Spherical Perspective - a brief tutorial
I'm here to undermine your world view. We always assume that
what we are taught about perspective 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 left and right they get really funky!
Have you 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. We say it looks like a
fish eye lens.
We were fish once.
Lets find out why this happens. Take the set up below. 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 points. This isn't
looking much like a railway track! Let's see what's really going on, we'll do
a whole set of tall narrow drawings and stitch them together.
This gives us the result below.
f we were a chicken - or even a fish- this is actually how we
would 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 draw a scene. Indeed your eyes
only do hardcore looking with a small patch of our retina called the fovea.
The brain then stitches all the bits together rather like our panorama. On top
of all this we turn our heads. In real life we can soak up a huge panorama of
visual information. Our problem as draughts people is to get some of this down
on a piece of flat paper or a screen.
So let's find out what's really going on. We'll 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.
Well this is a surprise. 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 it's 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! Let's try it with our polka dotted floor.
This is lots better all our major and minor axis are horizontal
and vertical. It also has the weird property of zooming in if you stretch it
horizontally. Try it double the width leaving the height the same... spooky.
Now don't relax... we are not out of the jungle just yet. Ever looked up at
a sky scraper? Well yes, it does that going to a point thing like our railway
tracks. Ever looked down a well? That circle of sky reflected in the water is
distinctly smaller than the top. Yep here's the bad news: it happens up and
down as well as side to side. The even worse news is that it won't go onto a
sheet of flat paper at all! But this problem has already been solved. How do
you get the world's land masses that have a distinct air of roundness into an
atlas that has more than a little flatness about it? Mercator solved this and
guess what? You can still use it to navigate. The only problems occur at the
pole which get sort of stretched out and it makes Canada look even bigger than
it really is, which seems unnecessary. If we include vertical perspective as
well it looks like this. Actually Mercator keeps all his verticals parallel
but I want vertical perspective as well, this causes four star shaped areas
that are holes in our universe!
I know this looks a mess of lines but give it time. This pattern
will repeat up and down and side to side as many time as you wish.
Here's a drawing done on the grid to show how it works.
Amazingly enough this the only only spherical perspective grid there is, every viewpoint can be found on here somewhere. As we near the horizon all the curves become straighter and straighter, traditional perspective hangs out here. Up at the top right down at the bottom... this is eyes on the top of the head territory tread with care! You don't have to use the grid with vertical perspective here's another grid with all the verticals left alone this would be cylindrical perspective with four vanishing points rather than the six you have in spherical perspective.
You can also play games with this set up here Escher has turned a cylindrical grid through 90 degrees.
Here's an example of the grid used in anger!
Link opens in separate window.
If you make the window smaller than the image and scroll side
to side and up and down it's like looking around.
Uses for this kind of projection are in animation and film backgrounds that
the camera has to pan across. I find it comes in handy for all sorts of architectural
and set concept drawings, because if any creature has a wider field of view
than a chicken in must be an art director!!