Coordinates transformation

Given a system of intersecting lines {OX, OY} and two directions {a, b} there is a coordinate system defined by projecting an arbitrary point P onto {OX, OY} to points {Px, Py}, such that {PPx, PPy} are respectively parallels to {a,b}.
Draw then from {Px, Py} parallels to {OY, OX} respectively intersecting at P'. Measuring the positions of {Px, Py} on the axes with a pair of numbers (x,y) defines:
(i) the [a,b]-related coordinate system, for which P is identified with (x,y).
(ii) the paralllel (to axes) coordinate system, for which P' is identified with (x,y).
To study the relations between the two coordinate systems, as well as the transformation F sending P to P'.

[1] To find the relations between the two coordinate systems apply the sine theorem to the two triangles with sides {PPx, y'} and {PPy, x'} respectively:

This, setting by=sin(b,OY), bx=sin(b,OX), ax=sin(a,OX) and ay=sin(a,OY) leads to the following linear relations between the coordinates:

In these equations A = ay/ax and B = bx/by. In these equations (x,y) express the [a,b]-related coordinates and (x',y') express the parallel coordinates of (the same) point P.
[2] Transformation P'=F(P), expressed in parallel coordinates (x',y'), has the same expression, since the coordinates (x'',y'') of P' are the same (x,y) with those of P in the [a,b]-related system, which are related to the (x',y') of P through the previous formula. Thus F in the parallel coordinate system is expressed through:

This shows that F(P)=P' is an affine mapping with fixed point O. In particular it maps lines onto lines. The picture above shows how to determine the image e'=F(e) of a line under this transformation: (i) Draw parallels to {a,b} from O to find their intersections {ae,be} with line e. (ii) Draw parallels to {b,a} from {ae,be} to find their intersections {ey,ex} with lines {OY,OX} respectively. Line e' passes through ex, ey.

Remarks
[1] There are two (yellow) lines through O, which are simultaneously harmonic conjugate with respect to the pairs of lines {OX, OY} and {Obe, Oae}. These correspond to the eigenvectors of the matrix representing F and are characterized by the fact that line PP' passes through O, i.e. they are invariant with respect to F.
[2] Lines PP' envelope a parabola. This follows from general considerations concerning the envelope of lines joining two points related by a homographic relation (see Chasles_Steiner_Envelope.html ). The fact that the conic is a parabola is due to the fact that F is an affinity (see AffinityGeneratedParabola.html ). It can also be verified directly by showing that it has only one point at infinity.
[3] There is obviously a preference in the use of {OX,OY} coordinates. One could consider two systems of parallel coordinates and extend {PPx, PPy} until they cut the respective axes {Obe, Oae} and do the same investigation now between two parallel systems. This is discussed in CoordinatesTransform2.html .