Absolute sizes of a two-dimensional square in perspective

Given is the following square in perspective view:

1. The square has an edge length of 500 feet.
2. The square is rotated by 45 degree.
3. The viewer is 1600 feet away from the square's center.

Needed are formulas for the following sizes in a two-dimensional view (trapezium):

• width w
• height h1
• height h2

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You still need to specify some things. Where is the $1600$ foot distance measured to? The near edge, the center, the far edge? You also have nothing to set the vertical scale: think of looking at it through a zoom lens. Assuming the $1600$ feet is to the near edge, the angle subtended by the near edge is $2 \arctan \frac{250}{1600}= 17.76^\circ$. The far edge is then $1600+250\sqrt{2}=1953.55$ feet away and the angle is then $14.59^\circ$. The horizontal angle subtended can be found from the law of cosines: $500^2=1600^2+1953.55^2-2*1600*1953.55 \cos \theta$, giving $11.46^\circ$

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Wow, thanks for your answer! You're right, I forgot to mention what distance it should be. Actually it is the distance to the square's center. Think of it like a rotated road sign ;) –  Elias Dec 22 '11 at 17:07
Can you update what I gave for that case? The near edge will then be $1600-250/\sqrt{2}=1422.23$ feet away. –  Ross Millikan Dec 22 '11 at 17:27
Yeah, I think that should be possible. But there is still one question I have. Where in these formulas does the rotation angle of 45° comes into play? –  Elias Dec 22 '11 at 17:42
That is where the $\sqrt{2}$ comes from. $\tan45^\circ=\sqrt{2}$, so the 500 foot width means the near edge is $500/\sqrt{2}$ feet nearer us than the rear edge. –  Ross Millikan Dec 22 '11 at 18:00
I just fiddled through your responses and ran into some problems. I think I didn't make myself clear enough what I meant with saying "absolute length". Lets say you make a photo of the square and load it on your pc, now I need the lengths as in the second graphic as if the square would be a two-dimensional trapezium. Do you know what I mean? –  Elias Dec 22 '11 at 22:10
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