# transition to skew-angled coordinates

Let the Euclidean plane  $\mathbb{R}$ be equipped with the rectangular coordinate system with the $x$ and $y$ coordinate   axes.  We choose new coordinate axes through the old origin and project () the new coordinates $\xi$, $\eta$ of a point orthogonally on the $x$ and $y$ axes getting the old coordinates expressed as

 $\displaystyle\begin{cases}x=\xi\cos\alpha+\eta\cos\beta,\\ y=\xi\sin\alpha+\eta\sin\beta,\end{cases}$ (1)

where $\alpha$ and $\beta$ are the angles which the $\xi$-axis and $\eta$-axis, respectively, form with the $x$-axis (positive if $x$-axis may be rotated anticlocwise to $\xi$-axis, else negative; similarly for rotating the $x$-axis to the $\eta$-axis).

The of (1) are got by solving from it for $\xi$ and $\eta$, getting

 $\xi=\frac{x\sin\beta-y\cos\beta}{\sin(\beta\!-\!\alpha)},\quad\eta=\frac{-x% \sin\alpha+y\cos\alpha}{\sin(\beta\!-\!\alpha)}.$
 $\displaystyle\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ (2)

and take its asymptote  $y=-\frac{b}{a}x$  for the $\xi$-axis and the asymptote  $y=+\frac{b}{a}c$  for the $\eta$-axis.  If $\omega$ is the angle formed by the latter asymptote with the $x$-axis, then  $\alpha=-\omega$,  $\beta=\omega$.  By (1) we get first

 $\displaystyle\begin{cases}x=\xi\cos\omega+\eta\cos\omega=(\eta\!+\!\xi)\cos% \omega,\\ y=-\xi\sin\omega+\eta\sin\omega=(\eta\!-\!\xi)\sin\omega.\end{cases}$

Since  $\displaystyle\tan\omega=\frac{b}{a}$,  we see that  $\displaystyle\cos\omega=\frac{a}{c}$,  $\displaystyle\sin\omega=\frac{b}{c}$,  where  $c^{2}=a^{2}+c^{2}$,  and accordingly

 $\frac{x}{a}=(\eta\!+\!\xi)\frac{a}{c}:a=\frac{\eta\!+\!\xi}{c},\quad\frac{y}{b% }=(\eta\!-\!\xi)\frac{b}{c}:b=\frac{\eta\!-\!\xi}{c}.$

Substituting these quotients in the equation of the hyperbola we obtain

 $(\eta\!+\!\xi)^{2}-(\eta\!-\!\xi)^{2}=c^{2},$

and after simplifying,

 $\displaystyle\xi\eta=\frac{c^{2}}{4}.$ (3)

This is the equation of the hyperbola (2) in the coordinate system  of its asymptotes.  Here, $c$ is the distance  of the focus (http://planetmath.org/Hyperbola2) from the nearer apex (http://planetmath.org/Hyperbola2) of the hyperbola.

If we, conversely, have in the rectangular coordinate system ($x,\,y$) an equation of the form (3), e.g.

 $\displaystyle xy=\mbox{\,constant},$ (4)

we can infer that it a hyperbola with asymptotes the coordinate axes. Since these are perpendicular    to each other, it’s clear that the hyperbola (4) is a rectangular (http://planetmath.org/Hyperbola2) one.

## References

• 1 L. Lindelöf: Analyyttisen geometrian oppikirja.  Kolmas painos.  Suomalaisen Kirjallisuuden Seura, Helsinki (1924).
Title transition to skew-angled coordinates TransitionToSkewangledCoordinates 2013-03-22 17:09:39 2013-03-22 17:09:39 pahio (2872) pahio (2872) 15 pahio (2872) Topic msc 51N20 RotationMatrix Hyperbola2 ConjugateDiametersOfEllipse skew-angled coordinate