transition to skew-angled coordinates

Let the Euclidean plane $ℝ$ 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 (http://planetmath.org/Projection) the new coordinates $\xi$, $\eta$ of a point orthogonally on the $x$ and $y$ axes getting the old coordinates expressed as

$\{\begin{array}{cc}x=\xi \cos \alpha +\eta \cos \beta ,\hfill & \\ y=\xi \sin \alpha +\eta \sin \beta ,\hfill & \end{array}$(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

$$\xi =\frac{x\sin \beta -y\cos \beta }{\sin (\beta -\alpha )},\eta =\frac{-x\sin \alpha +y\cos \alpha }{\sin (\beta -\alpha )}.$$

Example.  Let us consider the hyperbola (http://planetmath.org/Hyperbola2)

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

$\{\begin{array}{cc}x=\xi \cos \omega +\eta \cos \omega =(\eta +\xi )\cos \omega ,\hfill & \\ y=-\xi \sin \omega +\eta \sin \omega =(\eta -\xi )\sin \omega .\hfill & \end{array}$

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

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

Substituting these quotients in the equation of the hyperbola we obtain

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

and after simplifying,

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

$xy=\text{ constant},$(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.