Choose the foci and one of the circular directrices)
A ruler.
A string.
.
Take a pen and hold the string tight to the edge of the ruler.
Rotating (see the definition of a hyperbola by circular directrices).
Steiner generation of a hyperbola
The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section:
(see diagram). The parallel projection are points of the uniquely defined hyperbola.
Remarks:
in order to get more points, but the determination of the intersection points would become more inaccurate. A better idea is extending the points already constructed by symmetry (see animation).
The Steiner generation exists for ellipses and parabolas, too.
The Steiner generation is sometimes called a parallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.
Inscribed angles for hyperbolas y = a/(x − b) + c and the 3-point-form
with different x- and y uses the inscribed angle theorem for hyperbolas:
Analogous to the inscribed angle theorem for circles one gets the
A consequence of the inscribed angle theorem for hyperbolas is the
As an affine image of the unit hyperbola x2 − y2 = 1
Another definition of a hyperbola uses affine transformations:
Parametric representation
is a regular matrix (its determinant is mapped onto the hyperbola
a tangent vector at this point.
Vertices
are not which yields
The formulae were used.
The two vertices
Implicit representation
Hyperbola in space
to be vectors in space.
As an affine image of the hyperbola y = 1/x
, an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola
and the parameter of a vertex is
are the vertices of the hyperbola.
The following properties of a hyperbola are easily proven using the representation of a hyperbola introduced in this section.
Tangent construction
This means that
This property provides a way to construct the tangent at a point on the hyperbola.
This property of a hyperbola is an affine version of the 3-point-degeneration of Pascal's theorem.
Area of the grey parallelogram:
Point construction
(for simplicity the center is the origin) the following is true:
.
This property provides a possibility to construct points of a hyperbola if the asymptotes and one point are given.
This property of a hyperbola is an affine version of the 4-point-degeneration of Pascal's theorem.
Tangent–asymptotes triangle
.
The area (see rules for determinants of the hyperbola. Hence:
Reciprocation of a circle
The reciprocation of a circle B in a circle C always yields a conic section such as a hyperbola. 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. The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point.
The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then
Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle C.
This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle B, as well as the envelope of the polar lines of the points on B. Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.
Quadratic equation
in the plane,
satisfy the determinant condition
This determinant is conventionally called the discriminant of the conic section.
A special case of a hyperbola—the degenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero:
is sometimes called the discriminant of the conic section.
(the angle from the positive horizontal axis to the hyperbola's major axis) using the formulae:
These expressions can be derived from the canonical equation
by a translation and rotation of the coordinates
Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in Conic section#Eccentricity in terms of coefficients.
of the hyperbola may be determined from the formulae
the defining equation of the hyperbola can be written
-axis that is given by
-axis is aligned with the transverse axis brings the equation into its canonical form
are the roots of the quadratic equation
For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is
on the hyperbola is defined by the equation
are defined by
The normal line to the hyperbola at the same point is given by the equation
From the equation
For a point on the right branch,
and for a point on the left branch,
This can be proved as follows:
is a point on the hyperbola the distance to the left focal point is
To the right focal point the distance is
and
Subtracting these equations one gets
and
Subtracting these equations one gets
In Cartesian coordinates
Equation
If Cartesian coordinates are introduced such that the origin is the center of the hyperbola and the x-axis is the major axis, then the hyperbola is called east-west-opening and
the foci,
the vertices.
to obtain the equation of the hyperbola:
This equation is called the canonical form of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is congruent to the original (see below).
The axes of symmetry or principal axes are the transverse axis (containing the segment of length 2a with endpoints at the vertices) and the conjugate axis (containing the segment of length 2b perpendicular to the transverse axis and with midpoint at the hyperbola's center). on the conjugate axes are not on the hyperbola.
It follows from the equation that the hyperbola is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin.
Eccentricity
For a hyperbola in the above canonical form, the eccentricity is given by
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.
Asymptotes
. These two lines intersect at the center (origin) and are called asymptotes
With the help of the second figure one can see that
The perpendicular distance from a focus to either asymptote (the semi-minor axis).
From the Hesse normal form of the asymptotes and the equation of the hyperbola one gets:
The product of the distances from a point on the hyperbola to both the asymptotes which can also be written in terms of the eccentricity e
of the hyperbola (above) one can derive:
The product of the slopes of lines from a point P to the two vertices
In addition, from (2) above it can be shown that
The product of the distances from a point on the hyperbola to the asymptotes along lines parallel to the asymptotes
Semi-latus rectum
The length of the chord through one of the foci, perpendicular to the major axis of the hyperbola, is called the latus rectum. One half of it is the semi-latus rectum may also be viewed as the radius of curvature at the vertices.
Tangent
is to implicitly differentiate of the hyperbola. Denoting dy/dx as y′
A particular tangent line distinguishes the hyperbola from the other conic sections. Let f be the distance from the vertex V (on both the hyperbola and its axis through the two foci) to the nearer focus. 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 tangent to the hyperbola at P intersects that axis at point Q at an angle ∠PQV of greater than 45°.
Rectangular hyperbola
the hyperbola is called rectangular (or equilateral is a rectangular hyperbola.
Parametric representation with hyperbolic sine/cosine
Using the hyperbolic sine and cosine functions
Further parametric representations are given in the section Parametric equations below.
Conjugate hyperbola
, change the sign on the right to obtain the equation of the conjugate hyperbola:
A hyperbola and its conjugate may have diameters which are conjugate. In the theory of special relativity, such diameters may represent axes of time and space, where one hyperbola represents eventss at a given spatial distance from the center, and the other represents events at a corresponding temporal distance from the center.
also specify conjugate hyperbolas.
In polar coordinates
Origin at the focus
The polar coordinates used most commonly for the hyperbola are defined relative to the Cartesian coordinate system that has its origin in a focus and its x-axis pointing toward the origin of the "canonical coordinate system" as illustrated in the first diagram.
is called true anomaly.
Relative to this coordinate system one has that
and
Origin at the center
With polar coordinates relative to the "canonical coordinate system" (see second diagram) one has that
Eccentricity
approaches this limit, r approaches infinity and the denominator in either of the equations noted above approaches zero, hence:
Parametric equations
can be described by several parametric equations:
As a rational
With the tangent slope as parameter: and use formulae for the hyperbolic functions) are not covered by the representation. This description of the tangents of a hyperbola is an essential tool for the determination of the orthoptic of a hyperbola.
Hyperbolic functions
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. In a unit circle, the angle (in radians) is equal to twice the area of the circular sector which that angle subtends. The analogous hyperbolic angle is likewise defined as twice the area of a hyperbolic sector.
which simplifies to the area hyperbolic cosine and its inverse the area hyperbolic sine
Properties
Reflection property
This is called the optical property or reflection property of a hyperbola.
Proof:
. From the diagram and the triangle inequality.
Midpoints of parallel chords
The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram).
The points of any chord may lie on different branches of the hyperbola.
(see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas:
of a chord there exists a skew reflection.
and the asymptotes are given.
If the chord degenerates into a tangent, then the touching point divides the line segment between the asymptotes in two halves.
Orthogonal tangents – orthoptic
the intersection points of orthogonal. This circle is called the orthoptic of the given hyperbola.
The tangents may belong to points on different branches of the hyperbola.
there are no pairs of orthogonal tangents.
Pole-polar relation for a hyperbola
to be an arbitrary point different from the origin, then
, not through the center of the hyperbola.
This relation between points and lines is a bijection.
The inverse function maps
and
Such a relation between points and lines generated by a conic is called pole-polar relation or just polarity. The pole is the point, the polar the line. See Pole and polar.
By calculation one checks the following properties of the pole-polar relation of the hyperbola:
For a point (pole) on).
outside).
For a point within).
Remarks:
).
respectively belong to pairs of pole and polar.
Pole-polar relations exist for ellipses and parabolas, too.
Other properties
The following are concurrent: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.
The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes.
Since both the transverse axis and the conjugate axis are axes of symmetry, the symmetry group of a hyperbola is the Klein four-group.
The rectangular hyperbolas xy = constant admit group actions by squeeze mappings which have the hyperbolas as invariant sets.
The center of a rectangular hyperbola is the Poncelet point of any four points on it that do not form an orthocentric system.
Arc length
The arc length of a hyperbola does not have an elementary expression. The upper half of a hyperbola can be parameterized as
can be computed as:
, this can also be represented using the incomplete elliptic integral of the second kind:
Using only real numbers, this becomes
is the incomplete elliptic integral of the first kind is the Gudermannian function.
Derived curves
Several other curves can be derived from the hyperbola by inversion, the so-called inverse curves of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the lemniscate of Bernoulli; the lemniscate is also the envelope of circles centered on a rectangular hyperbola and passing through the origin. If the center of inversion is chosen at a focus or a vertex of the hyperbola, the resulting inverse curves are a limaçon or a strophoid, respectively.
Elliptic coordinates
A family of confocal hyperbolas is the basis of the system of elliptic coordinates in two dimensions. These hyperbolas are described by the equation
where the foci are located at a distance c from the origin on the x-axis, and where θ is the angle of the asymptotes with the x-axis. Every hyperbola in this family is orthogonal to every ellipse that shares the same foci. This orthogonality may be shown by a conformal map of the Cartesian coordinate system w = z + 1/z, where z= x + iy are the original Cartesian coordinates, and w=u + iv are those after the transformation.
Other orthogonal two-dimensional coordinate systems involving hyperbolas may be obtained by other conformal mappings. For example, the mapping w = z2 transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.
Conic section analysis of the hyperbolic appearance of circles
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. The viewer is typically a camera or the human eye and the image of the scene a central projection onto an image plane, that is, all projection rays pass a fixed point O, the center. The lens plane is a plane parallel to the image plane at the lens O.
The image of a circle c is (Special positions where the circle plane contains point O are omitted.)
These results can be understood if one recognizes that the projection process can be seen in two steps: 1) circle c and point O generate a cone which is 2) cut by the image plane, in order to generate the image.
One sees a hyperbola whenever catching sight of a portion of a circle cut by one's lens plane. The inability to see very much of the arms of the visible branch, combined with the complete absence of the second branch, makes it virtually impossible for the human visual system to recognize the connection with hyperbolas.
Applications
Sundials
Hyperbolas may be seen in many sundials. 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. At most populated latitudes and at most times of the year, this conic section is a hyperbola. 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.
Multilateration
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. 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. In particular, the set of possible positions of a point that has a distance difference of 2a from two given points is a hyperbola of vertex separation 2a whose foci are the two given points.
Path followed by a particle
The path followed by any particle in the classical Kepler problem is a conic section. In particular, if the total energy E of the particle is greater than zero (that is, if the particle is unbound), the path of such a particle is a hyperbola. This property is useful in studying atomic and sub-atomic forces by scattering high-energy particles; for example, the Rutherford experiment demonstrated the existence of an atomic nucleus by examining the scattering of alpha particles from gold atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive Coulomb force, which satisfies the inverse square law requirement for a Kepler problem.
Korteweg–de Vries equation
appears as one solution to the Korteweg–de Vries equation which describes the motion of a soliton wave in a canal.
Angle trisection
As shown first by Apollonius of Perga, a hyperbola can be used to trisect any angle, a well studied problem of geometry. Given an angle, first draw a circle centered at its vertex O, which intersects the sides of the angle at points A and B. Next draw the line segment with endpoints A and B. Construct a hyperbola of eccentricity e as directrix and B as a focus. Let P be the intersection (upper) of the hyperbola with the circle. Angle POB trisects angle AOB.
To prove this, reflect the line segment OP obtaining the point P' as the image of P. Segment AP' has the same length as segment BP due to the reflection, while segment PP' has the same length as segment BP due to the eccentricity of the hyperbola. As OA, OP', OP and OB are all radii of the same circle (and so, have the same length), the triangles OAP', OPP' and OPB are all congruent. Therefore, the angle has been trisected, since 3×POB = AOB.
Efficient portfolio frontier
In portfolio theory, the locus of mean-variance efficient portfolios (called the efficient frontier) is the upper half of the east-opening branch of a hyperbola drawn with the portfolio return's standard deviation plotted horizontally and its expected value plotted vertically; according to this theory, all rational investors would choose a portfolio characterized by some point on this locus.
Biochemistry
In biochemistry and pharmacology, the Hill equation and Hill-Langmuir equation respectively describe biological responses and the formation of protein–ligand complexes as functions of ligand concentration. They are both rectangular hyperbolae.
Hyperbolas as plane sections of quadrics
Hyperbolas appear as plane sections of the following quadrics:
See also
Other conic sections
Other related topics
Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolas.
Notes
References
External links
, Frans van Schooten, 1659