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Hyperbola

Family of hyperbolas with eccentricities {4., 2.81, 2.17, 1.76, 1.49, 1.28, 1.13, 1.01}, in order of light to dark shade. The left family share vertexes, the right are confocal.

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History

Differential Equations, Mechanics, and Computation

See the History section of Conic Sections page.

Description

Hyperbola describe a family of curves with single parameter. Together with ellipse and parabola, they make up the conic sections.

Hyperbola is commonly defined as the locus of points P such that the difference of the distances from P to two fixed points F1, F2 (called foci) are constant. That is, Abs[ distance[P,F1] - distance[P,F2] ] == 2 a, where a is a constant. Hyperbola as Difference of Distances.

The eccentricity is a number that describe the “flatness” of the hyperbola. Let the distance between foci be 2 c, then eccentricity e is defined by e := c/a. 1 < e. The larger the eccentricity, the more it resembles two parallel lines. As e approaches 1, the vertexes become more pointed.

The line passing through foci is the axis of the hyperbola. A line passing through center and perpendicular to the axis is the transverse axis. The vertexes are the intersections of the hyperbola and its axis.

A rectangular hyperbola is a hyperbola with eccentricity Sqrt[2]≈1.4142. Its asymptotes are mutually perpendicular. A simple Cartesian equation for rectangular hyperbola is x*y == 1.

Rectangular hyperbola have the property that when streched along one or both of its asymptotes, the curve remains the same. That is, the curve {t, 1/t n}, {t n, 1/t}, and {t, 1/t} Sqrt[n] are the same curve with various degrees of magnification.

Formulas

For a hyperbola with vertexes fixed at {±1,0} and eccentricity e, we have (foci is {±e, 0}):

Parametric: {-Sec[t], Sqrt[e^2-1]*Tan[t]}
Cartesian: x^2 - y^2/(e^2-1) == 1 hyperbola_eq.gcf

For a hyperbola with vertex at ±{a,0}, directrix at x==±a/e, asymptotes at y==±b/a*x, we have:

Parametric: {a*Sinh[t], b*Sinh[t]}
Cartesian: x^2/a^2 - y^2/b^2 == 1 hyperbola_eq2.gcf

The parametric form with Sinh is derived by replacing y in x^2/a^2-y^2/b^2==1 with b*Sinh[t], then solve for x. This is analogous to the ellipse case where we replace a variable in x^2/a^2+y^2/b^2==1 by Cos[t] and solve the other to find {a*Cos[t],b*Sin[t]}.

Properties

Point-wise construction

This method is derived from the paramteric formula for hyperbola {a Sec[t], b Tan[t]}, where a and b are the radiuses of the concentric circles. If a == b, then the curve traced is rectangular hyperbola.

Hyperboda Point-wise Construction; a:=1, b:=2; a:=b:=1; a:=2, b:=1

Let O be the origin. Let A and B be points on the positive x-axis.
Let there be a circle centered on O and passing A.
Let there be a circle centered on O and passing B.
Let there be a line a1, that passes A and parallel to y-axis.
Let there be a line b1, that passes B and parallel to y-axis.
Let there be a point D on one of the circle.
Let there be a line OD.
Let A1 := Intersection[Line[O,D],a1]
Let B1 := Intersection[Line[O,D],b1]
Let there be a circle centered on O passing A1. Let the intersection of this circle and positive x-axis be E. Let there be a line g, passing E and parallel to y-axis.
Let there be a line passing B1 and parallel to x-axis.
The locus of intersection g and B1, as D moves on the circle, is a hyperbola.

Optical Property

Light rays coming from one focus of a hyperbola will refract to the other focus.

The catacaustic and diacaustic of a hyperbola with eccentricity 2.5.

Pedal

The pedal of a hyperbola with respect to a focus is a circle. The pedal of a rectangular hyperbola with respect to its center is a lemniscate of Bernoulli.

Hyperbola Pedal

Inversion

The inversion of a rectangular hyperbola with respect to its center is again a lemniscate of Bernoulli. If the inversion point is at a vertex, the inversion curve is a right strophoid.