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Rectangular Hyperbola


RectangularHyperbola

A hyperbola for which the asymptotes are perpendicular, also called an equilateral hyperbola or right hyperbola. This occurs when the semimajor and semiminor axes are equal. This corresponds to taking a=b, giving eccentricity e=sqrt(2). Plugging a=b into the general equation of a hyperbola with semimajor axis parallel to the x-axis and semiminor axis parallel to the y-axis (i.e., vertical conic section directrix),

 ((x-x_0)^2)/(a^2)-((y-y_0)^2)/(b^2)=1
(1)

therefore gives

 (x-x_0)^2-(y-y_0)^2=a^2.
(2)

The rectangular hyperbola opening to the left and right has polar equation

 r^2=a^2sec(2theta),
(3)

and the rectangular hyperbola opening in the first and third quadrants has the Cartesian equation

 xy=a^2.
(4)

The parametric equations for the right branch of a rectangular hyperbola are given by

x=acosht
(5)
y=asinht,
(6)

where coshx is the hyperbolic cosine and sinhx is the hyperbolic sine. The curvature, arc length, and tangential angle for the above parametrization with a=b=1 are

kappa(t)=-1/(a[cosh(2t)]^(3/2))
(7)
s(t)=-iaE(it,sqrt(2))
(8)
=asqrt(cosh(2t))+([Gamma(3/4)]^2)/(sqrt(2pi))-sqrt(2)e^(-t)_2F_1(-1/4,1/2;3/4;-e^(4t))
(9)
=asqrt(cosh(2t))+([Gamma(3/4)]^2)/(sqrt(2pi))+1/4(i+1)B(-e^(4t);-1/4,1/2)
(10)
phi(t)=-tan^(-1)(tanht),
(11)

where E(phi,k) is an elliptic integral of the second kind, Gamma(z) is the gamma function, _2F_1(a,b;c;x) is a hypergeometric function, B(z;a,b) is an incomplete beta function, and tanhx is a hyperbolic tangent.

A parametrization which gives both branches is given by

x=asect
(12)
y=atant,
(13)

with t in (-pi,pi) and discontinuities at +/-pi/2.

The inverse curve of a rectangular hyperbola with inversion center at the center of the hyperbola is a lemniscate (Wells 1991).

RectangularHyperbolaTri

If the three vertices of a triangle DeltaABC lie on a rectangular hyperbola, then so does the orthocenter H (Wells 1991). Equivalently, if four points form an orthocentric system, then there is a family of rectangular hyperbolas through the points. Moreover, the locus of centers O of these hyperbolas is the nine-point circle of the triangle (Wells 1991).

If four points do not form an orthocentric system, then there is a unique rectangular hyperbola passing through them, and its center is given by the intersection of the nine-point circles of the points taken three at a time (Wells 1991).


See also

Hyperbola, Lemniscate, Nine-Point Circle, Orthocentric System

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 218-219, 1987.Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 76-77, 1996.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 118, 1969.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 209, 1991.

Referenced on Wolfram|Alpha

Rectangular Hyperbola

Cite this as:

Weisstein, Eric W. "Rectangular Hyperbola." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RectangularHyperbola.html

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