Proof of geometric perspective?

Question

So I'm struggling to draw squares in perspective. I came across this technique where the method used is by dividing in half the angle made by the vertices of the lowest horizontal line parallel to horizon and the lines connecting it to the vanishing point, you can draw the diagonals that would determine the upper side of the resultant square. I can kinda see it intuitively, since the halves made by a the diagonals of a square are congruent, so I'm assuming the degree to which they are foreshortened would be the same for both halves as well? Can this be proved, and if it can, can it be used for other vanishing points?

Link to technique: http://pekkanen.brinkster.net/circle/index.htm

Please make your question self-contained by including relevant context directly in your post. One shouldn’t have to chase links to understand what it is you’re asking in the first place. Moreover, this site is meant to be a Q&A archive. External links can and do go stale, so a future reader might not even be able to do that much. — amd, Jul 23 '19 at 05:26

amd · Accepted Answer · 2019-07-24T06:59:56.317

0

While it is true that the images of a square’s diagonals intersect at the image of its center, it’s not generally true that the images of those diagonals are the angle bisectors of the images of the square’s sides, even when a pair of sides parallels the horizon. If you have a square that’s positioned symmetrically within the scene as in the linked writeup, as it moves further from the viewpoint, its image becomes more and more foreshortened, with the angle between the horizontal sides of the square and its diagonals tending to zero.

edited Jul 24 '19 at 06:59

answered Jul 23 '19 at 05:29

amd

53,693

But wouldn't the degree of foreshortening affect both angles beside the diagonals equally, since they're both 45°? – lorenzo Jul 24 '19 at 18:54
@lorenzo The resulting trapezoid has bilateral symmetry, but it doesn’t follow that the diagonals are angle bisectors. When the square is far away, its image is very thin: the sides are still at a 45-degree angle, but the diagonals are almost parallel to top and bottom. – amd Jul 24 '19 at 19:48

score 0 · Answer 2 · answered Feb 08 '23 at 07:34

There are many accurate methods to draw a square or any other objects even houses in perspective whether it's one point-, two point- or three point perspective. Here's a method that doesn't care about vanishing points but rather uses Cartesian coordinate system. And here's another one using the same technique but with one little change, that's the mirror location as vertical. Cheers

Proof of geometric perspective?

2 Answers2