Hello moment

As I understand it, you need to find, using coordinate geometry, the positions of the vertices of rectangles when they are projected onto a series of lines. So, to be specific, I think you're saying that, given a point (representing a typical vertex of a rectangle) you need to find the position of the projection of onto a line .

If that's so, then the following may be helpful.

Suppose that has coordinates and that the equation of the line is . Suppose further that is the projection of onto , and that the coordinates of are .

Then, because lies on :

(1)

and because :

(2)

Substitute from (1) into (2):

(3)

Substitute into (1):

(4)

So equations (3) and (4) give us the coordinates of Q in terms of and .

Of course, you may be able to simplify these results if you are able to choose a particular origin of coordinates. For example, if the origin can be chosen to lie on the line , then , and equations (3) and (4) simplify to:

Or, if you could also choose two lines at to the axes, then you can make , and the equations would simplify even further to:

I hope this gives you some ideas.

Grandad