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 , giving eccentricity . Plugging 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),
(1)
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therefore gives
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The rectangular hyperbola opening to the left and right has polar equation
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and the rectangular hyperbola opening in the first and third quadrants has the Cartesian equation
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The parametric equations for the right branch of a rectangular hyperbola are given by
(5)
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(6)
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where is the hyperbolic cosine and is the hyperbolic sine. The curvature, arc length, and tangential angle for the above parametrization with are
(7)
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(8)
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(9)
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(10)
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(11)
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where is an elliptic integral of the second kind, is the gamma function, is a hypergeometric function, is an incomplete beta function, and is a hyperbolic tangent.
A parametrization which gives both branches is given by
(12)
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(13)
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with and discontinuities at .
The inverse curve of a rectangular hyperbola with inversion center at the center of the hyperbola is a lemniscate (Wells 1991).
If the three vertices of a triangle lie on a rectangular hyperbola, then so does the orthocenter (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 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).