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Drini-conjugatehyperbolas

A graph of the unit hyperbola (where x^2 - y^2 = 1, in blue) and its conjugate (y^2 - x^2 = 1, in green) as well as the asymptotes in red.

A hyperbola is a type of conic section. A hyperbola is a set of points where the distances between the line called the directrix and point called the focus remain in a constant ratio (this ratio is the hyperbola's eccentricity). A hyperbola is visually similar to a parabola, but with two mirrored sides. The formula for a hyperbola (assuming they are orientated facing left and right rather than up and down) is

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

with a being the distance between the vertices and the origin and \frac{a}{b} being equal to the slope of the asymptotes.

Similarly to how the unit circle defines trigonometric functions, the unit hyperbola, x^2 - y^2 = 1, defines hyperbolic functions.

Rectangular Hyperbola

The reciprocal function which shows one variable is inversely proportional to another also describes a hyperbola but oriented differently. It can be shown that this equation is in fact a hyperbola rotated 45 degrees about the origin. y=\frac{k}{x}

Start with the Cartesian form \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Rewrite in a parametric/vector form. \left (\begin{matrix}x
\\ 
y
\end{matrix}  \right )
=
\left (\begin{matrix}a \tan(t)
\\ 
b \sec(t)
\end{matrix}  \right )

Pre-multiply by rotation matrix representing a 45 degree anti-clockwise rotation about the origin from the x-axis (we are making a new equation).


\left (\begin{matrix}x
\\ 
y
\end{matrix}  \right )
=
\begin{pmatrix}
\cos{\pi/4} & -\sin{\pi/4} \\ 
\sin{\pi/4}  & \cos{\pi/4} 
\end{pmatrix}
\left (\begin{matrix}a \tan(t)
\\ 
b \sec(t)
\end{matrix}  \right )
\left (\begin{matrix}x
\\ 
y
\end{matrix}  \right )
=
\left (\begin{matrix}a\tan{t}\sqrt{2}/2-b\sec{t}\sqrt{2}/2
\\ 
a\tan{t}\sqrt{2}/2+b\sec{t}\sqrt{2}/2
\end{matrix}  \right )

Take out the factor of half square root 2 \left (\begin{matrix}x
\\ 
y
\end{matrix}  \right )
=
\frac{\sqrt{2}}{2}\left (\begin{matrix}a\tan{t}-b\sec{t}
\\ 
a\tan{t}+b\sec{t}
\end{matrix}  \right )

See also

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