Hyperbola

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History

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.

Formula

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

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

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.

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.

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 .

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.

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

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books .

Robert Yates: Curves and Their Properties .

See: Websites on Conic Sections .