Hyperbola

In mathematics, a hyperbola (/haɪˈpɜːrbələ/ hy-PUR-bə-lə) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse, with the circle being a special type of ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship \(xy = 1.\) In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others.

Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry 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 \(y(x) = 1/x\) the asymptotes are the two coordinate axes.

Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).

The word "hyperbola" derives from the Greek 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. 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 sections, the Conics. The names of the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than the segment or exceed the segment.

A hyperbola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:

The midpoint \(M\) of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis. It contains the vertices \(V_1, V_2\), which have distance \(a\) to the center. The distance \(c\) of the foci to the center is called the focal distance or linear eccentricity. The quotient \(\tfrac c a\) is the eccentricity \(e\).

The equation \(\left|\left|PF_2\right| - \left|PF_1\right|\right| = 2a\) can be viewed in a different way (see diagram):
If \(c_2\) is the circle with midpoint \(F_2\) and radius \(2a\), then the distance of a point \(P\) of the right branch to the circle \(c_2\) equals the distance to the focus \(F_1\): \[|PF_1|=|Pc_2|.\] \(c_2\) is called the circular directrix (related to focus \(F_2\)) of the hyperbola. In order to get the left branch of the hyperbola, one has to use the circular directrix related to \(F_1\). This property should not be confused with the definition of a hyperbola with help of a directrix (line) below.

If the xy-coordinate system is rotated about the origin by the angle \(+45^\circ\) and new coordinates \(\xi,\eta\) are assigned, then \(x = \tfrac{\xi+\eta}{\sqrt{2}},\; y = \tfrac{-\xi+\eta}{\sqrt{2}}\).
The rectangular hyperbola \(\tfrac{x^2-y^2}{a^2} = 1\) (whose semi-axes are equal) has the new equation \(\tfrac{2\xi\eta}{a^2} = 1\). Solving for \(\eta\) yields \(\eta = \tfrac{a^2/2}{\xi} \ .\)

Thus, in an xy-coordinate system the graph of a function \(f: x \mapsto \tfrac{A}{x},\; A>0\; ,\) with equation \[y = \frac{A}{x}\;, A>0\; ,\] is a rectangular hyperbola entirely in the first and third quadrants with

• the coordinate axes as asymptotes,
• the line \(y = x\) as major axis ,
• the center \((0,0)\) and the semi-axis \(a = b = \sqrt{2A} \; ,\)
• the vertices \(\left(\sqrt{A},\sqrt{A}\right), \left(-\sqrt{A},-\sqrt{A}\right) \; ,\)
• the semi-latus rectum and radius of curvature at the vertices \(p=a=\sqrt{2A} \; ,\)
• the linear eccentricity \(c=2\sqrt{A}\) and the eccentricity \(e=\sqrt{2} \; ,\)
• the tangent \(y=-\tfrac{A}{x_0^2}x+2\tfrac{A}{x_0}\) at point \((x_0,A/x_0)\; .\)

A rotation of the original hyperbola by \(-45^\circ\) 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 \(+45^\circ\) rotation, with equation \[y = -\frac{A}{x} \; , ~~ A>0\; ,\]

• the semi-axes \(a = b = \sqrt{2A} \; ,\)
• the line \(y = -x\) as major axis,
• the vertices \(\left(-\sqrt{A},\sqrt{A}\right), \left(\sqrt{A},-\sqrt{A}\right) \; .\)

Shifting the hyperbola with equation \(y=\frac{A}{x}, \ A\ne 0\ ,\) so that the new center is \((c_0,d_0)\), yields the new equation \[y=\frac{A}{x-c_0}+d_0\; ,\] and the new asymptotes are \(x=c_0\) and \(y=d_0\). The shape parameters \(a,b,p,c,e\) remain unchanged.

The two lines at distance \(d = \frac{a^2}c\) from the center and parallel to the minor axis are called directrices of the hyperbola (see diagram).

For an arbitrary point \(P\) of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity: \[\frac{|PF_1|}{|Pl_1|} = \frac{|PF_2|}{|Pl_2|} = e= \frac{c}{a} \, .\] The proof for the pair \(F_1, l_1\) follows from the fact that \(|PF_1|^2 = (x-c)^2+y^2,\ |Pl_1|^2 = \left(x-\tfrac{a^2}{c}\right)^2\) and \(y^2 = \tfrac{b^2}{a^2}x^2-b^2\) satisfy the equation \[|PF_1|^2-\frac{c^2}{a^2}|Pl_1|^2 = 0\ .\] The second case is proven analogously.

The inverse statement is also true and can be used to define a hyperbola (in a manner similar to the definition of a parabola):

For any point \(F\) (focus), any line \(l\) (directrix) not through \(F\) and any real number \(e\) with \(e > 1\) the set of points (locus of points), for which the quotient of the distances to the point and to the line is \(e\) \[H = \left\{P \, \Biggr| \, \frac{|PF|}{|Pl|} = e\right\}\] is a hyperbola.

(The choice \(e = 1\) yields a parabola and if \(e < 1\) an ellipse.)

The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see diagram: red curve). In order to prove the defining property of a hyperbola (see above) one uses two Dandelin spheres \(d_1, d_2\), which are spheres that touch the cone along circles \(c_1\), \(c_2\) and the intersecting (hyperbola) plane at points \(F_1\) and \(F_2\). It turns out: \(F_1, F_2\) are the foci of the hyperbola.

1. Let \(P\) be an arbitrary point of the intersection curve.
2. The generatrix of the cone containing \(P\) intersects circle \(c_1\) at point \(A\) and circle \(c_2\) at a point \(B\).
3. The line segments \(\overline{PF_1}\) and \(\overline{PA}\) are tangential to the sphere \(d_1\) and, hence, are of equal length.
4. The line segments \(\overline{PF_2}\) and \(\overline{PB}\) are tangential to the sphere \(d_2\) and, hence, are of equal length.
5. The result is: \(|PF_1| - |PF_2| = |PA| - |PB| = |AB|\) is independent of the hyperbola point \(P\), because no matter where point \(P\) is, \(A, B\) have to be on circles \(c_1\), \(c_2\), and line segment \(AB\) has to cross the apex. Therefore, as point \(P\) moves along the red curve (hyperbola), line segment \(\overline{AB}\) 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:

2. Choose the foci \(F_1,F_2\) and one of the circular directrices, for example \(c_2\) (circle with radius \(2a\))
3. A ruler is fixed at point \(F_2\) free to rotate around \(F_2\). Point \(B\) is marked at distance \(2a\).
4. A string gets its one end pinned at point \(A\) on the ruler and its length is made \(|AB|\).
5. The free end of the string is pinned to point \(F_1\).
6. Take a pen and hold the string tight to the edge of the ruler.
7. Rotating the ruler around \(F_2\) prompts the pen to draw an arc of the right branch of the hyperbola, because of \(|PF_1| = |PB|\) (see the definition of a hyperbola by circular directrices).

The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section:

For the generation of points of the hyperbola \(\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2} = 1\) one uses the pencils at the vertices \(V_1,V_2\). Let \(P = (x_0,y_0)\) be a point of the hyperbola and \(A = (a,y_0), B = (x_0,0)\). The line segment \(\overline{BP}\) is divided into n equally-spaced segments and this division is projected parallel with the diagonal \(AB\) as direction onto the line segment \(\overline{AP}\) (see diagram). The parallel projection is part of the projective mapping between the pencils at \(V_1\) and \(V_2\) needed. The intersection points of any two related lines \(S_1 A_i\) and \(S_2 B_i\) are points of the uniquely defined hyperbola.

Remarks:

• The subdivision could be extended beyond the points \(A\) and \(B\) in order to get more points, but the determination of the intersection points would become more inaccurate. A better idea is extending the points already constructed by symmetry (see animation).
• The Steiner generation exists for ellipses and parabolas, too.
• The Steiner generation is sometimes called a parallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.

A hyperbola with equation \(y=\tfrac{a}{x-b}+c,\ a \ne 0\) is uniquely determined by three points \((x_1,y_1),\;(x_2,y_2),\; (x_3,y_3)\) with different x- and y-coordinates. A simple way to determine the shape parameters \(a,b,c\) uses the inscribed angle theorem for hyperbolas:

Analogous to the inscribed angle theorem for circles one gets the

Inscribed angle theorem for hyperbolas, For four points \(P_i = (x_i,y_i),\ i=1,2,3,4,\ x_i\ne x_k, y_i\ne y_k, i\ne k\) (see diagram) the following statement is true:

The four points are on a hyperbola with equation \(y = \tfrac{a}{x-b} + c\) if and only if the angles at \(P_3\) and \(P_4\) are equal in the sense of the measurement above. That means if \[\frac{(y_4-y_1)}{(x_4-x_1)}\frac{(x_4-x_2)}{(y_4-y_2)}=\frac{(y_3-y_1)}{(x_3-x_1)}\frac{(x_3-x_2)}{(y_3-y_2)}\]

The proof can be derived by straightforward calculation. If the points are on a hyperbola, one can assume the hyperbola's equation is \(y = a/x\).

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 \(P_i=(x_i,y_i),\ i=1,2,3,\ x_i\ne x_k, y_i\ne y_k, i\ne k\) is the solution of the equation \[\frac{({\color{red}y}-y_1)}{({\color{green}x}-x_1)}\frac{({\color{green}x}-x_2)}{({\color{red}y}-y_2)}=\frac{(y_3-y_1)}{(x_3-x_1)}\frac{(x_3-x_2)}{(y_3-y_2)}\] for \({\color{red}y}\).

Another definition of a hyperbola uses affine transformations:

Because the unit hyperbola \(x^2-y^2=1\) is affinely equivalent to the hyperbola \(y=1/x\), an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola \(y = 1/x \,\):

\[\vec x = \vec p(t) = \vec f_0 + \vec f_1 t + \vec f_2 \tfrac{1}{t}, \quad t\ne 0\, .\]

\(M: \vec f_0\) is the center of the hyperbola, the vectors \(\vec f_1 , \vec f_2\) have the directions of the asymptotes and \(\vec f_1 + \vec f_2\) is a point of the hyperbola. The tangent vector is \[\vec p'(t)=\vec f_1 - \vec f_2 \tfrac{1}{t^2}.\] At a vertex the tangent is perpendicular to the major axis. Hence \[\vec p'(t)\cdot \left(\vec p(t) -\vec f_0\right) = \left(\vec f_1 - \vec f_2 \tfrac{1}{t^2}\right)\cdot\left(\vec f_1 t+ \vec f_2 \tfrac{1}{t}\right) = \vec f_1^2t-\vec f_2^2 \tfrac{1}{t^3} = 0\] and the parameter of a vertex is

\[t_0= \pm \sqrt[4]{\frac{\vec f_2^2}{\vec f_1^2}}.\]

\(\left|\vec f\!_1\right| = \left|\vec f\!_2\right|\) is equivalent to \(t_0 = \pm 1\) and \(\vec f_0 \pm (\vec f_1+\vec f_2)\) 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.

The reciprocation of a circle B in a circle C always yields a conic section such as a hyperbola. The process of "reciprocation in a circle C" consists of replacing every line and point in a geometrical figure with their corresponding pole and polar, respectively. The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point.

The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then

\[e = \frac{\overline{BC}}{r}.\]

Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle C.

This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle B, as well as the envelope of the polar lines of the points on B. Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.

A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates \((x, y)\) in the plane,

\[A_{xx} x^2 + 2 A_{xy} xy + A_{yy} y^2 + 2 B_x x + 2 B_y y + C = 0,\]

provided that the constants \(A_{xx},\) \(A_{xy},\) \(A_{yy},\) \(B_x,\) \(B_y,\) and \(C\) satisfy the determinant condition

\[D := \begin{vmatrix} A_{xx} & A_{xy} \\ A_{xy} & A_{yy} \end{vmatrix} < 0.\]

This determinant is conventionally called the discriminant of the conic section.

A special case of a hyperbola, the degenerate hyperbola consisting of two intersecting lines, occurs when another determinant is zero:

\[\Delta := \begin{vmatrix} A_{xx} & A_{xy} & B_x \\ A_{xy} & A_{yy} & B_y \\ B_x & B_y & C \end{vmatrix} = 0.\]

Condensed: the full section is in Wikipedia.

If Cartesian coordinates are introduced such that the origin is the center of the hyperbola and the x-axis is the major axis, then the hyperbola is called east-west-opening and

the foci are the points \(F_1=(c,0),\ F_2=(-c,0)\),

the vertices are \(V_1=(a, 0),\ V_2=(-a,0)\).

For an arbitrary point \((x,y)\) the distance to the focus \((c,0)\) is \(\sqrt{(x-c)^2 + y^2}\) and to the second focus \(\sqrt{(x+c)^2 + y^2}\). Hence the point \((x,y)\) is on the hyperbola if the following condition is fulfilled \[\sqrt{(x-c)^2 + y^2} - \sqrt{(x+c)^2 + y^2} = \pm 2a \ .\] Remove the square roots by suitable squarings and use the relation \(b^2 = c^2-a^2\) to obtain the equation of the hyperbola:

\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ .\]

This equation is called the canonical form of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is congruent to the original (see below).

The axes of symmetry or principal axes are the transverse axis (containing the segment of length 2a with endpoints at the vertices) and the conjugate axis (containing the segment of length 2b perpendicular to the transverse axis and with midpoint at the hyperbola's center). As opposed to an ellipse, a hyperbola has only two vertices: \((a,0),\; (-a,0)\). The two points \((0,b),\; (0,-b)\) on the conjugate axes are not on the hyperbola.

It follows from the equation that the hyperbola is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin.