transition to skew-angled coordinates

Let the Euclidean planeMathworldPlanetmath be equipped with the rectangular coordinate system with the x and y coordinateMathworldPlanetmathPlanetmath axes.  We choose new coordinate axes through the old origin and project ( the new coordinates ξ, η of a point orthogonally on the x and y axes getting the old coordinates expressed as

{x=ξcosα+ηcosβ,y=ξsinα+ηsinβ, (1)

where α and β are the angles which the ξ-axis and η-axis, respectively, form with the x-axis (positive if x-axis may be rotated anticlocwise to ξ-axis, else negative; similarly for rotating the x-axis to the η-axis).

The of (1) are got by solving from it for ξ and η, getting


Example.  Let us consider the hyperbolaMathworldPlanetmathPlanetmath (

x2a2-y2b2=1 (2)

and take its asymptoteMathworldPlanetmathy=-bax  for the ξ-axis and the asymptote  y=+bac  for the η-axis.  If ω is the angle formed by the latter asymptote with the x-axis, then  α=-ω,  β=ω.  By (1) we get first


Since  tanω=ba,  we see that  cosω=ac,  sinω=bc,  where  c2=a2+c2,  and accordingly


Substituting these quotients in the equation of the hyperbola we obtain


and after simplifying,

ξη=c24. (3)

This is the equation of the hyperbola (2) in the coordinate systemMathworldPlanetmath of its asymptotes.  Here, c is the distanceMathworldPlanetmath 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.

xy= constant, (4)

we can infer that it a hyperbola with asymptotes the coordinate axes. Since these are perpendicularMathworldPlanetmathPlanetmathPlanetmath 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).
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