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Linear Perspective

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The Mathematics of Linear Perspective

The idea behind linear perspective is simple: First, we assume that we are observing a physical scene (a basket of fruit, a sunset in over the ocean...) with one eye only. Now put up an imaginary vertical screen (called the picture plane) between the eye and the scene. For each object in the physical scene (an apple, a flower, a grain of sand...), there is a collection of light rays which are reflected off the object and then enter our eye. (That is how we normally see.) Suppose that on the way to our eyeball, each of these light rays makes a mark (with the same color as the light) on the picture plane. The resulting colored picture plane has the property that: wherever we place it, and as long as we stand at the same distance from it as before, the picture plane presents to our eye an image which is indistinguishable from the original physical scene (our basket of fruits, the sunset). It apears as if the space we saw now lies inside the flat picture plane.

Now, let's flip the idea around. If we can figure out the laws which govern the placement of the marks by the light rays reflecting off the physical scene, then we could start with a blank canvas, place marks on it according to the same laws, and produce an illusion of a three-dimensional physical scene contained within the canvas (whether or not the scene thus illustrated actually exists in the real world).

What are these laws? We name a few here. In the following:

Laws of Linear Perspective

1. With one type of exceptions (Exercise, which one?), straight physical lines project to straight pictorial lines.
2. Transversal lines project to the picture plane with no distortion, e.g. a horizontal (resp. vertical) transversal line projects to a horizontal (resp. vertical) pictorial line.

In particular, parallel transversal lines project to parallel pictorial lines.

3. Every non-transversal line projects to a pictorial line which terminates at a vanishing point on the picture plane. Moreover:
  • Any set of parallel non-transversals share the same vanishing point.
  • The vanishing points of horizonal non-transversals lie on the horizon of the picture plane. This is the horizontal line on the picture plane which is level with the viewer's eye.
  • The vanishing point every orthogonal line lies at the center of the horizon.

Of course, in practice, no one paints on an infinitely large canvas, so the "infinite" picture plane is necessarily cropped, and consequently vanishing points may lie outside of the painting.

It is remarkable that with only these simple rules, and a little bit of ingenuity in their application, some great visual results could be obtained. For example:

A tiled floor viewed in perspective:
The Virgin of chancellor Rolin (c. 1435)
Jan van Eyck

A series of columns receding into space:
Annunciation (1437-1446)
Fra Angelico

View of Molo (c. 1730s)

Human figures situated in space:

Delivery of the Keys (1481 - 1482)
Pietro Perugino

And foreshortened human figures:

The Lamentation over the Dead Christ (c. 1480)
Andrea Mantegna

Besides employing the geometric rules outlined above to depict objects viewed in perspective, there are various mechanical or semi-mechanical means to produce the same effect. In the following illustration by Albrecht Dürer, the "imaginary screen" we used in devising linear perspective is realized physically as a gridded net (grid #1) placed between a draftsman and the physical scene (in this case a reclining nude). Before the draftsman lies a drawing surface with a similar grid system (grid #2) overlaid upon it. Viewing the physical scene with one eye placed at a fixed point, various points of the scene lies directly behind points on grid #1. If, say, the tip of the nose of the woman lies behind the grid point on grid #1 which is two from the top and one from the left, then the draftsman would draw the tip of the nose on the drawing surface at the corresponding point on grid #2. Repeating this process for each point in the physical scene, a perspectival drawing of the scene is obtained.

Draftsman's Net (1525)
Albrecht Dürer

In fact, having a screen between the viewer and the physical scene which records all the instances of light rays reflecting off the scene is quite similar to the mechanism of the photographic camera, whose development traces back to a device called the camera obscura, which means "dark room" in Latin (and is indeed the etymological origin of the modern word "camera"). It works as follows: A pinhole is made on one side of a sealed box or room, and a screen is set up in the room facing the pinhole. Light rays which enter from outside the room then cast an inverted image of the physical scene outside. A draftsman could then trace the outlines of the outside scene directly onto the screen.

Drawings by Canaletto obtained with a Camera obscura
(Gallerie dell'Accademia, Venice, Italy)

Aside from the inversion (and various degrees of distortion depending on implementation, such as the use of a lens), the projected image produced by a camera obscura largely follows the same geometric laws of projection which underlie linear perspective. Consequently, it can be difficult to deduce whether a painter has employed linear perspective, a camera obscura, or some other optical means (a grid like the one illustrated by Dürer, a camera lucida, etc.). The following painting by Vermeer is the subject of the motion picture Tim's Vermeer, which documents one man's quest to determine if Vermeer had employed optical tools to produce the painting.

The Music Lesson (1662–1665)
Johannes Vermeer
One must be aware though, that any such analysis is invariably complicated by the fact that a painter is a human agent free to deviate from the rules of linear perspective or any image projected by a camera, and to combine or improvise whatever skills and techniques (both intellectual and instrumental) at their disposal, with the singular goal of fulfilling the needs of their artwork (or perhaps, in a less than ideal world, the needs of their patrons).

The Column Problem

So far, in our discussion of linear perspective, the basic setup was that the perspectival image is the intersection of the picture plane and the cone of light rays reflecting off the physical scene. As such, the perspectival image is an absolutely faithful imitation of the physical scene. Namely, an appropriately situated viewer of the picture sees an image that is optically identical with what they would see had they been physically in front of the actual scene at a correspondingly prescribed distance. We now examine what might happen when the viewer's eye is not fixed in space, in the context of what's commonly known as "The Column Problem."

Suppose we wish to depict a row of evenly spaced Greek columns parallel to the picture plane (i.e. front facing). Two rules of linear perspective relevant to this construction are as follows:

1. Physical ellipses poject to pictorial ellipses.
2. Linear perspectival projection preserves tangency. In other words, if a physical line is tangent to a physical object, then their pictorial projections are also tangent.
Due to these two rules, the bases of columns project to ellipses which becomes more and more elongated and slanted towards the left and right ends of the picture.

Perspective, Hans Vredeman de Vries

Moreover, the projected images of the columns would appear wider and wider, and spaced closer and closer, towards either end of the picture plane.

All of that is perfectly fine with regard to the rules of linear perspective, and would in fact create the correct optical illusion, provided that the spectator views the painting from a uniquely prescribed height and distance with one immobile eye.

In reality, that is almost never the case, and to a casual viewer strolling by, the columns would just appear distorted. This phenomenon has been a subject of serious consideration since the Renaissance, by such eminent figures as Piero della Francesca and Leonardo da Vinci.

Leonardo studied this problem of distortion to the roving viewer and concluded that it cannot be avoided, unless the picture plane is situated "at least 20 times as far off as the greatest width or height of the objects represented." In otherwords, if you want to incorporate into your painting a row of columns, and you have not either the means or intention of fixing your viewer's eye at one point in space, and you want to achieve a convincing visual illusion, then you should consider placing any front-facing row of columns far into the background.

In practice, painters who utilize linear perspective sometimes "break the rules" when it comes to circles and columns. Dora Norton opined in Freehand Perspective and Sketching that:

... cylindrical objects, however placed, should be drawn as if for those objects alone... But this does not apply to the straight-line portions of the picture..., nor to the placing of the cylindrical parts, nor to their height. These must be determined in the ordinary way [(i.e. using standard rules of linear perpsective)].

Hence, in a painting which otherwise adheres to the rule of linear perspective with absolute precision, the base of a cylindrical column off to the side may be depicted as a level ellipse, instead of a slanted one.

The Chess Game (1883)
Charles Bargue

Curvilinear Perspective

The basic optical mechanism of the human eye is not unlike that of a camera. The chief difference is that an inverted image of the physical scene is now projected onto the spherical (or more accurately quasi-spherical) surface of the retina. This projection is a type of curvilinear perspective. In the eyeball situation, the projected retinal image is subject to barrel distortion, where every physical line projects to a line which terminates at a vanishing point (whereas in the case of linear perspective, a transversal line has no vanishing point.) Barrel distortion is one type of curvilinear perspective, the other type being pin-cushion distortion.

In more concrete terms, barrel distortion is demonstrated in Leonardo da Vinci's observation that a long rectangular shape facing the viewer appears to diminish in height towards either end of the field of vision. Likewise, it is barrel distortion at work when a tall building appears narrower and narrower (however slightly) towards the top of our field of vision.

Hence, our visual experience of the world really obeys the laws of curvilinear perpsective, even though we may not think that is the case, since the deviation from linear perspective is often slight, and only clearly noticeable in our peripheral vision.


  1. ^ Leonardo da Vinci, A Treatise on Painting. Transl. John Francis Rigaud. [html]

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