PlanetMath: transition to skew-angled coordinates

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transition to skew-angled coordinates

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

$\begin{align*}\begin{cases}x = \xi\cos\alpha+\eta\cos\beta,\\ y = \xi\sin\alpha+\eta\sin\beta, \end{cases}\end{align*}$

(1)

where $\alpha$ and $\beta$ are the angles which the $\xi$ -axis and $\eta$ -axis, respectively, form with the

-axis (positive if

-axis may be rotated anticlocwise to $\xi$ -axis, else negative; similarly for rotating the

-axis to the $\eta$ -axis).

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

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

Example. Let us consider the hyperbola

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

(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

-axis, then $\alpha = -\omega$ , $\beta = \omega$ . By (1) we get first

$\begin{align*}\begin{cases}x = \xi\cos\omega+\eta\cos\omega = (\eta\!+\!\xi)\cos... ...\xi\sin\omega+\eta\sin\omega = (\eta\!-\!\xi)\sin\omega. \end{cases}\end{align*}$

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

, and accordingly

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

Substituting these quotients in the equation of the hyperbola we obtain

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

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,

is the distance of the focus from the nearer apex of the hyperbola.

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

$\displaystyle xy =$ constant $\displaystyle ,$

(4)

we can infer that it represents a hyperbola with asymptotes the coordinate axes. Since these are perpendicular to each other, it's clear that the hyperbola (4) is a rectangular one.

Bibliography

1: L. LINDELÖF: Analyyttisen geometrian oppikirja. Kolmas painos. Suomalaisen Kirjallisuuden Seura, Helsinki (1924).

"transition to skew-angled coordinates" is owned by pahio.

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skew-angled coordinate

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Cross-references: clear, perpendicular, distance, coordinate system, hyperbola, equation, quotients, asymptote, negative, positive, angles, point, origin, coordinate, rectangular coordinate, Euclidean plane
There are 3 references to this entry.

This is version 12 of transition to skew-angled coordinates, born on 2007-05-26, modified 2007-08-25.
Object id is 9472, canonical name is TransitionToSkewAngledCoordinates.
Accessed 1088 times total.

Classification:

AMS MSC:

51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

Pending Errata and Addenda

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