Putting Perspective into Perspective

by Jan Troost

Perspective is a simple concept. It is unfortunate that this is not clear from most literature readily available on the subject. In order to partly remedy the situation, we try to add weight to the cause of those demistifying the concept. (We include these remarks on this site since the art literature in particular may lead to unnecessary confusion.)

For practical purposes, it is fair to say that a perspective is a way to produce a two-dimensional picture from a three-dimensional object. In other words, we want for instance a drawing from an object in the real world.

It is not difficult to list the ingredients that we need to produce such a two-dimensional picture. We need an object, a plane in which we project (and in which we typically choose a rectangle that we can identify with our drawing, our painting, or our negative) and a point in space that will be the viewpoint.

It should be stressed that that's it. No more ingredients are needed. All other technical talk, mumbo jumbo, fancy phrasing and what not just refer to various aspects of the above procedure, or subcases thereof.

Actually, we have not given the procedure to project yet. It is fairly obvious. Draw a straight line from the viewpoint to a point on the rectangle on the projection screen, and prolong it until it hits the object. (If it does not, then we can ignore that line, and draw another one.) The point on the projection screen should be coloured in the colour of the point where the line hits the object. And so on, for every point in the rectangle on the projection screen.

That's it. No more.

Now, we wish to recall how various concepts that are usually associated to perspective relate to the above simple procedure, and how various mild modifications of the above procedure may be of interest.

Remarks:

* It is often the case that we want to draw a perspective picture of an object that includes two parallel lines. The image of two parallel lines on the projection plane will (apart from some very special cases) generically be two lines which are not parallel. These two lines will meet in a point in the projection plane. That point is called a vanishing point.

* It should be noted that if we have an object with three parallel lines, then they will have one vanishing point in the projection plane.

* It should be noted that if we have two more parallel lines in the object (not parallel to the original ones), then they will generically have a new vanishing point.

* So, should we have an object with twice two parallel lines (like a square), then generically, namely for a generic choice of projection plane and viewpoint, the resulting picture will have two vanishing points. This fact is misleadingly referred to as a two-point perspective. (Note that there is only one viewpoint on the one hand, and, on the other hand, that our definition of perspective did not change. Thus, the name is misleading.)

* Similarly, a generic cube in perspective gives rise to a three-point perspective.

* Exercise: think of objects that give rise to a four-point perspective.

* The above definition is precisely the way in which standard photography works.

* We can modify our definition to include the effects of limited resolution of images, for instance by averaging over small pixels on the projection screen, and prolonged pixels on the object. This is a technicality from the viewpoint of understanding the concept of perspective, and it should rightfully be ignored when thinking about the concept. Of course, one can debate the averaging procedure, the size of the pixels, the resolution, .., but this has little to do with the subject of perspective.

* One can distort the perspective in various ways. By not drawing straight lines to the object but curved ones, for instance, or by deforming the projection plane itself, or by choosing to project in a more general frame than a rectangle, or .. The possibilities are endless. Should we wish to do so, we could thus invent a huge list of perspectives, and we would be able to give a precise definition. (As an example: project onto a finite sized, connected subset of a conic section, and call this the "perspective on perspective perspective". Exercise: invent your own perspective, and give it our favorite name.)

* Our vision is more complicated. Basically, our brain processes the information of two viewpoints, which are combined to form an image about which we have more information than the sole image. (We have for instance information on the depth of an object, due to the differing viewpoints. We do use this information continuously in our lives.) Moreover, the viewpoints and our focus is rapidly changing. Much more can be said about vision science, but let us just remark here that the above explanation of perspective is a good first step towards understanding the information that our visual cortex receives.

* Objections like : but what about the background, like the sky in my photograph: that's not part of your procedure ?, are easily dealt with on second thought. (The sky is part of the object. The air molecules are. They are blue.)

* When the viewpoint is very very far ("at infinity") from the projection plane, then the projection lines are perpendicular and parallel to the projection plane. (This favorite subcase has acquired many names.)

* If need be, we can produce some formulas for the mathematically hungry that make the above well-defined procedure even more concrete. Put a viewpoint at Cartesian coordinates (x,y,z)=(0,0,0). Put the projection plane at x=1. Take a point of the object, of coordinates (a,b,c). Clearly, its projection on the screen will lie at (1,b/a,c/a), for non-zero values of 'a'. (If 'a' is zero, the point of the object does not appear in the projected image.) Exercises: put the screen at arbitrary distance from the viewpoint and give the appropriate formulas. Also, give the correct formulas for a viewpoint at infinity.

(to be continued)

A reasonable link explaining roughly the same thing differently, and with more images and history is by Andrejs Treiberg.

Some useful information is at Wikipedia on graphical perspective.

"Putting perspective into perspective", Jan Troost, 22-06-2006.

Copyright 2006, by Jan Troost