# Rotating a square

I want to rotate a square around it's vertical center line (red).

Looking at it from above (top right circle on image) I can tell that the object should, after the rotation, appear shorter length-wise by d1=d2 on each half.

After arbitrarily deciding on the position of the new vanishing point I drew the new, rotated square.

However, in the past I learned that the center point/line of any rectangle can be found using the crossing of it's diagonals. I drew those in (cross1) for a quick check and found that the apparent center of the rotated object completely deviated from the center line I intended to rotate it around.

Have I made a mistake in my construction, or is it not true that the diagonals mark the center of any rectangle? Or possibly both?

If relevant: The length of the original square is 2units, the desired degree of rotation is 30°, thus each half of the new, rotated object should be 1*cos(30°)= ~0.86 long horizontally, relatively speaking.

## 2 Answers

Your construction, as already told, is lousy. A systematic way is to draw sight lines from the observer to the corners of the objects, It must be separately horizontally and vertically.

To compare the looks of the original, straight on the face square and the rotated square let's construct the seen shapes. The observer is the little circle, sight lines are magenta, the image is constructed onto the same plane as the original square sits on.

The orange patterns are the seen squares.

When the rotated square has been mapped properly, the diagonals cross at the midpoint - no more contradiction.

Note: This method is not practical. It starts from the geometrical places of the objects, not from an artistic visio. For artists there is the perspective grid which quides to a consistent drawing.

The crossing of the diagonals and the vanishing ponit of parallel lines are mathematically proven aids to speed up consistent constructions, but they must be used properly. The vanishing point, the placements of the ojects, the place of the observer and the mapping plane cannot all be selected independently.

Your own mapped rotated square is right for the red square in the following image:

The rotation angle possibly can be 30 degrees, but from where it's seen? And is it really possible - I cannot say without proper calculations.

• There's too much artistic vision in my surroundings and too little facts, so at the very least this is very practical to me. Much appreciated. But if I understand your construction correctly, the object is built around the points that are crossings of the purple sight lines and the eye level (or what to call it). In that case, should the side of the rotated object that is further away from the observer not be more to the left where the sight line crosses?
– user55591
Commented Sep 24, 2017 at 5:54
• @vruvre Thanks, you had spotted an error. The diagonals still seemed to fit due the thick lines.. It's fixed now, The black solid line is the non-rotated square. I arbitarily took the same plane to be my mapping plane (=the plane whose crossing points with the sight lines form the image). The green lines only carry the horizontal and vertical coordinates of the image points.
– user82991
Commented Sep 24, 2017 at 7:54
• Alright, then it seems I've understood you correctly. I accepted your answer as it was helpful in realizing where and how my approach was not sufficient.
– user55591
Commented Sep 24, 2017 at 23:42

d1 != d2 in perspective, the perspective view is not linear. The diagonals mark the center of your rectangle, its just that your other assumptions are totaly wrong. Thus all your calculations are invalid in a perspective view.

To see this start by drawing a perspective grid with perspective circle on the floor.