In mathematics, a hyperbola ( ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse, with the circle being a special type of ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship 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 the asymptotes are the two coordinate axes.
Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).
Etymology and history The word "hyperbola" derives from the Greek , meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones.. The term hyperbola is believed to have been coined by Apollonius of Perga () in his definitive work on the conic sections, the Conics. The names of the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than the segment or exceed the segment.
Definitions
As locus of points
A hyperbola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:
The midpoint 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 , which have distance to the center. The distance of the foci to the center is called the focal distance or linear eccentricity. The quotient is the eccentricity .
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 the circular directrix (related to focus ) of the hyperbola.The German term for this circle is Leitkreis which can be translated as "Director circle", but that term has a different meaning in the English literature (see Director circle). 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.
Hyperbola with equation
If the xy-coordinate system is rotated about the origin by the angle and new coordinates are assigned, then . The rectangular hyperbola (whose semi-axes are equal) has the new equation . Solving for yields
Thus, in an xy-coordinate system the graph of a function with equation is a rectangular hyperbola entirely in the first and third quadrants with the coordinate axes as asymptotes, the line as major axis , the center and the semi-axis the vertices the semi-latus rectum and radius of curvature at the vertices the linear eccentricity and the eccentricity the tangent 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
the semi-axes the line as major axis, the vertices
Shifting the hyperbola with equation so that the new center is yields the new equation
and the new asymptotes are and . The shape parameters remain unchanged.
By the directrix property
The two lines at distance from the center and parallel to the minor axis are called directrices 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 equation
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 (focus), any line (directrix) not through and any real number with the set of points (locus of points), for which the quotient of the distances to the point and to the line is
is a hyperbola.
(The choice yields a parabola and if an ellipse.)
Proof Let and assume is a point on the curve. The directrix has equation . With , the relation produces the equations and The substitution yields
This is the equation of an ellipse () or a parabola () or a hyperbola (). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
If , introduce new parameters so that , and then the equation above becomes
which is the equation of a hyperbola with center , the x-axis as major axis and the major/minor semi axis .
Construction of a directrix Because of point of directrix (see diagram) and focus are inverse with respect to the circle inversion at circle (in diagram green). Hence point can be constructed using the theorem 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).
As plane section of a cone
The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see diagram: red curve). In order to prove the defining property of a hyperbola (see above) one uses two Dandelin spheres , which are spheres that touch the cone along circles and the intersecting (hyperbola) plane at points and It turns out: are the foci of the hyperbola. Let be an arbitrary point of the intersection curve. The generatrix 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.
Pin and string construction
The definition of a hyperbola by its foci and its circular directrices (see above) can be used for drawing an arc of it with help of pins, a string and a ruler: Frans van Schooten: Mathematische Oeffeningen, Leyden, 1659, p. 327
Choose the foci and one of the circular directrices, for example (circle with radius ) A ruler is fixed at point free to rotate around . Point is marked at distance . A string 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 a pen 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 by circular directrices).
Steiner generation of a hyperbola
The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section:
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 a parallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.
Inscribed angles for hyperbolas and the 3-point-form
A hyperbola with equation is uniquely determined by three points with different x- and y-coordinates. A simple way to determine the shape parameters uses the inscribed angle theorem for hyperbolas:
Analogous to the inscribed angle theorem for circles one gets the
A consequence of the inscribed angle theorem for hyperbolas is the
As an affine image of the unit hyperbola
Another definition of a hyperbola uses affine transformations:
Parametric representation An affine transformation of the Euclidean plane has the form , where is a regular matrix (its determinant 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.
Vertices In general the vectors are not perpendicular. That means, in general are not the vertices of the hyperbola. But point into the directions of the asymptotes. The tangent vector at point is
Because at a vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter of a vertex from the equation
and hence from
which yields
The formulae and were used.
The two vertices of the hyperbola are
Implicit representation Solving the parametric representation for by Cramer's rule and using , one gets the implicit representation
Hyperbola in space 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.
As an affine image of the hyperbola