This is the first instalment of a few posts on perspective and how we can use it in making images. This post is quite advanced but I do intend to do a “rule of thumb” perspective guide for those not of a mathematical bent. For now we are in the world of curved perspective which can be scary but explains a little about why even when we follow the construction rules of perspective things can look “wrong”. This is especially true when we use what in photographic terms is called a “wide angle”.
.
So here we go… We might assume from what we are taught about perspective that this is the way we actually see. But it’s not. In the outside world there are straight lines, so we put them that way into our pictures. We have developed complicated schemes of geometrical rules to guide us. We take photos with cameras that have lenses that carefully distort the world to make it fit with the expectation that straight line should be straight. But visually they are not. Have you ever tried to draw that really large checker board floor? Somehow at the far right and left it goes all stretched. Do the same thing with circles on the floor and it gets really wild. Just look at those ellipses on the far left they get really funky! Have you ever tried to stitch together that big panorama? They never quite fit do they. But if you take lots of pictures say every 5 degrees and just use the middle strip of each, it’s sort of easier. And when they’re all stuck together, well… those straight lines look distinctly curved. In camera terms we call this a Fish Eye lens effect.
.
Let’s find out why this happens. Take the set up above. Simple enough, a railway track, a station and you. Now perspective and our eyes tell us that things get smaller as they move further away. I’ve no problem with that. So here goes. If we look straight ahead Say “B” then the track is quite close. If you drew it it would go straight across the page left to right. If we look to our right say “C” then the track in the center of our vision is a lot further away. On the right the track vanishes at a point on the horizon. These pictures are both fine but you wouldn’t want to try and join them together! But wait a second, the track really is joined together. And we haven’t moved. We just looked to our right…If we looked to out left then we’d see the track go to a point at the horizon again. So what have we got? one set of parallel lines and two that meet at points. This isn’t looking much like a railway track!
.
Let’s see what’s really happening. Imagine if we do a whole set of tall narrow drawings turning our heads a little for each one and then stitch them together. This gives us the result above. If we were a chicken – or even a fish- this is actually how we might see it. We are not quite so different from them as you might think. You don’t, after all, keep your eyes still when you look at a scene. Indeed your eyes only do detailed looking with a small patch of our retina called the fovea. The brain then stitches all the bits together rather like you do in a photographic panorama. On top of all this we turn our heads. In real life we can soak up a huge vista of visual information and glue it all together seamlessly. Our problem as artists is to get some of this down on a piece of flat paper. So let’s try find out what’s really going on and how we might use it.
.
So if we take our camera and take tall thin pictures of an endless gridded floor and see what we get. I’m using a virtual one as endless gridded floors are a bit thin on the ground around here. The image above is the result in which everything joins up neatly. It looks pretty fisheye when it gets very close to us, but every paving stone joins every other where it should and is the right size for its distance from us. It’s also plain that It will repeat all the way round 360 degrees. That’s good too because it’s a well known fact that endless gridded floors do just that!
.
Here is the same method applied to our polkadot floor which looked so weird in the first example. This is a 180 degree view so very wide, but none of the ellipses are tilted and everything joins up in a logical manner. The problem of ellipses in perspective is a very old one which renaissance artists puzzled over a fair bit. This was due to their often needing to draw long rows of receding cylindrical columns which using linear perspective would look distinctly wonky on the far left and right. They devised a simple cheat which modern artists seem to have forgotten, but I will deal with that in the next instalment.
.
Here is our grid joined up so you can see how verticals work, you could easily imagine laying out a cityscape on this grid. You can repeat it endlessly if you mirror it left or right. This is the most useful curved perspective and is called Cylindrical perspective it is the equivalent of two point perspective, I will deal with the horrors of full Spherical perspective next! But first below an example of Cylindrical perspective by the wonderful M C Esher.
.
This is called House of Stairs and is made using the same grid I drew above, but swivelled through 90 degrees.
.
Now we are entering the strange world of Spherical perspective. The above grid can as before be duplicated endlessly, I know it looks like an impossible spiders web but it is the same as the cylindrical one except the verticals curve too. This is the equivalent of 3 point perspective as when you look up at a tall building and the top diminishes. The image above can be clicked for a larger version.
.
Here is the same grid used to place a few very simple forms. I takes a little while to get your head around it but if you print out the grid and scribble on top of it you will soon get the idea. For the mathematically minded the geometry of the Spherical grid is Hyperbolic whereas the traditional straight line perspective is termed Euclidian.
.
Here it is used in anger. You can click for a larger view. As with all spherical perspective the nearer you get to looking down at your toes the odder it looks!
.
Here it is finished. You can click for a larger view. The examples I have given are extreme ones in order to show the principles involved. We don’t often draw or paint 180 degree panoramas. But the same principles can be applied to good effect in more ordinary views where to the casual viewer the underlying curves would be to subtle to notice but they will unconsciously find the painting just that little bit more believable especially in townscapes and other subjects with a lot of man made rectilinear content.
