This article is about a geometric curve. For the term used in rhetoric, seeHyperbole.
A hyperbola is an open curve with two branches, the intersection of aplane with both halves of adouble 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
Inmathematics, ahyperbola is a type ofsmoothcurve lying in a plane, defined by its geometric properties or byequations for which it is the solution set. A hyperbola has two pieces, calledconnected components or branches, that are mirror images of each other and resemble two infinitebows. The hyperbola is one of the three kinds ofconic section, formed by the intersection of aplane and a doublecone. (The other conic sections are theparabola and theellipse. Acircle 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.
Besides being a conic section, a hyperbola can arise as thelocus of points whose difference of distances to two fixedfoci is constant, as a curve for each point of which the rays to two fixed foci arereflections across thetangent line at that point, or as the solution of certain bivariatequadratic equations such as thereciprocal relationship[1] In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of asundial'sgnomon, the shape of anopen orbit such as that of a celestial object exceeding theescape velocity of the nearest gravitational body, or thescattering trajectory of asubatomic particle, among others.
Eachbranch 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 theasymptote of those two arms. So there are two asymptotes, whose intersection is at the center ofsymmetry 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 the asymptotes are the twocoordinate axes.[1]
The word "hyperbola" derives from theGreekὑπερβολή, meaning "over-thrown" or "excessive", from which the English termhyperbole also derives. Hyperbolae were discovered byMenaechmus in his investigations of the problem ofdoubling the cube, but were then called sections of obtuse cones.[2] The term hyperbola is believed to have been coined byApollonius of Perga (c. 262 – c. 190 BC) in his definitive work on theconic sections, theConics.[3]The names of the other two general conic sections, theellipse and theparabola, 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.[4]
Hyperbola: definition by the distances of points to two fixed points (foci)Hyperbola: definition with circular directrix
A hyperbola can be defined geometrically as aset of points (locus of points) in the Euclidean plane:
Ahyperbola is a set of points, such that for any point of the set, the absolute difference of the distances to two fixed points (thefoci) is constant, usually denoted by:[5]
The midpoint of the line segment joining the foci is called thecenter of the hyperbola.[6] The line through the foci is called themajor axis. It contains thevertices, which have distance to the center. The distance of the foci to the center is called thefocal distance orlinear eccentricity. The quotient is theeccentricity.
The equation can be viewed in a different way (see diagram): If is the circle with midpoint and radius, then the distance of a point of the right branch to the circle equals the distance to the focus: is called thecircular directrix (related to focus) of the hyperbola.[7][8] In order to get the left branch of the hyperbola, one has to use the circular directrix related to. This property should not be confused with the definition of a hyperbola with help of a directrix (line) below.
Rotating the coordinate system in order to describe a rectangular hyperbola as graph of a functionThree rectangular hyperbolas with the coordinate axes as asymptotes red:A = 1; magenta:A = 4; blue:A = 9
If thexy-coordinate system isrotated about the origin by the angle and new coordinates are assigned, then. The rectangular hyperbola (whosesemi-axes are equal) has the new equation.Solving for yields
Thus, in anxy-coordinate system the graph of a function with equation is arectangular hyperbola entirely in the first and thirdquadrants with
the coordinate axes asasymptotes,
the line asmajor axis ,
thecenter and thesemi-axis
thevertices
thesemi-latus rectum andradius of curvature at the vertices
thelinear eccentricity and the eccentricity
thetangent at point
A rotation of the original hyperbola by results in a rectangular hyperbola entirely in the second and fourth quadrants, with the same asymptotes, center, semi-latus rectum, radius of curvature at the vertices, linear eccentricity, and eccentricity as for the case of rotation, with equation
thesemi-axes
the line as major axis,
thevertices
Shifting the hyperbola with equation so that the new center is, yields the new equationand the new asymptotes are and. The shape parameters remain unchanged.
Hyperbola: directrix propertyHyperbola: definition with directrix property
The two lines at distance from the center and parallel to the minor axis are calleddirectrices of the hyperbola (see diagram).
For an arbitrary point of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:The proof for the pair follows from the fact that and satisfy the equationThe second case is proven analogously.
Pencil of conics with a common vertex and common semi latus rectum
Theinverse 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 (focus), any line (directrix) not through and anyreal number with the set of points (locus of points), for which the quotient of the distances to the point and to the line isis a hyperbola.
Let and assume is a point on the curve.The directrix has equation. With, the relation produces the equations
and
The substitution yieldsThis is the equation of anellipse () or aparabola () or ahyperbola (). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
If, introduce new parameters so that, and then the equation above becomeswhich is the equation of a hyperbola with center, thex-axis as major axis and the major/minor semi axis.
Because of point of directrix (see diagram) and focus are inverse with respect to thecircle inversion at circle (in diagram green). Hence point can be constructed using thetheorem of Thales (not shown in the diagram). The directrix is the perpendicular to line through point.
Alternative construction of: Calculation shows, that point is the intersection of the asymptote with its perpendicular through (see diagram).
Hyperbola (red): two views of a cone and two Dandelin spheresd1,d2
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 twoDandelin spheres, which are spheres that touch the cone along circles, and the intersecting (hyperbola) plane at points and. It turns out: are thefoci of the hyperbola.
Let be an arbitrary point of the intersection curve.
Thegeneratrix of the cone containing intersects circle at point and circle at a point.
The line segments and are tangential to the sphere and, hence, are of equal length.
The line segments and are tangential to the sphere and, hence, are of equal length.
The result is: is independent of the hyperbola point, because no matter where point is, have to be on circles,, and line segment has to cross the apex. Therefore, as point moves along the red curve (hyperbola), line segment simply rotates about apex without changing its length.
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:[9]
Choose thefoci and one of thecircular directrices, for example (circle with radius)
Aruler is fixed at point free to rotate around. Point is marked at distance.
Astring gets its one end pinned at point on the ruler and its length is made.
The free end of the string is pinned to point.
Take apen and hold the string tight to the edge of the ruler.
Rotating the ruler around prompts the pen to draw an arc of the right branch of the hyperbola, because of (see the definition of a hyperbola bycircular directrices).
Given twopencils of lines at two points (all lines containing and, respectively) and a projective but not perspective mapping of onto, then the intersection points of corresponding lines form a non-degenerate projective conic section.
For the generation of points of the hyperbola one uses the pencils at the vertices. Let be a point of the hyperbola and. The line segment is divided into n equally-spaced segments and this division is projected parallel with the diagonal as direction onto the line segment (see diagram). The parallel projection is part of the projective mapping between the pencils at and needed. The intersection points of any two related lines and are points of the uniquely defined hyperbola.
Remarks:
The subdivision could be extended beyond the points and 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 aparallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.
Inscribed angles for hyperbolasy =a/(x −b) +c and the 3-point-form
A hyperbola with equation is uniquely determined by three points with differentx- andy-coordinates. A simple way to determine the shape parameters uses theinscribed angle theorem for hyperbolas:
In order tomeasure an angle between two lines with equations in this context one uses the quotient
Analogous to theinscribed angle theorem for circles one gets the
Inscribed angle theorem for hyperbolas[10][11]—For four points (see diagram) the following statement is true:
The four points are on a hyperbola with equation if and only if the angles at and are equal in the sense of the measurement above. That means if
The proof can be derived by straightforward calculation. If the points are on a hyperbola, one can assume the hyperbola's equation is.
A consequence of the inscribed angle theorem for hyperbolas is the
3-point-form of a hyperbola's equation—The equation of the hyperbola determined by 3 points is the solution of the equation for.
As an affine image of the unit hyperbolax2 −y2 = 1
An affine transformation of the Euclidean plane has the form, where is a regularmatrix (itsdeterminant is not 0) and is an arbitrary vector. If are the column vectors of the matrix, the unit hyperbola is mapped onto the hyperbola
is the center, a point of the hyperbola and a tangent vector at this point.
In general the vectors are not perpendicular. That means, in general arenot the vertices of the hyperbola. But point into the directions of the asymptotes. The tangent vector at point isBecause at a vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter of a vertex from the equationand hence fromwhich yields
The definition of a hyperbola in this section gives a parametric representation of an arbitrary hyperbola, even in space, if one allows to be vectors in space.
Because the unit hyperbola is affinely equivalent to the hyperbola, an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola:
is the center of the hyperbola, the vectors have the directions of the asymptotes and is a point of the hyperbola. The tangent vector isAt a vertex the tangent is perpendicular to the major axis. Henceand the parameter of a vertex is
is equivalent to and 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: asymptotes andP given → tangent
The tangent vector can be rewritten by factorization:This means that
the diagonal of the parallelogram is parallel to the tangent at the hyperbola point (see diagram).
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 ofPascal's theorem.[12]
Area of the grey parallelogram
The area of the grey parallelogram in the above diagram isand hence independent of point. The last equation follows from a calculation for the case, where is a vertex and the hyperbola in its canonical form
For simplicity the center of the hyperbola may be the origin and the vectors have equal length. If the last assumption is not fulfilled one can first apply a parameter transformation (see above) in order to make the assumption true. Hence are the vertices, span the minor axis and one gets and.
For the intersection points of the tangent at point with the asymptotes one gets the pointsThearea of the triangle can be calculated by a 2 × 2 determinant:(see rules fordeterminants). is the area of the rhombus generated by. The area of a rhombus is equal to one half of the product of its diagonals. The diagonals are the semi-axes of the hyperbola. Hence:
Thearea of the triangle is independent of the point of the hyperbola:
Thereciprocation of acircleB in a circleC always yields a conic section such as a hyperbola. The process of "reciprocation in a circleC" consists of replacing every line and point in a geometrical figure with their correspondingpole and polar, respectively. Thepole of a line is theinversion of its closest point to the circleC, whereas the polar of a point is the converse, namely, a line whose closest point toC 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 radiusr of reciprocation circleC. IfB andC represent the points at the centers of the corresponding circles, then
Since the eccentricity of a hyperbola is always greater than one, the centerB must lie outside of the reciprocating circleC.
This definition implies that the hyperbola is both thelocus of the poles of the tangent lines to the circleB, as well as theenvelope of the polar lines of the points onB. Conversely, the circleB is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines toB have no (finite) poles because they pass through the centerC of the reciprocation circleC; the polars of the corresponding tangent points onB are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circleB that are separated by these tangent points.
A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates in theplane,
provided that the constants and satisfy the determinant condition
This determinant is conventionally called thediscriminant of the conic section.[14]
A special case of a hyperbola—thedegenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero:
This determinant is sometimes called the discriminant of the conic section.[15]
The general equation's coefficients can be obtained from known semi-major axis semi-minor axis center coordinates, and rotation angle (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
For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is
The tangent line to a given point on the hyperbola is defined by the equation
where and are defined by
Thenormal line to the hyperbola at the same point is given by the equation
The normal line is perpendicular to the tangent line, and both pass through the same point
From the equation
the left focus is and the right focus is where is the eccentricity. Denote the distances from a point to the left and right foci as and For a point on the right branch,
and for a point on the left branch,
This can be proved as follows:
If is a point on the hyperbola the distance to the left focal point is
To the right focal point the distance is
If is a point on the right branch of the hyperbola then and
Subtracting these equations one gets
If is a point on the left branch of the hyperbola then and
If Cartesian coordinates are introduced such that the origin is the center of the hyperbola and thex-axis is the major axis, then the hyperbola is calledeast-west-opening and
For an arbitrary point the distance to the focus is and to the second focus. Hence the point is on the hyperbola if the following condition is fulfilledRemove the square roots by suitable squarings and use the relation to obtain the equation of the hyperbola:
This equation is called thecanonical 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 iscongruent to the original (seebelow).
The axes ofsymmetry orprincipal axes are thetransverse axis (containing the segment of length 2a with endpoints at the vertices) and theconjugate axis (containing the segment of length 2b perpendicular to the transverse axis and with midpoint at the hyperbola's center).[6] As opposed to an ellipse, a hyperbola has only two vertices:. The two points on the conjugate axes arenot on the hyperbola.
It follows from the equation that the hyperbola issymmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin.
Hyperbola: semi-axesa,b, linear eccentricityc, semi latus rectumpHyperbola: 3 properties
Solving the equation (above) of the hyperbola for yieldsIt follows from this that the hyperbola approaches the two linesfor large values of. These two lines intersect at the center (origin) and are calledasymptotes of the hyperbola[16]
With the help of the second figure one can see that
Theperpendicular distance from a focus to either asymptote is (the semi-minor axis).
From theHesse normal form of the asymptotes and the equation of the hyperbola one gets:[17]
Theproduct of the distances from a point on the hyperbola to both the asymptotes is the constant which can also be written in terms of the eccentricitye as
From the equation of the hyperbola (above) one can derive:
Theproduct of the slopes of lines from a point P to the two vertices is the constant
In addition, from (2) above it can be shown that[17]
The product of the distances from a point on the hyperbola to the asymptotes along lines parallel to the asymptotes is the constant
The length of the chord through one of the foci, perpendicular to the major axis of the hyperbola, is called thelatus rectum. One half of it is thesemi-latus rectum. A calculation showsThe semi-latus rectum may also be viewed as theradius of curvature at the vertices.
The simplest way to determine the equation of the tangent at a point is toimplicitly differentiate the equation of the hyperbola. Denotingdy/dx asy′, this producesWith respect to, the equation of the tangent at point is
A particular tangent line distinguishes the hyperbola from the other conic sections.[18] Letf be the distance from the vertexV (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°.
In the case the hyperbola is calledrectangular (orequilateral), because its asymptotes intersect at right angles. For this case, the linear eccentricity is, the eccentricity and the semi-latus rectum. The graph of the equation is a rectangular hyperbola.
Parametric representation with hyperbolic sine/cosine
Using thehyperbolic sine and cosine functions, a parametric representation of the hyperbola can be obtained, which is similar to the parametric representation of an ellipse:which satisfies the Cartesian equation because
Further parametric representations are given in the sectionParametric equations below.
Herea =b = 1 giving theunit hyperbola in blue and its conjugate hyperbola in green, sharing the same red asymptotes.
For the hyperbola, change the sign on the right to obtain the equation of theconjugate hyperbola:
(which can also be written as).
A hyperbola and its conjugate may havediameters which are conjugate. In the theory ofspecial relativity, such diameters may represent axes of time and space, where one hyperbola representsevents at a given spatial distance from thecenter, and the other represents events at a corresponding temporal distance from the center.
The polar coordinates used most commonly for the hyperbola are defined relative to the Cartesian coordinate system that has itsorigin in a focus and its x-axis pointing toward the origin of the "canonical coordinate system" as illustrated in the first diagram.
When using polar coordinates, the eccentricity of the hyperbola can be expressed as where is the limit of the angular coordinate. As approaches this limit,r approaches infinity and the denominator in either of the equations noted above approaches zero, hence:[19]: 219
A hyperbola with equation can be described by several parametric equations:
Through hyperbolic trigonometric functions
As arational representation
Through circular trigonometric functions
With the tangent slope as parameter: A parametric representation, which uses the slope of the tangent at a point of the hyperbola can be obtained analogously to the ellipse case: Replace in the ellipse case by and use formulae for thehyperbolic functions. One gets Here, is the upper, and the lower half of the hyperbola. The points with vertical tangents (vertices) are not covered by the representation. The equation of the tangent at point is This description of the tangents of a hyperbola is an essential tool for the determination of theorthoptic of a hyperbola.
A ray through theunit 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.
Let be twice the area between the axis and a ray through the origin intersecting the unit hyperbola, and define as the coordinates of the intersection point.Then the area of the hyperbolic sector is the area of the triangle minus the curved region past the vertex at:which simplifies to thearea hyperbolic cosineSolving for yields the exponential form of the hyperbolic cosine:From one getsand its inverse thearea hyperbolic sine:Other hyperbolic functions are defined according to the hyperbolic cosine and hyperbolic sine, so for example
Hyperbola: the tangent bisects the lines through the foci
The tangent at a point bisects the angle between the lines This is called theoptical property orreflection property of a hyperbola.[20]
Proof
Let be the point on the line with the distance to the focus (see diagram, is the semi major axis of the hyperbola). Line is the bisector of the angle between the lines. In order to prove that is the tangent line at point, one checks that any point on line which is different from cannot be on the hyperbola. Hence has only point in common with the hyperbola and is, therefore, the tangent at point. From the diagram and thetriangle inequality one recognizes that holds, which means:. But if is a point of the hyperbola, the difference should be.
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.
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.
The proof of the property on midpoints is best done for the hyperbola. Because any hyperbola is an affine image of the hyperbola (see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas: For two points of the hyperbola
the midpoint of the chord is
the slope of the chord is
For parallel chords the slope is constant and the midpoints of the parallel chords lie on the line
Consequence: for any pair of points of a chord there exists askew reflection with an axis (set of fixed points) passing through the center of the hyperbola, which exchanges the points and leaves the hyperbola (as a whole) fixed. A skew reflection is a generalization of an ordinary reflection across a line, where all point-image pairs are on a line perpendicular to.
Because a skew reflection leaves the hyperbola fixed, the pair of asymptotes is fixed, too. Hence the midpoint of a chord divides the related line segment between the asymptotes into halves, too. This means that. This property can be used for the construction of further points of the hyperbola if a point and the asymptotes are given.
If the chord degenerates into atangent, then the touching point divides the line segment between the asymptotes in two halves.
Any hyperbola can be described in a suitable coordinate system by an equation. The equation of the tangent at a point of the hyperbola is If one allows point to be an arbitrary point different from the origin, then
point is mapped onto the line, not through the center of the hyperbola.
This relation between points and lines is abijection.
Such a relation between points and lines generated by a conic is calledpole-polar relation or justpolarity. The pole is the point, the polar the line. SeePole and polar.
By calculation one checks the following properties of the pole-polar relation of the hyperbola:
For a point (pole)on the hyperbola the polar is the tangent at this point (see diagram:).
For a poleoutside the hyperbola the intersection points of its polar with the hyperbola are the tangency points of the two tangents passing (see diagram:).
For a pointwithin the hyperbola the polar has no point with the hyperbola in common. (see diagram:).
Remarks:
The intersection point of two polars (for example:) is the pole of the line through their poles (here:).
The foci and respectively and the directrices and respectively belong to pairs of pole and polar.
Pole-polar relations exist for ellipses and parabolas, too.
The following areconcurrent: (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.[21][22]
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.[22]
Since both the transverse axis and the conjugate axis are axes of symmetry, thesymmetry group of a hyperbola is theKlein four-group.
Several other curves can be derived from the hyperbola byinversion, the so-calledinverse curves of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is thelemniscate 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 alimaçon or astrophoid, respectively.
A family of confocal hyperbolas is the basis of the system ofelliptic coordinates in two dimensions. These hyperbolas are described by the equation
where the foci are located at a distancec from the origin on thex-axis, and where θ is the angle of the asymptotes with thex-axis. Every hyperbola in this family is orthogonal to every ellipse that shares the same foci. This orthogonality may be shown by aconformal map of the Cartesian coordinate systemw =z + 1/z, wherez=x +iy are the original Cartesian coordinates, andw=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 mappingw =z2 transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.
Conic section analysis of the hyperbolic appearance of circles
Central projection of circles on a sphere: The centerO 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 centerO wereon the sphere, all images of the circles would be circles or lines; seestereographic projection).
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 acentral projection onto an image plane, that is, all projection rays pass a fixed pointO, the center. Thelens plane is a plane parallel to the image plane at the lensO.
The image of a circle c is
acircle, if circlec is in a special position, for example parallel to the image plane and others (see stereographic projection),
anellipse, ifc hasno point with the lens plane in common,
aparabola, ifc hasone point with the lens plane in common and
ahyperbola, ifc hastwo points with the lens plane in common.
(Special positions where the circle plane contains pointO are omitted.)
These results can be understood if one recognizes that the projection process can be seen in two steps: 1) circle c and pointO 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.
Hyperbolas as declination lines on a sundialThe contact zone of a level supersonic aircraft'sshockwave on flat ground (yellow) is a part of a hyperbola as the ground intersects the cone parallel to its axis.
Hyperbolas may be seen in manysundials. On any given day, the sun revolves in a circle on thecelestial 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 thedeclination 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 apelekinon by the Greeks, since it resembles a double-bladed axe.
A hyperbola is the basis for solvingmultilateration 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 aLORAN orGPS 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.
The path followed by any particle in the classicalKepler problem is aconic section. In particular, if the total energyE 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, theRutherford experiment demonstrated the existence of anatomic nucleus by examining the scattering ofalpha particles fromgold atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsiveCoulomb force, which satisfies theinverse square law requirement for a Kepler problem.[24]
Trisecting an angle (AOB) using a hyperbola of eccentricity 2 (yellow curve)
As shown first byApollonius of Perga, a hyperbola can be used totrisect any angle, a well studied problem of geometry. Given an angle, first draw a circle centered at its vertexO, which intersects the sides of the angle at pointsA andB. Next draw the line segment with endpointsA andB and its perpendicular bisector. Construct a hyperbola ofeccentricitye=2 with asdirectrix andB as a focus. LetP be the intersection (upper) of the hyperbola with the circle. AnglePOB trisects angleAOB.
To prove this, reflect the line segmentOP about the line obtaining the pointP' as the image ofP. SegmentAP' has the same length as segmentBP due to the reflection, while segmentPP' has the same length as segmentBP due to the eccentricity of the hyperbola.[25] AsOA,OP',OP andOB are all radii of the same circle (and so, have the same length), the trianglesOAP',OPP' andOPB are all congruent. Therefore, the angle has been trisected, since 3×POB =AOB.[26]
Inportfolio theory, the locus ofmean-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.
^Boyer, Carl B.;Merzbach, Uta C. (2011),A History of Mathematics, Wiley, p. 73,ISBN9780470630563,It was Apollonius (possibly following up a suggestion of Archimedes) who introduced the names "ellipse" and "hyperbola" in connection with these curves.
^Eves, Howard (1963),A Survey of Geometry (Vol. One), Allyn and Bacon, pp. 30–31
^Apostol, Tom M.; Mnatsakanian, Mamikon A. (2012),New Horizons in Geometry, The Dolciani Mathematical Expositions #47, The Mathematical Association of America, p. 251,ISBN978-0-88385-354-2
^The German term for this circle isLeitkreis which can be translated as "Director circle", but that term has a different meaning in the English literature (seeDirector circle).
^Korn, Granino A;Korn, Theresa M. (2000).Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (second ed.). Dover Publ. p. 40.
^Heilbron, John L. (1968). "The Scattering of α and β Particles and Rutherford's Atom".Archive for History of Exact Sciences.4 (4):247–307.doi:10.1007/BF00411591.JSTOR41133273.
^Since 2 times the distance ofP to isPP' which is equal toBP by directrix-focus property
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