|
|
| Line 1: |
Line 1: |
| {{About|a geometric curve|the term used in rhetoric|Hyperbole}}
| | Have we ever heard which somebody said "My computer is getting slower, Help me?" Are we looking methods on how do I speed up my computer? Are we tired of spending too much time because the loading procedure is really slow? If you wish To recognize how to improve plus speed up computer performance, then this short article usually aid show we several concepts plus tricks "What is the cause?" plus How to prevent the computer getting slower?<br><br>Google Chrome crashes on Windows 7 by the corrupted cache contents plus difficulties with the stored browsing information. Delete the browsing data plus well-defined the contents of the cache to resolve this problem.<br><br>Perfect Optimizer furthermore has to remove junk files and is totally Windows Vista compatible. Most registry product merely don't have the time plus money to research Windows Vista errors. Because ideal optimizer has a big customer base, they do have the time, income and factors to support completely help Windows Vista.<br><br>There is several reasons why a computer may lose speed. Normal computer employ, including surfing the Internet will get your operating system inside a condition where it has no choice however, to slow down. The constant entering and deleting of temporary files which occur when you surf the Web leave the registries with thousands of false indicators in the running system's registry.<br><br>There are a great deal of [http://bestregistrycleanerfix.com/registry-reviver registry reviver] s. Which 1 is the best is not easy to be determined. But when we wish to stand out 1 among the multitude you must consider several products. These are features, scanning speed time, total mistakes detected, total errors repaired, tech support, Boot time performance and price. According to these items Top Registry Cleaner for 2010 is RegCure.<br><br>Why this problem occurs frequently? What are the causes of it? In fact, there are 3 main causes which will lead to the PC freezing issue. To resolve the problem, we have to take 3 procedures in the following paragraphs.<br><br>To accelerate the computer, we merely should be able to do away with all these junk files, allowing your computer to find just what it wants, whenever it wants. Luckily, there's a tool that enables us to do this easily and swiftly. It's a tool called a 'registry cleaner'.<br><br>Before you buy a whole new system; it really is time to receive the old one cleaned up thus you can begin getting more done online today! Visit our site under and access the most reputable registry cleaner software accessible. |
| [[File:Hyperbola (PSF).svg|right|thumb|210px|A hyperbola is an open curve with two branches, the intersection of a [[plane (geometry)|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.|alt=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.]]
| |
| [[File:HyperbolaeDrawnByHalogenLamp.jpg|right|thumb|210px|Hyperbolas in the physical world: three cones of light of different widths and intensities are generated by a (roughly) downwards-pointing halogen lamp and its holder. Each cone of light is drawing a branch of a hyperbola on a nearby vertical wall.]]
| |
| [[File:PSM V36 D057 Hyperbolas produced by interference of waves.jpg|thumb|right|Hyperbolas produced by [[interference (wave propagation)|interference of waves]]]] | |
| In [[mathematics]], a '''hyperbola''' (plural ''hyperbolas'' or ''hyperbolae'') is a type of [[smooth function|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 component (graph theory)|connected component]]s or branches, that are mirror images of each other and resemble two infinite [[bow (weapon)|bows]]. The hyperbola is one of the four kinds of [[conic section]], formed by the intersection of a [[plane (mathematics)|plane]] and a double [[cone (geometry)|cone]]. (The other conic sections are the [[parabola]], the [[ellipse]], and the [[circle]]; the circle is a special case of the 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.
| |
| | |
| Hyperbolas arise in many ways: as the curve representing the function <math>f(x) = 1/x</math> in the [[Cartesian plane]], as the appearance of a [[circle]] viewed from within it, as the path followed by the shadow of the tip of a sundial, as the shape of an [[open orbit]] (as distinct from a closed elliptical orbit), such as the orbit of a [[spacecraft]] during a [[gravity assist]]ed swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, as the path of a single-apparition [[comet]] (one travelling too fast to ever return to the solar system), as the [[Rutherford scattering|scattering trajectory]] of a [[subatomic particle]] (acted on by repulsive instead of attractive forces but the principle is the same), and so on.
| |
| | |
| 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 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 <math>f(x) = 1/x</math> 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 object]]s have their origin in the hyperbola, such as [[hyperbolic paraboloid]]s (saddle surfaces), [[hyperboloid]]s ("wastebaskets"), [[hyperbolic geometry]] ([[Nikolai Lobachevsky|Lobachevsky]]'s celebrated [[non-Euclidean geometry]]), [[hyperbolic function]]s (sinh, cosh, tanh, etc.), and [[gyrovector space]]s (a geometry used in both relativity and quantum mechanics which is not Euclidean).
| |
| | |
| ==History==
| |
| The word "hyperbola" derives from the [[Greek language|Greek]] {{lang|grc|ὑπερβολή}}, 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.<ref>{{citation|title=Apollonius of Perga: Treatise on Conic Sections with Introductions Including an Essay on Earlier History on the Subject|first=Sir Thomas Little|last=Heath|publisher=Cambridge University Press|year=1896|contribution=Chapter I. The discovery of conic sections. Menaechmus|pages=xvii–xxx|url=http://books.google.com/books?hl=en&lr=&id=B0k0AQAAMAAJ&pg=PR17}}.</ref> The term hyperbola is believed to have been coined by [[Apollonius of Perga]] (c. 262–c. 190 BC) in his definitive work on the [[conic section]]s, the ''Conics''.<ref>{{citation|title=A History of Mathematics|first=Carl B.|last=Boyer|first2=Uta C.|last2=Merzbach|publisher=Wiley|year=2011|isbn=9780470630563|url=http://books.google.com/books?id=bR9HAAAAQBAJ&pg=RA2-PT73|page=73|quote= It was Apollonius (possibly following up a suggestion of Archimedes) who introduced the names “ellipse” and “hyperbola” in connection with these curves.}}</ref> For comparison, the other two general conic sections, the [[ellipse]] and the [[parabola]], derive from the corresponding Greek words for "deficient" and "comparable"; these terms may refer to the [[eccentricity (mathematics)|eccentricity]] of these curves, which is greater than one (hyperbola), less than one (ellipse) and exactly one (parabola).
| |
| | |
| ==Nomenclature and features==
| |
| [[File:Hyperbola properties.svg|right|frame|The hyperbola consists of the red curves. The asymptotes of the hyperbola are shown as blue dashed lines and intersect at the center of the hyperbola, ''C''. The two focal points are labeled '''F'''<sub>1</sub> and '''F'''<sub>2</sub>, and the thin black line joining them is the transverse axis. The perpendicular thin black line through the center is the conjugate axis. The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, ''D''<sub>1</sub> and ''D''<sub>2</sub>. The eccentricity ''e'' equals the ratio of the distances from a point '''P''' on the hyperbola to one focus and its corresponding directrix line (shown in green). The two vertices are located on the transverse axis at ±''a'' relative to the center. So the parameters are:
| |
| | |
| ''a'' — distance from center ''C'' to either vertex<br>
| |
| ''b'' — length of a segment perpendicular to the transverse axis drawn from each vertex to the asymptotes<br>
| |
| ''c'' — distance from center ''C'' to either Focus point, '''F'''<sub>1</sub> and '''F'''<sub>2</sub>, and <br>
| |
| θ — angle formed by each asymptote with the transverse axis.]]
| |
| | |
| Similar to a [[parabola]], a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an [[ellipse]] does. A hyperbola consists of two disconnected [[curve]]s called its '''arms''' or '''branches'''.
| |
| | |
| The points on the two branches that are closest to each other are called the [[Vertex (curve)|vertices]]; they are the points where the curve has its smallest [[Radius of curvature (mathematics)|radius of curvature]]. The line segment connecting the vertices is called the ''transverse axis'' or ''major axis'', corresponding to the major diameter of an ellipse. The midpoint of the transverse axis is known as the hyperbola's ''center''. The distance ''a'' from the center to each vertex is called the [[semi-major axis]]. Outside of the transverse axis but on the same line are the two ''[[focus (geometry)|focal points (foci)]]'' of the hyperbola. The line through these five points is one of the two principal axes of the hyperbola, the other being the [[bisection|perpendicular bisector]] of the transverse axis. The hyperbola has [[reflection symmetry|mirror symmetry]] about its principal axes, and is also symmetric under a 180° turn about its center.
| |
| | |
| At large distances from the center, the hyperbola approaches two lines, its [[asymptote]]s, which intersect at the hyperbola's center. A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them; however, a [[Degenerate conic|degenerate hyperbola]] consists only of its asymptotes. Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the ''x''-axis of a [[Cartesian coordinate system]], the slopes of the asymptotes are equal in magnitude but opposite in sign, ±{{frac|b|a}}, where ''b''=''a''×tan(θ) and where θ is the angle between the transverse axis and either asymptote. The distance ''b'' (not shown) is the length of the perpendicular segment from either vertex to the asymptotes.
| |
| | |
| A ''conjugate axis'' of length 2''b'', corresponding to the ''minor axis'' of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices. Because of the minus sign in some of the formulas below, it is also called the ''imaginary axis'' of the hyperbola.
| |
| | |
| If {{nowrap|''b'' {{=}} ''a''}}, the angle 2θ between the asymptotes equals 90° and the hyperbola is said to be ''rectangular'' or ''equilateral''. In this special case, the rectangle joining the four points on the asymptotes directly above and below the vertices is a square, since the lengths of its sides ''2a'' {{=}} ''2b''.
| |
| | |
| If the transverse axis of any hyperbola is aligned with the ''x''-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as
| |
| | |
| :<math>
| |
| \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.
| |
| </math>
| |
| | |
| A hyperbola aligned in this way is called an "East-West opening hyperbola". Likewise, a hyperbola with its transverse axis aligned with the ''y''-axis is called a "North-South opening hyperbola" and has equation
| |
| | |
| :<math>
| |
| \frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1.
| |
| </math>
| |
| | |
| Every hyperbola is [[congruence (geometry)|congruent]] to the origin-centered East-West opening hyperbola sharing its same eccentricity ε (its shape, or degree of "spread"), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be [[translation (geometry)|translated]] (rigidly moved in the plane) so that it is centered at the origin. For convenience, hyperbolas are usually analyzed in terms of their centered East-West opening form.
| |
| | |
| If <math>c</math> is the distance from the center to either focus, then <math>a^2+b^2=c^2</math>.
| |
| | |
| [[File:Drini-conjugatehyperbolas.svg|thumb|300px|left|Here {{nowrap|''a'' {{=}} ''b'' {{=}} 1}} giving the [[unit hyperbola]] in blue and its conjugate hyperbola in green, sharing the same red asymptotes.]]
| |
| The shape of a hyperbola is defined entirely by its [[eccentricity (mathematics)|eccentricity]] ε, which is a dimensionless number always greater than one. The distance ''c'' from the center to the foci equals ''a''ε. The eccentricity can also be defined as the ratio of the distances to either focus and to a corresponding line known as the [[Directrix (conic section)|directrix]]; hence, the distance from the center to the directrices equals ''a''/ε. In terms of the parameters ''a'', ''b'', ''c'' and the angle θ, the eccentricity equals
| |
| | |
| :<math>
| |
| \varepsilon = \frac{c}{a} = \frac{\sqrt{a^{2} + b^{2}}}{a} = \sqrt{1 + \frac{b^{2}}{a^{2}}} = \sec \theta .
| |
| </math>
| |
| | |
| For example, the eccentricity of a rectangular hyperbola {{nowrap|(θ {{=}} 45°}}, {{nowrap|''a'' {{=}} ''b'')}} equals the [[square root]] of two: ε = <math>\sqrt{2}</math>.
| |
| | |
| Every hyperbola has a ''conjugate hyperbola'', in which the transverse and conjugate axes are exchanged without changing the asymptotes. The equation of the conjugate hyperbola of <math>\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1</math> is <math>\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = -1</math>. If the graph of the conjugate hyperbola is rotated 90° to restore the east-west opening orientation (so that ''x'' becomes ''y'' and vice versa), the equation of the resulting rotated conjugate hyperbola is the same as the equation of the original hyperbola except with ''a'' and ''b'' exchanged. For example, the angle θ of the conjugate hyperbola equals 90° minus the angle of the original hyperbola. Thus, the angles in the original and conjugate hyperbolas are complementary angles, which implies that they have different eccentricities unless θ = 45° (a rectangular hyperbola). Hence, the conjugate hyperbola does ''not'' in general correspond to a 90° rotation of the original hyperbola; the two hyperbolas are generally different in shape.
| |
| | |
| A few other lengths are used to describe hyperbolas. Consider a line perpendicular to the transverse axis (i.e., parallel to the conjugate axis) that passes through one of the hyperbola's foci. The line segment connecting the two intersection points of this line with the hyperbola is known as the ''latus rectum'' and has a length <math>\frac{2b^{2}}{a}</math>. The ''semi-latus rectum'' ''l'' is half of this length, i.e., <math>l=\frac{b^{2}}{a}</math>. The ''focal parameter'' ''p'' is the distance from a focus to its corresponding directrix, and equals <math>p=\frac{b^{2}}{c}</math>.
| |
| | |
| ==Mathematical definitions==
| |
| A hyperbola can be defined mathematically in several equivalent ways.
| |
| | |
| ===Conic section===
| |
| [[File:Conic sections 2.png|thumb|right|300px|Three major types of conic sections.]]
| |
| | |
| A hyperbola may be defined as the curve of [[intersection (set theory)|intersection]] between a right circular [[conical surface]] and a [[plane (geometry)|plane]] that cuts through both halves of the cone. The other major types of conic sections are the [[ellipse]] and the [[parabola]]; in these cases, the plane cuts through only one half of the double cone. If the plane passes through the central apex of the double cone a [[Degenerate conic|degenerate hyperbola]] results — two straight lines that cross at the apex point.
| |
| | |
| ===Difference of distances to foci===
| |
| A hyperbola may be defined equivalently as the [[locus of points]] where the absolute value of the ''difference'' of the distances to the two [[focus (geometry)|foci]] is a constant equal to 2''a'', the distance between its two vertices. This definition accounts for many of the hyperbola's applications, such as [[trilateration]]; this is the problem of determining position from the ''difference'' in arrival times of synchronized signals, as in [[Global Positioning System|GPS]].
| |
| | |
| This definition may be expressed also in terms of [[tangent circles]]. The center of any circles externally tangent to two given circles lies on a hyperbola, whose foci are the centers of the given circles and where the vertex distance 2''a'' equals the difference in radii of the two circles. As a special case, one given circle may be a point located at one focus; since a point may be considered as a circle of zero radius, the other given circle—which is centered on the other focus—must have radius 2''a''. This provides a simple technique for constructing a hyperbola, as shown [[#Geometrical constructions|below]]. It follows from this definition that a tangent line to the hyperbola at a point '''P''' bisects the angle formed with the two foci, i.e., the angle '''F'''<sub>1</sub>'''P''' '''F'''<sub>2</sub>. Consequently, the feet of perpendiculars drawn from each focus to such a tangent line lies on a circle of radius ''a'' that is centered on the hyperbola's own center.
| |
| | |
| A proof that this [[characterization (mathematics)|characterization]] of the hyperbola is equivalent to the conic-section characterization can be done without coordinate geometry by means of [[Dandelin spheres]].
| |
| | |
| ===Directrix and focus===
| |
| A hyperbola can be defined as the locus of points for which the [[ratio]] of the distances to one focus and to a [[line (geometry)|line]] (called the directrix) is a constant <math>\epsilon</math> that is larger than 1. This constant is the [[eccentricity (mathematics)|eccentricity]] of the hyperbola. By symmetry a hyperbola has two directrices, which are parallel to the conjugate axis and are between it and the tangent to the hyperbola at a vertex.
| |
| | |
| ===Reciprocation of a circle===
| |
| The [[reciprocation (geometry)|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 [[inversive geometry#Circle inversion|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
| |
| | |
| :<math>
| |
| \epsilon = \frac{\overline{BC}}{r}
| |
| </math>
| |
| | |
| 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 (mathematics)|locus]] of the poles of the tangent lines to the circle ''B'', as well as the [[envelope (mathematics)|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===
| |
| A hyperbola can also be defined as a second-degree equation in the [[Cartesian coordinates]] (''x'', ''y'') of the [[plane (geometry)|plane]]
| |
| | |
| :<math>
| |
| A_{xx} x^{2} + 2 A_{xy} xy + A_{yy} y^{2} + 2 B_{x} x + 2 B_{y} y + C = 0
| |
| </math>
| |
| | |
| provided that the constants ''A''<sub>''xx''</sub>, ''A''<sub>''xy''</sub>, ''A''<sub>''yy''</sub>, ''B''<sub>''x''</sub>, ''B''<sub>''y''</sub>, and ''C'' satisfy the determinant condition
| |
| | |
| :<math>
| |
| D = \begin{vmatrix} A_{xx} & A_{xy}\\A_{xy} & A_{yy} \end{vmatrix} < 0\,
| |
| </math>
| |
| | |
| A special case of a hyperbola—the ''degenerate hyperbola'' consisting of two intersecting lines—occurs when another determinant is zero
| |
| | |
| :<math>
| |
| \Delta := \begin{vmatrix} A_{xx} & A_{xy} & B_{x} \\A_{xy} & A_{yy} & B_{y}\\B_{x} & B_{y} & C \end{vmatrix} = 0
| |
| </math>
| |
| | |
| This determinant Δ is sometimes called the [[discriminant]] of the conic section.<ref>Korn, Granino A. and Korn, Theresa M. ''Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review'', Dover Publ., second edition, 2000: p. 40.</ref>
| |
| | |
| 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 parameters of the quadratic form]].
| |
| | |
| The center (''x''<sub>''c''</sub>, ''y''<sub>''c''</sub>) of the hyperbola may be determined from the formulae
| |
| | |
| :<math>
| |
| x_{c} = -\frac{1}{D} \begin{vmatrix} B_{x} & A_{xy} \\B_{y} & A_{yy} \end{vmatrix}
| |
| </math>
| |
| | |
| :<math>
| |
| y_{c} = -\frac{1}{D} \begin{vmatrix} A_{xx} & B_{x} \\A_{xy} & B_{y} \end{vmatrix}
| |
| </math>
| |
| | |
| In terms of new coordinates, {{nowrap|ξ {{=}} ''x'' − ''x''<sub>''c''</sub>}} and {{nowrap|η {{=}} ''y'' − ''y''<sub>''c''</sub>}}, the defining equation of the hyperbola can be written
| |
| | |
| :<math>
| |
| A_{xx} \xi^{2} + 2A_{xy} \xi\eta + A_{yy} \eta^{2} + \frac{\Delta}{D} = 0
| |
| </math>
| |
| | |
| The principal axes of the hyperbola make an angle Φ with the positive ''x''-axis that equals
| |
| | |
| :<math>
| |
| \tan 2\Phi = \frac{2A_{xy}}{A_{xx} - A_{yy}}
| |
| </math>
| |
| | |
| Rotating the coordinate axes so that the ''x''-axis is aligned with the transverse axis brings the equation into its '''canonical form'''
| |
| | |
| :<math>
| |
| \frac{{x}^{2}}{a^{2}} - \frac{{y}^{2}}{b^{2}} = 1
| |
| </math>
| |
| | |
| The major and minor semiaxes ''a'' and ''b'' are defined by the equations
| |
| | |
| :<math>
| |
| a^{2} = -\frac{\Delta}{\lambda_{1}D} = -\frac{\Delta}{\lambda_{1}^{2}\lambda_{2}}
| |
| </math>
| |
| | |
| :<math>
| |
| b^{2} = -\frac{\Delta}{\lambda_{2}D} = -\frac{\Delta}{\lambda_{1}\lambda_{2}^{2}}
| |
| </math>
| |
| | |
| where λ<sub>1</sub> and λ<sub>2</sub> are the [[root of a function|roots]] of the [[quadratic equation]]
| |
| | |
| :<math>
| |
| \lambda^{2} - \left( A_{xx} + A_{yy} \right)\lambda + D = 0
| |
| </math>
| |
| | |
| For comparison, the corresponding equation for a degenerate hyperbola is
| |
| | |
| :<math>
| |
| \frac{{x}^{2}}{a^{2}} - \frac{{y}^{2}}{b^{2}} = 0
| |
| </math>
| |
| | |
| The tangent line to a given point (''x''<sub>0</sub>, ''y''<sub>0</sub>) on the hyperbola is defined by the equation
| |
| | |
| :<math>
| |
| E x + F y + G = 0
| |
| </math>
| |
| | |
| where ''E'', ''F'' and ''G'' are defined
| |
| | |
| :<math>
| |
| E = A_{xx} x_{0} + A_{xy} y_{0} + B_{x}
| |
| </math>
| |
| | |
| :<math>
| |
| F = A_{xy} x_{0} + A_{yy} y_{0} + B_{y}
| |
| </math>
| |
| | |
| :<math>
| |
| G = B_{x} x_{0} + B_{y} y_{0} + C
| |
| </math>
| |
| | |
| The [[Normal (geometry)|normal line]] to the hyperbola at the same point is given by the equation
| |
| | |
| :<math>
| |
| F \left( x - x_{0} \right) - E \left( y - y_{0} \right) = 0
| |
| </math>
| |
| | |
| The normal line is perpendicular to the tangent line, and both pass through the same point (''x''<sub>0</sub>, ''y''<sub>0</sub>).
| |
| | |
| From the equation
| |
| :<math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \qquad 0 < b \leq a</math>
| |
| the basic property that with <math> r_1 \,\!</math> and <math> r_2 \,\!</math> being the distances from a point <math> (x,y) \,\!</math> to the left focus <math> (-a e , 0) \,\!</math> and the right focus <math> (a e , 0) \,\!</math> one has for a point on the right branch that
| |
| | |
| :<math> r_1 - r_2 =2 a\,\!</math>
| |
| | |
| and for a point on the left branch that
| |
| | |
| :<math> r_2 - r_1 =2 a\,\!</math>
| |
| | |
| can be proved as follows:
| |
| | |
| If x,y is a point on the hyperbola the distance to the left focal point is
| |
| | |
| :<math> r_1^2 =(x+a e)^2 + y^2 = x^2 + 2 x a e + a^2 e^2 + (x^2-a^2)(e^2-1)=
| |
| (e x + a)^2</math>
| |
| | |
| To the right focal point the distance is
| |
| :<math> r_2^2 = (x-a e)^2 + y^2 = x^2 - 2 x a e + a^2 e^2 + (x^2-a^2)(e^2-1)=
| |
| (e x - a)^2</math>
| |
| | |
| If x,y is a point on the right branch of the hyperbola then <math>e x > a\,\!</math> and
| |
| :<math> r_1 =e x + a\,\!</math>
| |
| :<math> r_2 =e x - a\,\!</math>
| |
| | |
| Subtracting these equations one gets
| |
| | |
| :<math> r_1 - r_2 =2 a\,\!</math>
| |
| | |
| If x,y is a point on the left branch of the hyperbola then <math>e x < -a\,\!</math> and
| |
| :<math> r_1 = -e x - a\,\!</math>
| |
| :<math> r_2 = -e x + a\,\!</math>
| |
| | |
| Subtracting these equations one gets
| |
| | |
| :<math> r_2 - r_1 =2 a\,\!</math>
| |
| | |
| ==True anomaly==
| |
| [[File:Hyperbola polar coordinates.svg|thumb|right|300px| The angle shown is the true anomaly of the indicated point on the hyperbola.]]
| |
| In the section above it is shown that using the coordinate system in which the equation of the hyperbola takes its '''canonical form'''
| |
| | |
| :<math>
| |
| \frac{{x}^{2}}{a^{2}} - \frac{{y}^{2}}{b^{2}} = 1
| |
| </math>
| |
| | |
| the distance <math>r</math> from a point <math> (x\ ,\ y)</math> on the left branch of the hyperbola to the left focal point <math> ( -e a\ ,\ 0)</math> is
| |
| | |
| :<math> r = -e x - a\,\!</math>.
| |
| | |
| Introducing [[polar coordinates]] <math> ( r\ ,\ \theta)</math> with origin at the left focal point the coordinates relative the canonical coordinate system are
| |
| :<math> x\ =\ -ae+r \cos \theta</math>
| |
| :<math> y\ =r \sin \theta</math>
| |
| | |
| and the equation above takes the form
| |
| :<math> r = -e (-ae+r \cos \theta) - a\,\!</math>
| |
| | |
| from which follows that
| |
| | |
| :<math>r = \frac{a(e^2-1)}{1+e\cos \theta}</math>
| |
| | |
| This is the representation of the near branch of a hyperbola in polar coordinates with respect to a focal point.
| |
| | |
| The polar angle <math>\theta</math> of a point on a hyperbola relative the near focal point as described above is called the '''true anomaly''' of the point.
| |
| | |
| [[File:Hyperbola construction - parallelogram method.gif|thumb|left|Hyperbola construction using the [https://de.wikipedia.org/wiki/Hyperbel_%28Mathematik%29#Steiner-Erzeugung_einer_Hyperbel parallelogram method]]]
| |
| | |
| ==Geometrical constructions==
| |
| Similar to the [[ellipse]], a hyperbola can be constructed using a taut thread. A straightedge of length ''S'' is attached to one focus '''F'''<sub>1</sub> at one of its corners '''A''' so that it is free to rotate about that focus. A thread of length ''L'' = ''S'' - 2''a'' is attached between the other focus '''F'''<sub>2</sub> and the other corner '''B''' of the straightedge. A sharp pencil is held up against the straightedge, sandwiching the thread tautly against the straightedge. Let the position of the pencil be denoted as '''P'''. The total length ''L'' of the thread equals the sum of the distances ''L''<sub>2</sub> from '''F'''<sub>2</sub> to '''P''' and ''L''<sub>B</sub> from '''P''' to '''B'''. Similarly, the total length ''S'' of the straightedge equals the distance ''L''<sub>1</sub> from '''F'''<sub>1</sub> to '''P''' and ''L''<sub>B</sub>. Therefore, the difference in the distances to the foci, {{nowrap|''L''<sub>1</sub> − ''L''<sub>2</sub>}} equals the constant 2''a''
| |
| | |
| :<math>
| |
| L_{1} - L_{2} = \left( S - L_{B} \right) - \left( L - L_{B} \right) = S - L = 2a
| |
| </math>
| |
| | |
| A second construction uses intersecting circles, but is likewise based on the constant difference of distances to the foci. Consider a hyperbola with two foci '''F'''<sub>1</sub> and '''F'''<sub>2</sub>, and two vertices '''P''' and '''Q'''; these four points all lie on the transverse axis. Choose a new point '''T''' also on the transverse axis and to the right of the rightmost vertex '''P'''; the difference in distances to the two vertices, {{nowrap|QT − PT}} = 2''a'', since 2''a'' is the distance between the vertices. Hence, the two circles centered on the foci '''F'''<sub>1</sub> and '''F'''<sub>2</sub> of radius QT and PT, respectively, will intersect at two points of the hyperbola.
| |
| | |
| A third construction relies on the definition of the hyperbola as the reciprocation of a circle. Consider the circle centered on the center of the hyperbola and of radius ''a''; this circle is tangent to the hyperbola at its vertices. A line ''g'' drawn from one focus may intersect this circle in two points '''M''' and '''N'''; perpendiculars to ''g'' drawn through these two points are tangent to the hyperbola. Drawing a set of such tangent lines reveals the [[envelope (mathematics)|envelope]] of the hyperbola.
| |
| | |
| A fourth construction is using the parallelogram method. It is similar to such method for [[parabola]] and [[ellipse]] construction: certain equally spaced points lying on parallel lines are connected with each other by two straight lines and their intersection point lies on the hyperbola.
| |
| | |
| ==Reflections and tangent lines==
| |
| The ancient Greek geometers recognized a reflection property of hyperbolas. If a [[ray (geometry)|ray]] of light emerges from one focus and is [[Reflection (mathematics)|reflected]] from the hyperbola, the light-ray appears to have come from the other focus. Equivalently, by reversing the direction of the light, rays directed at one of the foci from the exterior of the hyperbola are reflected towards the other focus. This property is analogous to the property of [[ellipse]]s that a ray emerging from one focus is reflected from the ellipse directly ''towards'' the other focus (rather than ''away'' as in the hyperbola). Expressed mathematically, lines drawn from each focus to the same point on the hyperbola intersect it at equal angles; the tangent line to a hyperbola at a point '''P''' bisects the angle formed with the two foci, F<sub>1</sub>PF<sub>2</sub>.
| |
| | |
| Tangent lines to a hyperbola have another remarkable geometrical property. If a tangent line at a point '''T''' intersects the asymptotes at two points '''K''' and '''L''', then '''T''' bisects the line segment KL, and the product of distances to the hyperbola's center, OK×OL is a constant.
| |
| | |
| ==Hyperbolic functions and equations==
| |
| [[Image:Hyperbola parametrized.svg|thumb|right|The points <math>
| |
| ( - a\ \cosh\ \mu_k \ ,\ b\ \sinh\ \mu_k)</math> with <math>\mu_k\ =\ 0.3\ k </math> for <math> k=-5,-4, \cdots ,5</math>]]
| |
| Just as the [[sine]] and [[cosine]] functions give a [[parametric equation]] for the [[ellipse]], so the [[hyperbolic function|hyperbolic sine]] and [[hyperbolic function|hyperbolic cosine]] give a parametric equation for the hyperbola.
| |
| | |
| As
| |
| | |
| <math>
| |
| \cosh^2 \mu - \sinh^2 \mu= 1
| |
| </math>
| |
| | |
| one has for any value of <math>\mu</math> that the point
| |
| | |
| :<math>
| |
| x = a\ \cosh\ \mu
| |
| </math>
| |
| :<math>
| |
| y = b\ \sinh\ \mu
| |
| </math>
| |
| | |
| satisfies the equation
| |
| | |
| :<math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1</math>
| |
| | |
| which is the equation of a hyperbola relative its canonical coordinate system.
| |
| | |
| When ''μ'' varies over the interval <math>-\infty < \mu < \infty </math> one gets with this formula all points <math>(x\ ,\ y)</math> on the right branch of the hyperbola.
| |
| | |
| The left branch for which <math>x < 0</math> is in the same way obtained as
| |
| | |
| :<math>
| |
| x = -a\ \cosh\ \mu
| |
| </math>
| |
| :<math>
| |
| y = b\ \sinh\ \mu
| |
| </math>
| |
| | |
| In the figure the points <math>(x_k\ ,\ y_k)</math> given by
| |
| :<math>
| |
| x_k = -a\ \cosh \mu _k
| |
| </math>
| |
| :<math>
| |
| y_k = b\ \sinh \mu _k
| |
| </math>
| |
| | |
| for
| |
| :<math>\mu_k\ =\ 0.3\ k \quad k=-5,-4, \cdots ,5</math>
| |
| | |
| on the left branch of a hyperbola with eccentricity 1.2 are marked as dots.
| |
| | |
| ==Relation to other conic sections==
| |
| There are three major types of conic sections: hyperbolas, [[ellipse]]s and [[parabola]]s. Since the parabola may be seen as a limiting case poised exactly between an ellipse and a hyperbola, there are effectively only two major types, ellipses and hyperbolas. These two types are related in that formulae for one type can often be applied to the other.
| |
| | |
| The canonical equation for a hyperbola is
| |
| | |
| :<math>
| |
| \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.
| |
| </math>
| |
| | |
| Any hyperbola can be rotated so that it is east-west opening and positioned with its center at the origin, so that the equation describing it is this canonical equation.
| |
| | |
| The canonical equation for the hyperbola may be seen as a version of the corresponding ellipse equation
| |
| | |
| :<math>
| |
| \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1
| |
| </math>
| |
| | |
| in which the [[semi-minor axis]] length ''b'' is imaginary. That is, if in the ellipse equation ''b'' is replaced by ''ib'' where ''b'' is real, one obtains the hyperbola equation.
| |
| | |
| Similarly, the parametric equations for a hyperbola and an ellipse are expressed in terms of [[hyperbolic function|hyperbolic]] and [[trigonometric function]]s, respectively, which are again related by an imaginary number, e.g.,
| |
| | |
| :<math>
| |
| \cosh \mu = \cos i\mu
| |
| </math>
| |
| | |
| Hence, many formulae for the ellipse can be extended to hyperbolas by adding the imaginary unit ''i'' in front of the semi-minor axis ''b'' and the angle. For example, the [[arc length]] of a segment of an ellipse can be determined using an [[Elliptic integral#Incomplete elliptic integral of the second kind|incomplete elliptic integral of the second kind]]. The corresponding arclength of a hyperbola is given by the same function with imaginary parameters ''b'' and μ, namely, ''ib E(iμ, c)''.
| |
| | |
| ==Conic section analysis of the hyperbolic appearance of circles==
| |
| [[File:hyperbo.svg|thumb|250px|Figure 10: The hyperbola as a circle on the ground seen in perspective while gazing down slightly, showing the circle's (non-parallel) tangents as asymptotes. The portion above the horizon is normally invisible. When gazing straight ahead the tangents will be parallel and therefore intersect at the horizon instead of above, described further in the text.]]
| |
| 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 a circle, or more generally an ellipse. The viewer is typically a camera or the human eye. In the simplest case the viewer's lens is just a pinhole; the role of more complex lenses is merely to gather far more light while retaining as far as possible the simple pinhole geometry in which all rays of light from the scene pass through a single point. Once through the lens, the rays then spread out again, in air in the case of a camera, in the [[vitreous humor]] in the case of the eye, eventually distributing themselves over the film, imaging device, or retina, all of which come under the heading of [[image plane]]. The '''lens plane''' is a plane parallel to the image plane at the lens; all rays pass through a single point on the lens plane, namely the lens itself.
| |
| | |
| When the circle directly faces the viewer, the viewer's lens is on-axis, meaning on the line normal to the circle through its center (think of the axle of a wheel). The rays of light from the circle through the lens to the image plane then form a cone with circular cross section whose apex is the lens. The image plane concretely realizes the abstract cutting plane in the conic section model.
| |
| | |
| When in addition the viewer directly faces the circle, the circle is rendered faithfully on the image plane without perspective distortion, namely as a scaled-down circle. When the viewer turns attention or gaze away from the center of the circle the image plane then cuts the cone in an ellipse, parabola, or hyperbola depending on how far the viewer turns, corresponding exactly to what happens when the surface cutting the cone to form a conic section is rotated.
| |
| | |
| A parabola arises when the lens plane is tangent to (touches) the circle. A viewer with perfect 180-degree wide-angle vision will see the whole parabola; in practice this is impossible and only a finite portion of the parabola is captured on the film or retina.
| |
| | |
| When the viewer turns further so that the lens plane cuts the circle in two points, the shape on the image plane becomes that of a hyperbola. The viewer still sees only a finite curve, namely a portion of one branch of the hyperbola, and is unable to see the second branch at all, which corresponds to the portion of the circle behind the viewer, more precisely, on the same side of the lens plane as the viewer. In practice the finite extent of the image plane makes it impossible to see any portion of the circle near where it is cut by the lens plane. Further back however one could imagine rays from the portion of the circle well behind the viewer passing through the lens, were the viewer transparent. In this case the rays would pass through the image plane before the lens, yet another impracticality ensuring that no portion of the second branch could possibly be visible.
| |
| | |
| The tangents to the circle where it is cut by the lens plane constitute the asymptotes of the hyperbola. Were these tangents to be drawn in ink in the plane of the circle, the eye would perceive them as asymptotes to the visible branch. Whether they converge in front of or behind the viewer depends on whether the lens plane is in front of or behind the center of the circle respectively.
| |
| | |
| If the circle is drawn on the ground and the viewer gradually transfers gaze from straight down at the circle up towards the horizon, the lens plane eventually cuts the circle producing first a parabola then a hyperbola on the image plane as shown in Figure 10. As the gaze continues to rise the asymptotes of the hyperbola, if realized concretely, appear coming in from left and right, swinging towards each other and converging at the horizon when the gaze is horizontal. Further elevation of the gaze into the sky then brings the point of convergence of the asymptotes towards the viewer.
| |
| | |
| By the same principle with which the back of the circle appears on the image plane were all the physical obstacles to its projection to be overcome, the portion of the two tangents behind the viewer appear on the image plane as an extension of the visible portion of the tangents in front of the viewer. Like the second branch this extension materializes in the sky rather than on the ground, with the horizon marking the boundary between the physically visible (scene in front) and invisible (scene behind), and the visible and invisible parts of the tangents combining in a single X shape. As the gaze is raised and lowered about the horizon, the X shape moves oppositely, lowering as the gaze is raised and vice versa but always with the visible portion being on the ground and stopping at the horizon, with the center of the X being on the horizon when the gaze is horizontal.
| |
| | |
| All of the above was for the case when the circle faces the viewer, with only the viewer's gaze varying. When the circle starts to face away from the viewer the viewer's lens is no longer on-axis. In this case the cross section of the cone is no longer a circle but an ellipse (never a parabola or hyperbola). However the principle of conic sections does not depend on the cross section of the cone being circular, and applies without modification to the case of eccentric cones.
| |
| | |
| It is not difficult to see that even in the off-axis case a circle can appear circular, namely when the image plane (and hence lens plane) is parallel to the plane of the circle. That is, to see a circle as a circle when viewing it obliquely, look not at the circle itself but at the plane in which it lies. From this it can be seen that when viewing a plane filled with many circles, all of them will appear circular simultaneously when the plane is looked at directly.
| |
| | |
| A common misperception about the hyperbola is that it is a mathematical curve rarely if ever encountered in daily life. The reality is that one sees a hyperbola whenever catching sight of portion of a circle cut by one's lens plane (and a parabola when the lens plane is tangent to, i.e. just touches, the circle). 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 such as ''y'' = 1/''x'' where both branches are on display simultaneously.
| |
| | |
| ==Derived curves==
| |
| {{Sinusoidal_spirals.svg}}
| |
| Several other curves can be derived from the hyperbola by [[inversive geometry#Circle inversion|inversion]], the so-called [[inverse curve]]s 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.
| |
| | |
| ==Coordinate systems==
| |
| | |
| ===Cartesian coordinates===
| |
| An east-west opening hyperbola centered at (''h'',''k'') has the equation
| |
| :<math>\frac{\left( x-h \right)^2}{a^2} - \frac{\left( y-k \right)^2}{b^2} = 1.</math>
| |
| 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),<ref>In some literature the value of ''a'' is taken negative for a hyperbola (the negative of half the distance between the two arms of the hyperbola measured along the major axis). This allows some formulas to be applicable to ellipses as well as to hyperbolas.</ref> and ''b'' is the [[semi-minor axis]] (half the distance between the asymptotes along a line tangent to the hyperbola at a vertex).
| |
| | |
| If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, 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'' despite the names ''minor'' and ''major''.
| |
| | |
| 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 (mathematics)|eccentricity]] is given by
| |
| :<math>\varepsilon = \sqrt{1+\frac{b^2}{a^2}} = \sec\left(\arctan\left(\frac{b}{a}\right)\right) = \cosh\left(\operatorname{arsinh}\left(\frac{b}{a}\right)\right)</math>
| |
| | |
| If ''c'' equals the distance from the center to either focus, then
| |
| :<math>\varepsilon = \frac{c}{a}</math>
| |
| where
| |
| :<math>c = \sqrt{a^2 + b^2}</math>.
| |
| The distance ''c'' is known as the '''linear eccentricity''' of the hyperbola. The distance between the foci is 2''c'' or 2''aε''.
| |
| | |
| The foci for an east-west opening hyperbola are given by
| |
| :<math>\left(h\pm c, k\right)</math>
| |
| and for a north-south opening hyperbola are given by
| |
| :<math>\left( h, k\pm c\right)</math>.
| |
| | |
| The directrices for an east-west opening hyperbola are given by
| |
| :<math>x = h\pm a \; \cos\left(\arctan\left(\frac{b}{a}\right)\right)</math>
| |
| and for a north-south opening hyperbola are given by
| |
| :<math>y = k\pm a \; \cos\left(\arctan\left(\frac{b}{a}\right)\right)</math>.
| |
| | |
| ===Polar coordinates===
| |
| 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 towards the origin of the "canonical coordinate system" as illustrated in the figure of the section "True anomaly".
| |
| | |
| Relative to this coordinate system one has that
| |
| | |
| :<math>r = \frac{a(e^2-1)}{1+e\cos \theta}</math>
| |
| | |
| and the range of the true anomaly <math> \theta</math> is:
| |
| | |
| :<math>-\arccos {\left(-\frac{1}{e}\right)} < \theta < \arccos {\left(-\frac{1}{e}\right)} </math>
| |
| | |
| With polar coordinate relative to the "canonical coordinate system"
| |
| | |
| :<math>x = R\, \cos t</math>
| |
| :<math>y = R\, \sin t</math>
| |
| | |
| one has that
| |
| | |
| :<math>R^2 =\frac{b^2}{e^2 \cos^2 t -1} \,</math>
| |
| | |
| For the right branch of the hyperbola the range of <math> t </math> is:
| |
| | |
| :<math>-\arccos {\left(\frac{1}{e}\right)} < t < \arccos {\left(\frac{1}{e}\right)} </math>
| |
| | |
| ===Parametric equations===
| |
| ''East-west opening hyperbola:''
| |
| :<math>\begin{matrix}
| |
| x = a\sec t + h \\
| |
| y = b\tan t + k \\
| |
| \end{matrix}
| |
| \qquad \mathrm{or} \qquad\begin{matrix}
| |
| x = \pm a\cosh t + h \\
| |
| y = b\sinh t + k \\
| |
| \end{matrix}
| |
| </math>
| |
| | |
| ''North-south opening hyperbola:''
| |
| :<math>\begin{matrix}
| |
| x = b\tan t + h \\
| |
| y = a\sec t + k \\
| |
| \end{matrix}
| |
| \qquad \mathrm{or} \qquad\begin{matrix}
| |
| x = b\sinh t + h \\
| |
| y = \pm a\cosh t + k \\
| |
| \end{matrix}
| |
| </math>
| |
| | |
| In all formulae (''h'',''k'') are the center coordinates of the hyperbola, ''a'' is the length of the semi-major axis, and ''b'' is the length of the semi-minor axis.
| |
| | |
| ===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
| |
| | |
| :<math>
| |
| \left(\frac{x}{c \cos\theta}\right)^2 - \left(\frac{y}{c \sin\theta}\right)^2 = 1
| |
| </math>
| |
| | |
| 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'' = ''z''<sup>2</sup> transforms the [[Cartesian coordinate system]] into two families of orthogonal hyperbolas.
| |
| | |
| ==Rectangular hyperbola with horizontal/vertical asymptotes (Cartesian coordinates)==
| |
| [[File:Rectangular hyperbola.svg|thumb|330px|A graph of the rectangular hyperbola <math>y=\tfrac{1}{x}</math>, the [[Multiplicative inverse|reciprocal]] function]]
| |
| | |
| A '''rectangular hyperbola''', '''equilateral hyperbola''', or '''right hyperbola''' is a hyperbola for which the asymptotes are [[perpendicular]].<ref>Weisstein, Eric W. "Rectangular Hyperbola." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RectangularHyperbola.html</ref>
| |
| | |
| Rectangular hyperbolas with the coordinate axes parallel to their asymptotes have the equation
| |
| :<math>(x-h)(y-k) = m \, \, \, </math>.
| |
| | |
| Rectangular hyperbolas have eccentricity <math>\varepsilon = \sqrt 2</math> with semi-major axis and semi-minor axis given by <math>a=b=\sqrt{2m}</math>.
| |
| | |
| The simplest example of rectangular hyperbolas occurs when the center (''h'', ''k'') is at the origin:
| |
| :<math>y=\frac{m}{x}\, </math>
| |
| describing quantities ''x'' and ''y'' that are [[inversely proportional]]. By rotating the coordinate axes counterclockwise by 45 degrees, with the new coordinate axes labelled <math>(x',y')</math> the equation of the hyperbola is given by canonical form
| |
| :<math>\frac{(x')^2}{(\sqrt{2m})^2}-\frac{(y')^2}{(\sqrt{2m})^2}=1</math>.
| |
| If the scale factor ''m''=1/2, then this canonical rectangular hyperbola is the [[unit hyperbola]].
| |
| | |
| ==Other properties of hyperbolas==
| |
| | |
| *If a line intersects one branch of a hyperbola at M and N and intersects the asymptotes at P and Q, then MN has the same midpoint as PQ.<ref name=web4>[http://mathafou.free.fr/themes_en/hyperb.html]</ref><ref name=Spain/>{{rp|p.49,ex.7|date=November 2012}}
| |
| | |
| *The following are [[concurrent lines|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.<ref name=web4/><ref name=web1>[http://www.ul.ie/~rynnet/swconics/HP%27s.htm]</ref>
| |
| | |
| *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.<ref name=web1/>
| |
| | |
| *The product of the distances from a point P to one of the asymptotes along a line parallel to the other asymptote, and to the second asymptote along a line parallel to the first asymptote, is independent of the location of point P on the hyperbola.<ref name=web1/> If the hyperbola is written in canonical form <math>\frac{x^2}{a^2} - \frac{y^2}{b^2}=1</math> then this product is <math>\frac{a^2+b^2}{4}</math>.
| |
| | |
| *The product of the perpendicular distances from a point P on the hyperbola <math>\frac{x^2}{a^2} - \frac{y^2}{b^2}=1</math> or on its conjugate hyperbola <math>\frac{x^2}{a^2} - \frac{y^2}{b^2}=-1</math> to the asymptotes is a constant independent of the location of P: specifically, <math>\frac{a^2b^2}{a^2+b^2}</math>, which also equals <math>(b/e)^2</math> where ''e'' is the eccentricity of the hyperbola <math>\frac{x^2}{a^2} - \frac{y^2}{b^2}=1</math>.<ref>Mitchell, Douglas W., "A property of hyperbolas and their asymptotes", ''Mathematical Gazette'' 96, July 2012, 299-301.</ref>
| |
| | |
| *The product of the slopes of lines from a point on the hyperbola to the two vertices is independent of the location of the point.<ref name=web2>[http://home.scarlet.be/~ping1339/hyperbola.htm#Geometric-property-o]</ref>
| |
| | |
| *A line segment between the two asymptotes and tangent to the hyperbola is bisected by the tangency point.<ref name=Spain>Spain, Barry. ''Analytical Conics''. Dover Publ., 2007.</ref>{{rp|p.49,ex.6|date=November 2012}}<ref name=web2/><ref name=web3>[http://www.nabla.hr/Z_Pre-CalculusHyperbolaAndLine_3.htm]</ref>
| |
| | |
| *The area of a triangle two of whose sides lie on the asymptotes, and whose third side is tangent to the hyperbola, is independent of the location of the tangency point.<ref name=Spain/>{{rp|p.49,ex.6|date=November 2012}} Specifically, the area is ''ab'', where ''a'' is the semi-major axis and ''b'' is the semi-minor axis.<ref name=web3/>
| |
| | |
| *The distance from either focus to either asymptote is ''b'', the semi-minor axis; the nearest point to a focus on an asymptote lies at a distance from the center equal to ''a'', the semi-major axis.<ref name=web4/> Then using the [[Pythagorean theorem]] on the right triangle with these two segments as legs shows that <math> a^2 + b^2 =c^2 </math>, where ''c'' is the semi-focal length (the distance from a focus to the hyperbola's center).
| |
| | |
| ==Applications==
| |
| [[File:Akademia Ekonomiczna w Krakowie Pawilon C.JPG|thumb|right|Hyperbolas as declination lines on a sundial]]
| |
| | |
| ===Sundials===
| |
| Hyperbolas may be seen in many [[sundial]]s. 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 2''a'' from two given points is a hyperbola of vertex separation 2''a'' 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 (i.e., 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 [[Geiger-Marsden experiment|Rutherford experiment]] demonstrated the existence of an [[atomic nucleus]] by examining the scattering of [[alpha particle]]s from [[gold]] atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive [[Coulomb's law|Coulomb force]], which satisfies the [[inverse square law]] requirement for a Kepler problem.
| |
| | |
| ===Korteweg-de Vries equation===
| |
| The hyperbolic trig function <math>\operatorname{sech}\, x</math> appears as one solution to the [[Korteweg-de Vries equation]] which describes the motion of a soliton wave in a canal.
| |
| | |
| ===Angle trisection===
| |
| [[File:Hyperbola angle trisection.svg|thumb|Trisecting an angle (AOB) using a hyperbola of eccentricity 2 (yellow curve)]]
| |
| | |
| As shown first by [[Apollonius of Perga]], a hyperbola can be used to [[angle trisection|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 through '''A''' and '''B''' and its perpendicular bisector <math>\ell</math>. Construct a hyperbola of [[eccentricity (mathematics)|eccentricity]] ε=2 with <math>\ell</math> as [[Directrix (conic section)|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''' about the line <math>\ell</math> 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'''.<ref>This construction is due to [[Pappus of Alexandria]] (circa 300 A.D.) and the proof comes from {{harvtxt|Kazarinoff|1970|loc=pg. 62}}.</ref>
| |
| | |
| ===Efficient portfolio frontier===
| |
| In [[modern portfolio theory#The efficient frontier with no risk-free asset|portfolio theory]], the locus of [[mean variance efficiency|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.
| |
| | |
| ==Extensions==
| |
| The three-dimensional analog of a hyperbola is a [[hyperboloid]]. Hyperboloids come in two varieties, those of one sheet and those of two sheets. A simple way of producing a hyperboloid is to rotate a hyperbola about the axis of its foci or about its symmetry axis perpendicular to the first axis; these rotations produce hyperboloids of two and one sheet, respectively.
| |
| | |
| ==See also==
| |
| <div style="-moz-column-count:3; column-count:4">
| |
| ===Other conic sections===
| |
| *[[Circle]]
| |
| *[[Ellipse]]
| |
| *[[Parabola]]
| |
| | |
| ===Other related topics===
| |
| *[[Apollonius of Perga]], the Greek geometer who gave the ellipse, parabola, and hyperbola the names by which we know them.
| |
| *[[Elliptic coordinates]], an orthogonal coordinate system based on families of [[ellipse]]s and hyperbolas.
| |
| *[[Hyperbolic angle]]
| |
| *[[Hyperbolic function]]
| |
| *[[Hyperbolic growth]]
| |
| *[[Hyperbolic partial differential equation]]
| |
| *[[Hyperbolic sector]]
| |
| *Hyperbolic structure
| |
| *[[Hyperbolic trajectory]]
| |
| *[[Hyperboloid]]
| |
| *[[Multilateration]]
| |
| </div>
| |
| | |
| ==Notes==
| |
| {{Reflist}}<!--added above External links/Sources by script-assisted edit-->
| |
| | |
| ==References==
| |
| * {{citation|last=Kazarinoff|first=Nicholas D.|title=Ruler and the Round|year=2003|publisher=Dover|location=Mineola, N.Y.|isbn=0-486-42515-0}}
| |
| | |
| ==External links==
| |
| {{commons category|Hyperbolas}}
| |
| * {{springer|title=Hyperbola|id=p/h048230}}
| |
| *[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=196&bodyId=204 Apollonius' Derivation of the Hyperbola] at [http://mathdl.maa.org/convergence/1/ Convergence]
| |
| * {{planetmath reference|id=5996|title=Unit hyperbola}}
| |
| * {{planetmath reference|id=3584|title=Conic section}}
| |
| * {{planetmath reference|id=6241|title=Conjugate hyperbola}}
| |
| * {{MathWorld |id=Hyperbola}}
| |
| | |
| [[Category:Conic sections]]
| |
| [[Category:Analytic geometry]]
| |
| [[Category:Curves]]
| |