Solving the Points on a Rotated Rectangle

Question

I am working in AS3.

I have a generic rectangle. This rectangle can have any length, any width and any rotation. I am trying to solve for the x and y coordinates of the four corners of the rectangle. I know the coordinates of the centre of the rectangle, I know its width, its height, the y distance between the highest and lowest point and the x distance between the farthest left and farthest right point as well as knowing the rotation.

My code currently looks like this (Object, of course, being the rectangle in question, keep in mind that when I apply this it can have any dimensions - This is just one possibility. Initial width and height are the actual length and width, while width and height referenced later are the x and y distances between the highest and lowest points and the farthest left and right points, rotation is of course rotation, and x and y are the object's centre coordinates).

import flash.events.Event;
addEventListener(Event.ENTER_FRAME, Rotate, false, 0, true);

var Radius:Number = Math.sqrt(((Object.height / 2) * (Object.height / 2)) + ((Object.width / 2) * (Object.width / 2)));

function Rotate(event:Event)
{

    Object.rotation += 1;
    Marker1.x = Math.sqrt((Radius * Radius) - ((Object.height / 2) * (Object.height / 2))) + Object.x;
    Marker2.x = - Math.sqrt((Radius * Radius) - ((Object.height / 2) * (Object.height / 2))) + Object.x;
    Marker3.y = Math.sqrt((Radius * Radius) - ((Object.width / 2) * (Object.width / 2))) + Object.y;
    Marker4.y = - Math.sqrt((Radius * Radius) - ((Object.width / 2) * (Object.width / 2))) + Object.y;

    Marker1.y = Object.y + (Object.height / 2);
    Marker2.y = Object.y - (Object.height / 2);
    Marker3.x = Object.x + (Object.width / 2);
    Marker4.x = Object.x - (Object.width / 2);

}

As you can see I am attempting to use circle geometry to place four small circles (Markers 1-4) at the corners of the rectangle, just for testing purposes to confirm that I have gathered the correct coordinates. Problem is, the coordinates will always be placed in either +x and +y or -x and -y, but never the other two quadrants of the graph. I can't figure out a simple way of dynamically simulating the +- of the quadratic equation in the program. Does anyone know of a way to find these four points with and length, width and rotation of the rectangle?

Answer 1

If you represent the coordinates of the corners as offsets from the midpoint of the rectangle you can easily rotate them anti-clockwise by an angle θ with

dx' = dx × cos θ - dy × sin θ
dy' = dx × sin θ + dy × cos θ

You can then add the rotated offsets to the midpoint to recover the new coordinates of the corners.