How to rotate a rectangle until one of the edges is parallel to the x-axis (with unknown rotation angle)?

Hello,

Within a (Cartesian) coordinate system, I have many tilted rectangles that I need to rotate so that one edge will be parallel to the x-axis. For simplicity let's just consider one rectangle. For each rectangle drawn in the system, I also have a separate point which I will also have to rotate. So, for example, if I had a tilted rectangle with these coordinates

x = [4 16 14 2];
y = [2 6 12 8];

and a separate point p1 = [5,7], then I need to rotate the rectangle until the edge which connects the vertices [4,2] and [16,6] is parallel to the x-axis, and also map the point p1. I don't mind where the vertices will end up as long as that edge is parallel to the x-axis.

Mapping the rectangle using simple arithmetics (Pythagoras etc) is easy, but then I don't know how to map the point p1 (or at least I've thought about this for so long, I can't think straight anymore!). I don't have a rotation angle, so I couldn't find a way how to calculate new points using a standard rotation matrix. I was thinking about something like this below, but I'm not sure how to implement this, so any advice would be great.

while slopeOfEdge~=0;
   %keep rotating; 
end

Thanks!!!

[dpb]

I don't follow what "map" means in your context and how point p1 enters into the mix.

But, you can easily derive the angle of the lower edge boundary from the coordinates to derive the rotation matrix.

I guess, is the question to rotate around p1, maybe???

Answer 1

[Massimo Zanetti]

To find the angle between the edge of the rectangle and the x-axis do simply (using your notation for vertex as x,y):

c=[x(1),1;x(2),1]\[y(1);y(2)];
alpha=atan2(c(1),1);

Then you can just build a rotation matrix using this alpha:

R=[cos(-alpha),-sin(-alpha);
   sin(-alpha),cos(-alpha)];

To obtain the new coordinates of the aligned rectangle:

W=R*[x;y];
newX=W(1,:);
newY=W(2,:);
plot(x,y,'b',newX,newY,'r'); axis equal

Notice that, you have to adjust the rotatation direction of R accounting for the sign of alpha, that's why in sin and cos the argument is -alpha.

Of course, you can use the same rotation matrix to map the point p.

[xiuJe]

This is exactly what I was looking for! Thank you!

And yes, although I wasn't very clear, you understood correctly that I also need to map the point p (rather than using p as a point for rotation). Thanks!

[SHREYASI BHATTACHARYA]

If I don't know any vertices of rectangle,then how I find rotation angle? Please help me by giving answer.