How To Construct A Rotated Box
This demo shows how to construct a rotated box using ellipses.
It was inspired by this video and resulted in this 3D model:
You can step through the construction by using the arrow buttons at the top. The images below were derived from the 3D model.
First, Draw a box. The top and bottom side have to be square, because we want to construct circles (ellipses in perspective) in these faces.
Draw the crosses in the top and bottom planes to find the middle of these square planes.
Draw plusses through the found center points. The lines should go to the same vanishing points as the ribs of the box.
Construct a circle in the top and bottom planes. The ellipses should touch the box ribs at the points where the plusses touch them, and the ellipses should also touch it parallel to these ribs.
Construct a box with the corners on the circles, but also on the lines that create the crosses we drew earlier.
This is the box we are going to rotate.
Do the same as we did with the bigger rectangle, and draw the smaller circle inside the smaller box in the same way.
All we will need to draw that exact box rotated around its center at any angle is these two ellipses.
Now for the fun part: we're going to draw a rotated box.
Choose some point on the top outer circle.
Now draw a line from that point in such a way that it touches in parallel to the inner circle, and stop the line when it reaches the outer circle on the other side again.
That's the first rib of the new, rotated box!
Continue the process, going from the last found point on the outer circle, and reaching to the next point, and so on, until you have the top square of the new, rotated box.
Pull lines down and make them touch the outer circle on the bottom side. It might be a good idea to use the initial box to figure out the direction of these vertical lines, as they need to go to the same vanishing point.
Close the box on the underside by drawing the ribs between the four points of the box on the outer circle. These lines should also touch the inner circle, and they where they touch, it should be parallel and in a single point.
