transition to skew-angled coordinates
Let the Euclidean plane be equipped with the rectangular coordinate system with the and coordinate axes. We choose new coordinate axes through the old origin and project (http://planetmath.org/Projection) the new coordinates , of a point orthogonally on the and axes getting the old coordinates expressed as
where and are the angles which the -axis and -axis, respectively, form with the -axis (positive if -axis may be rotated anticlocwise to -axis, else negative; similarly for rotating the -axis to the -axis).
The of (1) are got by solving from it for and , getting
Example. Let us consider the hyperbola (http://planetmath.org/Hyperbola2)
and take its asymptote for the -axis and the asymptote for the -axis. If is the angle formed by the latter asymptote with the -axis, then , . By (1) we get first
Since , we see that , , where , and accordingly
Substituting these quotients in the equation of the hyperbola we obtain
and after simplifying,
This is the equation of the hyperbola (2) in the coordinate system of its asymptotes. Here, 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 () an equation of the form (3), e.g.
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.
- 1 L. Lindelöf: Analyyttisen geometrian oppikirja. Kolmas painos. Suomalaisen Kirjallisuuden Seura, Helsinki (1924).
|Title||transition to skew-angled coordinates|
|Date of creation||2013-03-22 17:09:39|
|Last modified on||2013-03-22 17:09:39|
|Last modified by||pahio (2872)|