Calculate the R2 change of basis matrix to transform two lines that intersect into two parallel lines perpendicular to x axis.

Question

Image: Calculate the R2 change of basis to transform two lines that intersect into two parallel lines perpendicular to x axis.

Calculate the R2 change of basis to transform two lines that intersect into two parallel lines perpendicular to x axis.

The idea is going from lines f and g to lines h and i. B is in f equivalent to E in h, D is in g equivalent to F in i.

I think Points A, C, H, G, J and I are important to construct the vectors of the change of basis.

Its a kind of rotation with scale transform. The intersection point between f and g should end up on the infinity.

I think a good solution could be to change the space to a higher dimension space like R4 or R5 with an ortonormal component to the plane xy. Calculate a basis, apply the change of basis and then proyect the higher space to xy plane. Using the points A, B, C and D filling z, v and w with 0's and an unitary vector for w axis.

Answer 1

Knowing the images of four points uniquely define a projective transformation, as long as no three of them lie on a line. An older post of mine has details on how to compute that.

Note that embedding the plane into a 3d space can help dealing with protective transformations by using homogeneous coordinates. But it's not a simple 3d rotation or scaling. Both rotations and scalings preserve parallelism, so you need a larger type of transformation to turn intersecting lines into parallel lines. A projective transformation is likely the tool you need. So when you speak about a change of basis, you would need to use a projective basis, and when you speak about the vectors of that basis, likely they should be homogeneous vectors.

So in your case you could map $B\mapsto E, D\mapsto F, C\mapsto C, A\mapsto A$ as one possible solution.

But I'm not sure that $A$ and $C$ have to be fixed points. From what you have written, you might as well map any second point on $f$ other than $B$ to any second point on $h$ other than $E$, and any second point on $g$ other than $D$ to any second point on $i$ other than $F$. That would still map the lines as required.