Perspective distortion is especially noticeable using wide-angle lenses near a subject that extends away from the camera. Wide lenses (e.g. 24 or 18 mm focal length for a 35 mm camera) magnify apparent distance, causing objects close to the camera to appear considerably larger than objects farther away, and causing parallel lines to appear to converge.

In comparison, longer lenses (85 mm and greater for a 35 mm camera) appear to shorten distance between objects, and compress the perspective.

For this reason, lens choice in film and photography can influence perception of a scene. The general assumption that "undoctored" photos can not distort a scene is thus incorrect. This is particularly noticeable in en face portraits. Taken with wide-angle lenses, they generally give an unpleasant impression; making the nose appear far too large, and distorting the facial expression. For good results, portraits should generally be taken using a moderate telephoto lens. Again, for a 35 mm camera, a lens of focal length 85 or 105 mm is considered to be a portrait objective.

Below, a series of four photos shows an object framed as nearly the same as possible while taken with four different lenses. As a result of the different angle of view of each lens, the photographer moved closer to the object with each photo. Note that the angle of view changes significantly (compare the background in each photo), and the distance between objects appears greater with each succeeding image. In the fourth lower right image, taken with the widest lens, the building behind the object appears much further away than in reality.

The process described above is similar to the in-camera special effect known as the Hitchcock zoom, where the zoom lens zooms out at the same time as the camera moves towards the subject in such a way as to keep the subject the same size in the frame while the background "changes size" relative to the subject. The Hitchcock zoom is an unsettling visual illusion that appears to undermine normal visual perception by visibly stretching distance. It was first used in Alfred Hitchcock's film Vertigo (when looking down a stairway) and has since been used in Jaws (when the main character sees the shark) and Goodfellas (in the cafe towards the end, where Henry Hill gets the unsavory assignment), Lord of the Rings 1 (where Frodo first senses the ringwraiths and yells "get off the road!") and many other films.

   

   

Photos taken using a 35 mm camera with a 100 mm, a 70 mm, a 50 mm, and a 28 mm lens.

Table of contents

1 Why perspective distortion occurs 1.1 Distortion 1.2 Conclusion

Why perspective distortion occurs

Artificial perspective projection is the name given by Leonardo da Vinci to what today is called classical perspective projection. natural perspective projection is the name given by Leonardo to the projection that produces the image beheld by the human eye. Both types of projection involve a distortion; parallel lines never intersect in nature, but they always intersect in perspective projections, with the exception being the special situation wherein both the surface of projection is planar and the spatially parallel lines are parallel to the plane of projection.

The difference between the images of the same object produced by artificial perspective projection and by natural perspective projection is called perspective distortion. This article preserves the usage of the Leonardo designation, artificial projection, in place of classical projection so as to acknowledge that Leonardo recognized that classical perspective projection indeed does produce an image different from that beheld by the eye.

The physiological basis of foreshortening went undefined until the year 1000 when the Arabian mathematician and philosopher, Alhazen, in his Perspectiva, first explained that light projects conically into the eye. A method for presenting foreshortened geometry systematically on a plane surface was unknown for another 300 years. The artist Giotto may have been the first to recognize that the image beheld by the eye is distorted---that to the eye, parallel lines appear to intersect (in the manner of receding railroad tracks) whereas in "undistorted" nature, they do not.

Distortion

Figure 1

Figure 1 shows the principle of an artificial perspective projection. The artificial perspective projection image appears upon the projection plane. Imagine a human eye is placed at the station point. This human eye views both the object and the image of the object as if the projection plane did not exist. That is to say, to the eye, from the station point (only) the image of the object is indistinguishable from the object itself.

In the special case---and the only instance---in which an artificial perspective image appears the same to the eye as does a natural perspective image, the eye must view the artificial perspective image from precisely the spatial station point, i.e., from the station point as it is located in actual space.

Figure 2

To illustrate, Figure 2 shows the object in Figure 1 to be pivoted about the imagined eye to a new position, while maintaining its original face toward the eye. The new object image will appear the same to the eye as at the original position. But take note that the artificial projection image of the object in its new position is different from the original artificial projection image. In other words, the two artificial projection images are different because the angles of the respective projectors as they intersect the picture plane are different. The pivoted image thus exhibits an obvious distortion over the unpivoted image. However, both images appear identical to the eye because the eye is located at the station point and the angle of viewing from the station point to the new image cancels out the new angles of the projectors. Or to say it another way, by looking at the distorted image in a distorted (skewed) direction the image appears undistorted to the eye.

It is not immediately apparent on a day-to-day basis that there is distortion in the image beheld by the eye because we experience nothing else visually throughout our lives, but on reflection there is an obvious contradiction that has not been fully explained: Why do straight parallel lines in space appear to intersect twice? The following thought experiment illustrates the contradiction:

Imagine that on an infinite plane there is an infinitely long and infinitely straight railroad, and that you are standing between the parallel rails. As you peer down the track in one direction the rails appear to intersect (on the horizon). As you peer in the opposite direction they again appear to intersect. You look at your feet and the rails are far apart. It logically follows that if they intersect, as it would appear, one or both of the rails are curved. To determine which, you put your eye onto one rail and sight down it. You discover it is straight in the sighted direction. You then sight down the same rail in the opposite direction. You discover it is straight also in the opposite direction. It logically follows, then, that the other rail must be curved. You similarly test the other rail and discover it too is straight. How can this happen?

Figure 3

The explanation lies in an extension of Alhazen's discovery: the conical projections of spatially, parallel, straight lines create intersecting straight-line images on surfaces of projection. However, for the spatially parallel lines to appear straight and also to appear to intersect twice, the surface of projection must be spherical and viewed from the center. See Fig. 3. (The brick wall in the background serves as a representative object to be projected.) This would produce a true simulation of that which the eye beholds.

To understand the process, be reminded that:

• perspective projections of straight lines that are in space, through the center of a sphere, projectively create great circles upon the picture sphere, and
• these great circles appear as straight lines when viewed from the center of the sphere.

These are the reasons why the natural perspective projection of the rails of our thought experiment actually produce two intersecting (at opposite poles of the sphere) circular images upon the picture Sphere (retina) and yet appear straight when viewed from the center. For this to be consistent with our normal vision requires that the retina (macula?) be spherical in shape. The image is inverted at the focal point before striking the retina, but which has no bearing on the theory.

Therefore, to produce the image which the retina beholds, using projective geometry techniques, substitute a spherical surface of projection for a plane of projection, i.e., use a "picture sphere" rather than a "picture plane" and place the eye, or "station point," at the center of the picture sphere.

Figure 4

For a step-by-step example, see Figures 4-8.

Figure 4 shows that the perspective projection of the horizontal top line of a wall intersects at the retina (actually, in true life, at the focal point slightly in front of the retina and from where it is inverted). The intersecting lines define a plane that intersects the sphere of projection in a great circle.

Figure 5

Figure 5 shows the projection of the horizontal bottom line of the wall and its great circle of intersection.

Figure 6

Figure 6 shows that the two great circles of intersection intersect. The two points of intersection are two (vanishing) points on the horizon line of the plane of the wall as seen from the center of the picture sphere. The points of intersection are analogous to the two points of intersection as one peers down a railroad track in both directions.

Figure 7

Figure 7 shows a similar determination of the two vanishing points for the spatially vertical lines of the wall. Note that this correct "four point" perspective is analogous to the distorted "three point" perspective of classical perspective projection.

Figure 8

Observe at this stage that a horizon line of the wall is not yet determined. Figure 8 demonstrates that a horizon line is determined by vanishing points only.

Conclusion