Draw then from {P

(i) the

(ii) the

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 {PP

This, setting b

In these equations A = a

[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 {a

[1] There are two (yellow) lines through O, which are simultaneously harmonic conjugate with respect to the pairs of lines {OX, OY} and {Ob

[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

Chasles_Steiner_Envelope.html

CoordinatesTransform2.html

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