Hyperbola

hyperbola 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.

Mathematica Notebook for This Page.

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}):

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

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.

hyperbola point const hyperbola hyperbola hyperbola
Hyperboda Point-wise Construction; a:=1, b:=2; a:=b:=1; a:=2, b:=1
  1. Let O be the origin. Let A and B be points on the positive x-axis.
  2. Let there be a circle centered on O and passing A.
  3. Let there be a circle centered on O and passing B.
  4. Let there be a line a1, that passes A and parallel to y-axis.
  5. Let there be a line b1, that passes B and parallel to y-axis.
  6. Let there be a point D on one of the circle.
  7. Let there be a line OD.
  8. Let A1 := Intersection[Line[O,D],a1]
  9. Let B1 := Intersection[Line[O,D],b1]
  10. 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.
  11. Let there be a line passing B1 and parallel to x-axis.
  12. 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.

hyperbola catacaustic hyperbola diacaustic
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 hyperbola
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.

hyperbola hyperbola
Left: hyperbola and lemniscate of Bernoulli (both in red dots). Right: Hyperbola (purple) and a right strophoid (red dots).

The inversion of any hyperbola with respect to a focus is a limacon of Pascal.

hyperbola
Hyperbola and Limacon of Pascal (both in red dots).

Quadric Surfaces

Equations of polynomials of degree 2 with 2 variables have cross sections that are hyperbola, ellipse, parabola in various ways.

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

Robert Yates: Curves and Their Properties.

Hyperbola.

See: Websites on Conic Sections.

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