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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 $x$ 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
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(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 inverse formulas of (1) are got by solving from it for $\xi$ and $\eta$ , getting $$\xi = \frac{x\sin\beta-y\cos\beta}{\sin(\beta\!-\!\alpha)},\quad \eta = \frac{-x\sin\alpha+y\cos\alpha}{\sin(\beta\!-\!\alpha)}.$$
Example. Let us consider the hyperbola
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(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
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,
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(3) |
This is the equation of the hyperbola (2) in the coordinate system of its asymptotes. Here, $c$ 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.
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(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.
- 1
- L. LINDELÖF: Analyyttisen geometrian oppikirja. Kolmas painos. Suomalaisen Kirjallisuuden Seura, Helsinki (1924).
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"transition to skew-angled coordinates" is owned by pahio.
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Cross-references: clear, perpendicular, conversely, 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 2000 times total.
Classification:
| AMS MSC: | 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry) |
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Pending Errata and Addenda
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