This article is about a geometric curve. For the term used in rhetoric, see Hyperbole.
In mathematics, a hyperbola is a type of smoothcurve 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. A circle 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 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 [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 the asymptotes are the two coordinate axes.[1]
The word "hyperbola" derives from the Greekὑπερβολή, 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.[2] 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.[3] 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.[4]
A hyperbola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:
A hyperbola is a set of points, such that for any point of the set, the absolute difference of the distances to two fixed points (the foci) is constant, usually denoted by :[5]
The midpoint of the line segment joining the foci is called the center of the hyperbola.[6] 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.[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.
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.
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.
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 .
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).
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.
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 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).
Given two pencils 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 a parallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.
Inscribed angles for hyperbolas y = a/(x − b) + c and the 3-point-form
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:
In order to measure an angle between two lines with equations in this context one uses the quotient
Analogous to the inscribed 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 hyperbola x2 − y2 = 1
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.
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 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 is At a vertex the tangent is perpendicular to the major axis. Hence and 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.
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 of Pascal's theorem.[12]
Area of the grey parallelogram
The area of the grey parallelogram in the above diagram is and 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 points The area of the triangle can be calculated by a 2 × 2 determinant: (see rules for determinants). 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:
The area of the triangle is independent of the point of the hyperbola:
The reciprocation of a circleB in a circle C always yields a conic section such as a hyperbola. The process of "reciprocation in a circle C" consists of replacing every line and point in a geometrical figure with their corresponding pole and polar, respectively. The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point.
The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then
Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle C.
This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle B, as well as the envelope of the polar lines of the points on B. Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.
A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates in the plane,
provided that the constants and satisfy the determinant condition
This determinant is conventionally called the discriminant of the conic section.[14]
A special case of a hyperbola—the degenerate 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
The normal 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 the x-axis is the major axis, then the hyperbola is called east-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 fulfilled Remove the square roots by suitable squarings and use the relation