An exercise in perspective / Tom Axford 20230528

The photo below is of the North Warehouse at Gloucester Docks, a Grade II listed building. However, I took the photo not so much to show the building, but more as an illustration of some of the quirks of perspective.

Look at the right-hand corner of the building, nearest the camera, with the bell hanging part-way up.

Do the two sides that meet at that corner appear to be at ninety degrees to each other? What affects our judgement of that angle?

I already know the answer to this question, but I am posting this in case others are also interested in the complexities of perspective and may enjoy thinking about this and working it out for themselves.

Alan, I'm not sure what you mean by saying "we CAN see 3D". Our eyes see only 2-D images, although binocular vision does give us a rangefinder for fairly close objects. However, most of the cues we use for depth perception work with monocular vision which is strictly 2-D. So our brains get very good at depth perception in 2-D images because we are doing it all the time as part of ordinary vision. Binocular vision only works for things closer than about 50 metres.

You're right that there is slight barrel distortion present, but it is best seen in the curvature of the vertical line at the right corner of the building. I think the distortion of the bell is mainly wide-angle perspective distortion.

Try enlarging the image I posted to a sufficient size that you can view it from a distance of half the image width (with your eye in line with the centre of the image). At that distance you will see it with the "correct" perspective; i.e. it will appear exactly as if you were looking at the building from the camera position. There will still be a very small amount of barrel distortion from the lens, but all the perspective distortion should disappear.

It really is worth doing this if you can as it is quite instructive and almost magical to see the perspective distortion suddenly disappear and the image look exactly as if you were there looking at the scene from the camera position. You do have to get your eye into exactly the right position for it to work.

In this shot we have 3 point perspective with 3 vanishing points. Thus the bell support does not appear at 90° as we would expect. With two point perspective the effect would be much the same.

As we get closer to a building the walls at 90° appear to close at the top. It is not too marked here.

There is a clue in the orientation of the cobblestones at lower right. The orientation tells me not to expect to see the "missing" 90-degree wall from that camera location.

Yes, the camera is exactly in line with the face of that wall. Indeed, if you look carefully, you can see the window sills which protrude an inch or two out from the wall, but you cannot see the wall itself.

But does the corner appear to you to be 90 degrees?

To me it looks more like 60 degrees. So there is definitely an optical illusion there.
Of course there are buildings with corners less than 90 degrees, such as the Flatiron in New York. Or this one in Brighton.

It looks much like that to me.

It's possible to work out the angle geometrically. Consider the triangle ABC, where A is the camera position, B is the far corner of the building and C is the near corner of the building. We want to know the angle BCA. The angle of the corner is then 180 degrees minus angle BCA. In reality, angle BCA is 90 degrees, but what does it appear to be when we look at the image?

If you measure the height of the building in the image, it is about 4 times higher at the near corner than at the far corner. So the brain interprets that as meaning that the far corner appears to be approximately 4 times as far away as the near corner. The angle BAC can be measured at the eye of the viewer and will depend on the size of the image and the viewing distance.

Then we have a triangle ABC in which side AB is 4 units, side AC is 1 unit and angle BAC is known but depends on the viewing distance. It is then possible to work out the angle BCA. I'll leave it to someone else to do it (my geometry is a bit rusty). However, it should be obvious that the angle BCA will increase as the angle BAC decreases. Remember, the angle BCA is the apparent angle, not the real angle.

If you view the image with the correct perspective, angle BAC is around 75 degrees, and angle BCA is 90 degrees.