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.

Title	transition to skew-angled coordinates
Canonical name	TransitionToSkewangledCoordinates
Date of creation	2013-03-22 17:09:39
Last modified on	2013-03-22 17:09:39
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Topic
Classification	msc 51N20
Related topic	RotationMatrix
Related topic	Hyperbola2
Related topic	ConjugateDiametersOfEllipse
Defines	skew-angled coordinate