Perspective projection distortion

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This page is about graphical perspective projection, specifically about distortion

Perspective projection distortion is the mechanism that permits a draftsman or artist to produce linear perspective. It is accomplished by a geometric protocol that exhibits the inevitable distortion of three-dimensional space when “projected," i.e., drawn, on a two-dimensional surface. The words projected/projection here refer to the use of graphics’ lines in the protocol to simulate light ray traces from a Station Point (a supposed observer´s location) to the edges and corners of an object in space, creating thereby or by their extension, an image at the lines’ points of intersection with a Projection Plane.[1]

No type of projection can perfectly map the imagery of three dimensional space onto a projection plane because of the image's (mapped on the retina-sphere) undevelopability.[2] This is a distortion of the drawing in itself called perspective projection distortion, and refers to the difference between the drawing and the way the objects depicted on it would look if it was real, but there is another distortion caused by the difference between the location of the supposed observer situated for the drawing process and the location of the real observer of the drawing. These two distortions exist simultaneously. In the special case — and the only instance — in which perspective imagery appears affected only by the perspective projection distortion, the real observer must view the perspective imagery from precisely the supposed station point of the perspective.

Real human vision and perspective projection should (unless it is otherwise desired) look the same. The difference should be imperceptible. The base to rate the quality of the perspective projection is the real vision and the difference between them is the perspective projection distortion. Normal human vision should not be considered to present any distortion unless a disturbing factor is involved. Distortion in human vision appears when there is a visual problem involved. The use of lenses can also cause, modify or avoid these distortions. In photography a lens may magnify distortion.

Historical development[edit]

Fig. 5A. Jesus Before the Caïf, by Giotto (1305). The ceiling rafters show the Giotto’s introduction of convergent perspective. B. Detailed analysis, however, reveals that the ceiling has an inconsistent vanishing point and that the Caïf’s dais is in parallel perspective, with no vanishing point.

The physiological basis of visual foreshortening was 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 onto 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 apparently distorted (foreshortened): to the eye, parallel lines appear to intersect (like the distant edges of a path or road), whereas in "un−distorted" nature, they do not. One of the first uses of perspective was in Giotto’s Jesus Before the Caïf, more than 100 years before Filippo Brunelleschi’s perspectival demonstrations galvanized the proper widespread use of convergent perspective of the Renaissance.

A people mover at Frankfurt International Airport showing perspective distortion as parallel lines appear to converge.

"Artificial perspective projection" was the name given by Leonardo da Vinci to what today is called "classical perspective projection" and, as noted above, is the result of a geometric protocol. 'Artificial perspective projection' is used here rather than 'classical perspective projection' in order to recognize Leonardo's priority in developing the concept.

"Natural perspective projection" is the name given by Leonardo to the projective., i.e., reflected light, image beheld by the human eye and which is impossible to replicate on a plane surface.

Cause[edit]

Fig.1 Comparison of human sight with perspective projection. Object and image are indistinguishable to the viewer at the Station Point.

Figures 1-2 illustrate the principle of an artificial perspective projection. The artificial perspective projection image appears upon the projection plane (P). A supposed human eye is placed at the station point (S).

In Figure 1 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 the image of the object is almost indistinguishable from the object itself (the only difference is a minimal perspective projection distortion).

Fig.2 Pivoting an object about the Station Point, keeping the same face toward the viewer, should produce an unchanged image to the viewer but produces a distorted image on the plane of projection.

Figure 2 shows the object to be pivoted about the imagined eye to a new position, while maintaining its original face toward the eye. In a natural view, if the pivoting angle is not wide, even if the observer would not turn together with the object, the new object image would appear the same to the eye as it is at the original position because it is projected to the retina, which is concave. Or to say it another way, even if the observer looks at the image in a distorted (skewed) direction the image appears undistorted to the eye.

But 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 intersection angles of the projectors, as they intersect the projection plane, are different. The pivoted image thus exhibits an obvious distortion over the unpivoted image.

In artificial projection, to avoid additional distortion, the observer is supposed to turn along with the object. This means that the line of sight of the observer, that was directed to the center of the object in its first position, now has to be directed to the center of the object in the second position. As the projection plane should always be perpendicular to the line of sight, it should turn along with it, and thus the object would appear exactly as in the first position. Now, even if the object would not keep its face towards the observer but the observer´s line of sight would follow it, the object would appear rotated but still with minimal projection distortion (provided that the projection plane was also rotated).

The difference between the images of the same object produced by "artificial" perspective projection and by "natural" perspective projection is called "perspective distortion." It should be noted that both "artificial" and "natural" projection (the image beheld by the eye) foreshorten real objects and that parallel lines appear to intersect, so that is not the difference between them. The distortion is produced by the difference between the concave retina and the flat projection plane. If an artificial perspective would be projected onto a spherical sector, like in a 360° film projection, distortion would be minimized.

It logically follows that all film photography (now almost in disuse) distorted the image beheld by the eye, among other reasons because the film surface was flat in the manner of the picture plane. Artifactual characteristics of a camera lens may aggravate the distortion. This is demonstrated with a pinhole camera which has no lens but which produces the same distortion as described herein.

The difference of the projection on the two different surfaces consists in that the distortion in the drawing (with respect to what is seen by the human eye) increases as the object(s) of the view are at a larger distance measured across from the observer´s line of sight. In fact, the only point in the perspective with absolutely zero distortion is the point of intersection of the projection plane with the line of sight. And thus, if the view is not symmetrically placed around it, there will be objects staying unnecessarily farer from it and will be more distorted than they could be (like the pivoted object in figure 2).

If the purpose of the perspective is to represent an image as close as possible to the real view, the line of sight should be directed to its center. On a plan drawing, the line of sight should correspond to the bisector of the angle containing the extreme projections of the object(s) to the station point. The area on a plan or on an elevation or section, between the pair of lines that correspond to the extremes of the view and converge at the station point is called the drawing´s visual angle and its breadth depends on the distance from the observer to the object(s). The line of sight should always bisect the observer´s visual angle in order to minimize distortion.

On a longitudinal section or elevation, the line of sight should also be determined the same way, but usually it is placed horizontally because this position permits a much faster and an easier drawing process, keeping the projection plane in a vertical position.

Vanishing points are manifested in perspective projection by the convergence toward an apparent intersection at an infinite distance of spatially parallel lines that obviously never intersect. This phenomenon is also considered a reality in human vision because it is all one experiences throughout life. It consists in an apparent gradual reduction in object´s size as the objects recede from the observer until they can no more be distinguished (among other optical effects). Parallel lines never intersect in nature, but if sufficiently extended they nearly always seem to intersect in perspective projections and invariantly in human vision. The rare one exception is in the former wherein a plane of a projected object is parallel to the projection plane and thus its edges have no vanishing points.

The horizon is depicted in a perspective by a line which designates the infinite limit of a horizontal plane at the observer eyes height. In a single scene there may exist an unlimited number of differing sets of parallel lines which, of course, are not necessarily horizontal defining an unlimited number of planes and an accompanying unlimited number of vanishing points (which correspond to their infinite extensions), which can be on, over or under the horizon.[3]

If one would have bilateral vision so as to view 180°, one would see the parallel lines appear to intersect in opposite directions simultaneously even if viewed from between the lines, but this bilateral vision is impossible, both in human vision and in perspective. We must not forget that the visual angle, the line of sight, the projection plane and thus the horizon in a perspective, must all be coordinated and centered in order to minimize distortion. When the observer looks to another place, all of them must be changed simultaneously. As vanishing points are related to this whole system they are automatically relocated too.

On the other hand, the distance used to calculate a perspective´s size (scale) is based not on the actual distance from the viewer to the object (in Figure M1, the viewer is "S"), but on the perpendicular distance measured from the observer to the picture plane ("P"), along with the line of sight and independently of the single or multiple object´s location. The projection plane can be placed at any place, not only between the object and the observer, but also behind the object, causing the apparent scale of the object. The scale doesn´t impact in any way the perspective´s distortion.

Figure M1 Small-angle approximation resulting from projecting onto a flat surface.

Figure M1 illustrates why this occurs. The two lines, "x" are the same length and are the same distance, "z," away from the picture plane, "P." When the two lines are projected onto the picture plane, toward the viewer, "S," the size of the lines represented on the picture plane is identical, "y." because both lines are projected onto one single picture plane and its distance to the observer is measured perpendicularly. This is the case even though it is clear that the left line "x" is actually further away from "S" than the right line "x." This distortion is enhanced because the lines "x" are in a parallel position with respect to the projection plane, and thus have no vanishing point (it is a frontal perspective). And besides that, as the visual angle covers both lines, but the line of sight is wrongly directed to the first line´s middle point and is not obtained bisecting the visual angle (which can be deduced by the position of the projection plane), the distortion is greatly exaggerated.

An example on how to contrast this distortion, is a view where one is standing facing north towards a road which runs perfectly east-west. In an artificial frontal perspective projection, every car on the road would be drawn at the same size, even though it is clear in reality that the farther away from the center of the picture that a car is, the farther away from the viewer that car would be. However, this seeming incongruity is cancelled out if the perspective meets three conditions: A) That the line of sight bisects the angle of view, B) That the angle of view is close to 30° (this will be explained below) and C) That it is viewed from the same point as the generated perspective.

Mathematical description[edit]

Mathematically, the difference between artificial perspective projection (perspective projection onto a flat surface) and natural perspective projection (perspective projection onto a spherical surface) is a distortion resulting from the difference in the size of the projection on a concave surface like the retina, on which real objects are projected, vs. a flat one situated frontally and at a unitary distance from an observer, depending on the visual angle that encompasses the object(s), in the following overall percentages:

DISTORTION PERCENTAGES PER VISUAL ANGLE
VISUAL ANGLE TANGENT OF THE SEMI-ANGLE x 2 CIRCULAR ARC'S LENGTH (unitary radius) OVERALL DISTORTION (%)
10° Tan 5° x 2 = 0.175 u. 1/18 ¶ = 0.17453 rad 0.27%
20° Tan 10° x 2 = 0.3526 u. 1/9 ¶ = 0.34906 rad 1.01%
30° Tan 15° x 2 = 0.5358 u. 1/6 ¶ = 0.52359 rad 2.33%
40° Tan 20° x 2 = 0.728 u. 2/9 ¶ = 0.69812 rad 4.28%
50° Tan 25° x 2 = 0.9326 u. 5/18 ¶ = 0.87265 rad 6.87%
60° Tan 30° x 2 = 1.1548 u. 1/3 ¶ = 1.04718 rad 10.28%
70° Tan 35° x 2 = 1.4004 u. 7/18 ¶ = 1.22171 rad 14.62%
80° Tan 40° x 2 = 1.6782 u. 4/9 ¶ = 1.39624 rad 20.19%
90° Tan 45° x 2 = 2 u. 1/2 ¶ = 1.57077 rad 27.32%

With the trigonometric formula to find the dimension of the opposed face to an acute angle in a right triangle (tan A), supposing the adjacent leg (that is the line of sight) as unitary (1 x tan A) and doubling it (because the line of sight is the bisector of the visual angle: 1 x tan A x 2), the dimension of a view in a visual angle, projected on a flat surface, can be found.

As the dimension of the circular arc that delimits a circular sector of ¼ of a circle (90 grades) is equivalent to ½ ¶ radian multiplied by the radius, if the radius is, as in the first formula, unitary, the projection of a circular arc of 90 grades on a concave surface is still ½ ¶ radian (1.57077 units). To obtain fractions of this circular arc it is possible to multiply the ½ ¶ radian by the fraction desired. Another way to achieve this result is to multiply the constant .017453 (or ¶ /180 or tau/360) by the radius (in this case 1) and by the visual angle.

A division of the dimension of an object projected on a flat surface between the one projected on a concave surface, will result in the difference between them and the consequent overall percentage of distortion.

The distortion is not uniform through the whole view. It increases accumulatively towards the sides of the view. The maximum punctual distortion occurs precisely at the extremes of the view. As the magnitude of the projection of a line on a plane increases or decreases in direct proportion to its distance to the station point, the difference between the hypotenuse of a right triangle supposing the adjacent leg has the length of the line of sight and has a unitary value (projection on a flat surface) and the radius of a circular arc, also with unitary value (projection on a concave spherical surface), results in the maximum punctual distortion.

MAXIMUM PUNCTUAL DISTORTION PERCENTAGE
VISUAL ANGLE TANGENT OF THE SEMI-ANGLE HYPOTENUSE (unitary adjacent leg) MINUS RADIUS (unitary) MAXIMUM PUNCTUAL DISTORTION (%)
10° Tan 5° = 0.0875 u. 0.00382 0.38%
20° Tan 10° = 0.1763 u. 0.01495 1.54%
30° Tan 15° = 0.2679 u. 0.03526 3.53%
40° Tan 20° = 0.364 u. 0.06418 6.42%
50° Tan 25° = 0.4663 u. 0.10337 10.34%
60° Tan 30° = 0.5774 u. 0.15472 15.47%
70° Tan 35° = 0.7002 u. 0.22077 22.08%
80° Tan 40° = 0.8391 u. 0.3054 30.54%
90° Tan 45°= 1 u. 0.41421 41.42%

It is important to notice that a change of 10% in an object´s dimension is clearly perceivable at first sight, as it is the overall distortion with a 60° visual angle or the maximum punctual distortion with a 50° visual angle. To avoid this excessive distortion, the visual angle of the observer should be always under 40°, and the closer it is to 30°, the better.

See also[edit]

References[edit]

  1. ^ webster.com/dictionary/plane%20of%20projection [dead link]
  2. ^ Thomas Ewing French. A Manual of Engineering Drawing for Students and Draftsmen, Chapter VII, Developed Surfaces and Intersections (p. 98): “The operation of laying out the complete surface on one plane is called the development of the surface”.
  3. ^ "Robert Kelso Sr". scribd.com.