Basic Perspective 

You can draw very accurately by eyeballing and measuring angles and proportions.  However, no basic drawing course would be complete without some coverage of perspective and prespective grids, but we'll keep it brief  because it really isn't that practical in terms of everyday drawing.  It is useful, however, to comprehend some of the basic concepts behind perspective.

But please don't feel obliged to completely and meticulously understand all that I've written here!  Just having a feel for the gist of it will be sufficient for your purposes.  If you want to come back and read this again later, you can do so, or, if your interest in perspective persists, you can go further afield online or in books published specifically dealing with the subject. 

Perspective is apparently created geometrically as a form of 'projection'.  Projection is a very accurate description of what is happening because it mimics geometrically  what is happening as I use the 'projector' in the classroom.  There are various types of perspective 'projections' mentioned below, from one point, two point, three point, and parallel projections.  There are many more used in fields such as architecture and 3d computer modelling, and there are 'projections' that are used in map making that help unfold the topography of the earth's sphere onto a flat surface with varying degrees of usefullness.  There is an underlying mathematics to them all to which I am, alas, oblivious, and fortunately for me, and for you, in this course I only need to present perspective informally and very basically.

Below is a diagram from Wikipedia that beautifully illustrates a 'projection' of two point perspective of a 3d cube onto a 2d surface or plane.  As well as representing a 3d object on a 2d surface, 'projecting' also has the useful quality of scaling down a large object to a managable size.  Hence we can percieve something enormous like a mountain through a camera, on our retina, or in a painting.  Projections can project a small image to make it large, as with my classroom projector, or work like looking down the wrong end of a telescope to make a large object small.

The iris in our eye works similarly to a pin hole camera that 'projects' a scaled down and  upside down version of the world on our retinas. The projection is thrown upside down on the back of our retinas because if you extend the projection lines (in the diagram below) leftward from the object, through the 2D plane and converging at the circle representing our eye, you'll notice that by extending those projection lines beyond the convergion point (the iris) they flip around.  Projection lines that previously were on top are now below the projection lines that were below and are now on top...

Wikipedia image.

diagram illustrating a projection of perspective by Casagne

One way artists, over the centuries have often assisted themselves to create accurate drawing and painting is to create a grid with which to look through and see the world.  It's not quite the same thing as projecting perspective, but it's similar.  If you imagine a grid on the 2D surfaces in the two diagrams above, you can see how an artist could quite accurately trace or assist his or her observation of a view or object to more accurately draw it.  Many plein air artists make little frames with grids of wire or fishing line criss crossing throught their centre, with which they can view a subject and, sketching the grid on their paper, more accurately draw a subject.

A way of creating a large drawing from a small drawing, of 'scaling it up', is to draw a grid through through the small drawing, redrawing the grid in the same ratios on a larger piece of paper or canvas, and then using the grid to assist with accurately enlarging the drawing. For example, if your grid squares are one inch and you scale them up to one foot in your new grid, your new drawing will be...bigger.  I'm reluctant to say it will be 12 times bigger, because my knowledge of math is so dismal.  But it will bigger.  Much bigger.

Accurate perspective in a drawing creates an approximate representation of three dimensions on the flat two dimensional surface of your paper.  The most obvious aspects of perspective are that objects get smaller the further they recede from our eye or from the surface of a two dimensional picture plane, and that individual objects are  foreshortened.  That is, an individual object will become smaller or shrink the further away it is.  The foreshortening of an object is often a big stumbling block when artists try to render an accurate drawing.

Perspective grids are imaginary lines radiating from a vanishing point on the horizon.  These imaginary lines are parallel to each other, but appear to merge at the edge of infinity on the horizon. They are often drawn and carefully constructed in elaborately rendered drawings for realistic painting. 

One point perspective, above.  One point perspective has us imagining only lines radiating from one point on the horizon, and we can only correctly place objects directly in the middle of our line of vision to the vanishing point.  If one of our boxes is below the horizon line we see the top and sides, as in the bottom box in the sketch above.  If the box is on the horizon line all we can see is just one side of the box, or if the box is 'open', as I have it, you can see the receding interior.  One point perspective is limited in it's usefulness because we can't correctly rotate objects and see them an aspect showing two sides.  Nevertheless, taking note of one point perspective provides some understanding how how perspective in general works and can actually be useful drawing in some situations.  For example, from the middle of the inside of a room, an elevator shaft, or in between rails on a railway.

Below is an example of how, once an object like our boxes are off centre and not directly between us and the vanishing point, problems arise with one point perspective.  The centre box works, but we have to end up estimating or guessing at the angle edges that are not on the vanishing point side of the box (see dotted line).

This problem can be solved with two point perspective as in the sketch below.  By creating two vanishing points we can accurately draw both edges of the cube of the box.  Note how in both of these sketches I've put a rough perspective grid to guide my point of view in general. 

We are drawing boxes in our first class, but boxes could well be buildings, and cars and machinary can be imagined as made up of cube components or modules, which help us draw perspective correctly.  In fact, as we will see later, even the body can be envisioned to be within cube modules to help us draw it correctly.  Below you can see the 'box' shape of a building.  I'm 'eyeballing' the perspective of the edge of one side of the cube rather than doing a perspective grid.  

If I wanted to create an actual two point perspective grid to make an accurate drawing I would need an extremely wide sheet of paper so that I could place the two vanishing points far enough apart to make the angle of the box/building edges appropriate to my point of view in the photo above.  This is the problem with two point perspective.  The only times I've actually ever bothered with perspective grids and lines from vanishing points was when occasionally creating more complex animation background layout drawings to match storyboard panels.  Animation layout artists would have to get a great big ruler and a big sheet of paper, or tape smaller peices together to create the grid out beyond the object of interest.  Or rough the perspective linework on the drawing desk itself, beyond the sheet of paper you were working on.  Below you can see just how much extra paper is needed beyond the view in the photo to create an accurate two point perspective drawing of the view above. 

If using a formal perspective grid and vanishing points is unwieldy for your general drawing practice, it's nevertheless useful to understand how various perspective templates work; for example sometime you'll find yourself using one point perspective just to check the edges of one side of several buildings by imagining or sometimes even drawing in those lines that head back to the vanishing point.

Below is an example of three point perspective.  This is useful when you are significantly above or below your box.  In the other perspective templates above, all vertical axis' are straight up and down.  With three point perspective we create another vanishing point in the sky, below the ground, or off to the left or right depending on how we want to view our box/cube/object. The closer we move the vanish points together the closer we get to our object, and the further away from the centre of the picture we move them the further away our point of view is.  In the sketch below we are looking up at our box/building at a pretty acute angle because the vanish points are close to the object.  The perspective is 'increased', 'sharpened', made more dramatic as a result.

Below are three drawings of a cube that is 'transparent', which we can see through to the other side, allowing us to see examples of the three perspective templates applied.

Below are a couple of representations of a cube pulled from Google SketchUp which show two point perspective on the left, and a parallel perspective, on the right.  In 3d reality, the two lines that define opposite edges of the cube are parallel, but in the two point perspective they merge together at their respective vanishing points to create the illusion of depth and foreshortening as we actually percieve things.  In a parallel projection of perspective, we actually render the object on a two dimensional surface with the opposite edges still parallel.  As a result the object has an odd appearance, often appearing wider at the back than at the front.  In fact, if we saw all the world like this it would be a world that is even odder than the world we have, one so cluttered with objects there would be no horizon in sight.

As an aside, parallel perspective projections are often used in architectural rendering, in early video gaming objects, and you can also see it used in early attempts at perspective during the mediaval period prior to the Renaissance.

I recall during art history studies reading of how the Byzantine Emporer in Constantinople had a palace built where his throne was at one end of a Basillica-like hall and the entrance was at the other.  The elongated cube that was the hall was actually slightly smaller at the Emporer's end which of course had the effect of making the Emporer appear to his audience as larger than life.  This was  long before platform soles were invented.

Google SketchUp image

It's worth remembering that perspective grids are just one way of simulating three dimensional reality.  Its hard to know exactly how we actually see.  The above perspective templates don't allow for any curving distortion as found in the fish eyed len's of cameras.  Do we actually have some fish eye lensing in our own vision?  I don't know the answer to that.  It's fair to say though, that although our eyes see like a camera, what we visually percieve is altered and stitched together by our brains in ways that don't necessarily have a lot in common with optics.

Often you'll look at a painting, and you'll realize that the drawing underneath is basically copied from a photograph, the painting shows the peculiar qualities of a camera lens perspective, such as fish eye lensing, an effect that increases as you move to the edge of the image.  That is why experienced representational artists using photo reference might completely redraw the image rather than just trace it, so no tell tale distortions associated with a camera lens are carried into the painting.

Understanding basic perspective templates is extremely useful to understanding the drawing process; more useful conceptually than practically, most of the time.  Again, don't feel it is absulutely necessary to understand everything that has been written about perspective.  Heck, I don't.

Here are a few examples of how an understanding of basic perspective can help in your drawing, and therefore painting, of landscapes. In the photo above, by eyeballing, we can accurately estimate the angle of the shore line at various points toward the horizon and vanishing point.  Look at how straight the shoreline is on that prominatory of land and trees!  It is dead straight. You can come to that conclusion by just eyeballing with your brush or pencil, or you can create an elaborate and time consuming perspective grid.

Here's a quick sketch of the subject which has a rough one point perspective grid upon it to help visualize perspective in the beach scene. When drawing the clouds, the few stones and the shore I've tried to keep the imaginary grid in mind. I've tried to image the clouds in space in front of and behind each other. The shoreline becomes close to horizontal by the time it reaches the middle of the picture. In the nearby stones you can see their roundness as emerge from the sand; the further ones have become horizontal at the sandline.

Here's a photo with a very high point of view and a rough sketch with a one point perspective grid on it (above). The original lightly sketched grid corresponding with the surface of the lake which you can see in the foreground is actually almost 1000m below us, a long way away. So the grid on the surface of the land we are standing on would be so huge that we wouldn't actually see it as a grid. Each grid square would be a couple of hundred metres diameter.

Exercise: Drawing Cylinders; Containers and Drinking vessels

Still life is one of the classic art genres, and still life with glasses, cups and containers present the problem of accurately  drawing the foreshortened circle we see on the top of cylinder shaped objects viewed from their sides.  This is called an ellipse by artists. Below is technically what an ellipse is in geometry.  Apparently a plane intersecting a cone is where you encounter an actual ellipse, below left.  In the diagram of an ellipse on the right, the one with all the unintellegible markings on it, you'll notice that the the shape is symmetrical on an axis from 12 o'clock down to 6 o'clock, as well as form 9 to 3.  This ellipse would work on a cylinder drawn in parallel projection, but this won't  work for our purposes because it doesn't show foreshortening.

above images from Wikipedia

 What we're looking for in an 'ellipse' is that foreshortened circle that we see every time we stare across the top of our beer mug or wine glass before we purposefully raise it to our lips and quaff or slurp.  What we're looking for is what we're seeing when I make a cylinder in SketchUp and rotate the point of view from a downshot to a side view, see below. 

Google SketchUp images

Below you can see a series of foreshortenings of a circle as the top falls away from us.  You can see that the ratio of  the distance between the central horizontal axis of the 'cross hairs' and each side changes rapidly as the circle falls away.  The far half of the circle becomes smaller in proportion to the near side.  You will have to mentally estimate or actually measure this proportion to get convincing foreshortened circles in for your ellipses in still life cylinders.

Below you can see how the near half of the foreshortened circle, b, becomes wider than the far half, a.

Below a quick sketch of a perspective grid with some cylinders sketched in at various distances.  In this sketch you can see how cylinders and the foreshortened circles of their 'ellipses' can be built within the box, so to speak.  Building a cylinder within a cube can be helpful in visualizing how the cylinder should look in perspective.  Notice how, as the cylinders move away from the centre, the line between our point of view and the vanishing point in this single point perspective grid, the ellipses are not just stretching front to back, but also stretching from side to side as a result of my attempt to informally create a two point perspective. Also note that as the cylinders move back in the distance toward the horizon the squashing of the cicle or ellipse becomes heightened.

Finally, below, is an arrangement of cylinders of various sizes that I have just drawing and estimated the elipses or foreshortened circles.  This is what you'll end up doing most of the time; estimating ellipses.  At first you'll do a lot of measuring, but eventually you'll get better and better at just winging it.