The paper presents a unique approach in associating perspective projection with the image beheld by the eye and demonstrates that all graphical and photographic perspective projections must contain distortion when compared to the image beheld by the eye

Robert P. Kelso Sr.

Kelso Sr., Robert P.

Engineering Design Graphic Journal (ASEE - American Society for Engineering Education)

K e l s o  -   2 7A u t u m n  2 0 0 8INTRODUCTION Th is paper deﬁ nes distortion in perspective projection as the diﬀ erence between normal vision and the emulation of normal vision by graphics or photography.Section (1.0) ﬁ rst analyzes the distortion of perspective projection produced on a plane-of-projection and demonstrates that such distortion is due to the lack of perpendicularity among the projectors to the plane-of-projection. Th e paper further oﬀ ers an observation on problems associ-ated with viewing perspective pictures.Section (2.0) then investigates an alternate surface, the sphere, which permits all projectors toward its center to be perpendicular to a surface of projection.Section (3.0) demonstrates that a perspective projection onto a spherical surface of projection is analogous to light rays into the eye, which rays are similarly perpendicular to the spherical-shaped retina, and reproduces thereon and there-by the image beheld by the eye. Section (4.0) concludes that there is no reme-dy available to correct for the intrinsic distortion in perspective projection because it is impossible to rectify onto a planar surface, without distor-tion, an image on a sphere.1.0 DISTORTION ON THE PLANE-OF-PROJECTIONFig. 1 and Fig. 2 demonstrate a) that distor-tion is due to the lack of perpendicularity among the projectors to the plane-of-projection and ac-knowledges b) that  only a single projector of sight, namely, the axis of the cone of vision, from the object to the eye may be orthogonal to a plane-of-projection and c), concludes there-fore, that all projectors other than b) result in the distortion of the image. Th is is also true of the camera: only a single ray of light, namely, the principal optical axis, may be orthogonal to the ﬁ lm and, therefore, all other rays of light result in the distortion of the photograph. Fig. 1 identiﬁ es the elements of Fig. 2. As an aid to the reader’s intuition the illustrations use a human form as an analog for the Station Point (Focal Point). Th e object projected is a base molding for a building column.Fig. 2 compares a projection onto a plane from one direction with a projection onto the same plane of the same object from a diﬀ erent direction—thereby creating diﬀ erent angles among the projectors to the plane-of-projection. Pivoting of the object eliminates diﬀ ering view-ing directions as a factor which might cause a diﬀ erence in the images on the Projection Plane. Th e diﬀ erence between the two images, as seen in the front view, is apparent.Distortion in Perspective ProjectionRobert P. Kelso Sr.Professor (retired) Louisiana Tech University, kelso@coes.latech.eduAbstractTh e paper presents a unique approach in associating perspective projection with the image beheld by the eye and demonstrates that all graphical and photographic perspective projections must contain distortion when compared to the image beheld by the eye.___________________________________________________________________________________2 8  -  E n g i n e e r i n g  D e s i g n  G r a p h i c s  J o u r n a lv  o  l  u  m e    7 2    n  u  m b  e  r    3       FIGURE 1:  Orthographic Top Front and Ilsometric ViewsFig. 1 is an orthographic top-front, and iso-metric view of a pictorial analog of a standard graphical/photographic perspective projection. P designates the planar surface-of-projection. In photography, the projection plane (ﬁ lm) is locat-ed to the rear of the Station Point, S. Th is place-ment does not alter the geometric principles or the characteristics of the developed image. Note that an eye at S views both the object and the image of the object as if the projection plane did not exist. Th at is to say, to the eye, from the Sta-tion Point, the image of the object on the plane-of-projection is indistinguishable from the object itself. Fig. 2 demonstrates the basis of perspective distortion on the projection plane. Th e object in Fig. 1 is pivoted by an arbitrary amount about the Station Point, S, so that at all times the object appears unchanged to the eye. Th is causes the new image to appear the same to the eye as does the original image, namely, indis-tinguishable from the object itself. But the Front View in Fig. 2 reveals the new image to be clearly diﬀ erent from the original image. Th erefore,  be-cause the only diﬀ erence in the two projections is the angles they form with the plane-of-projec-tion, this paper concludes that the two images are diﬀ erent because of the diﬀ erences in the angles which the respective projectors form with the projection plane. Th is logic extends to Fig. 1: the lines of sight from the object in its original position, at an angle other than perpendicular to the Projection Plane, produces a distortion of the image. Th is essen-tially encompasses all of the lines of sight in the projection other than the central axis of vision, which is to say all the lines of projection other than the axis of the cone of vision. Th erefore, the paper concludes that any perspective projection of any image onto any plane-of-projection is nec-essarily distorted. FIGURE 2:  Pivoting the ModelK e l s o  -   2 9A u t u m n  2 0 0 8[Note: An exception to the phenomenon of distortion in graphics/photography would seem to be the case in which spatially parallel straight lines in nature are parallel to the plane-of-pro-jection:  they project as parallel line images onto the plane-of-projection. Th is is true on a plane-of projection, but because—as will be demonstrat-ed—the retina is spherical in shape, straight lines that are parallel to a plane-of-projection cannot be parallel to a spherical surface, e.g., the retina. In other words, the exception applies to planar projection but not to vision.] From an analysis of the above, it follows that an inspecting eye of a standard perspective im-age may avoid the distortion on Th e Plane-of-Projection by viewing the image from the exact position of its related Station Point/Focal Point. Th e likelihood of a casual viewer looking from precisely that point—the point at which the plane-of-projection distortion disappears—is vanishingly small. Th erefore, an inspecting eye it-self is challenged with an intrinsic distortion dif-ferent from the intrinsic distortion on the plane-of-projection: the amount of the eyeball’s chance distance from the related Station Point. 2.0 SPHERICAL SURFACE OF THE PROJECTIONTh e eye distorts reality as evidenced by spatial-ly-parallel straight lines appearing to intersect—in the manner of receding straight railroad tracks. Th e author takes the concept a step further by asserting that an observer standing between the rails will see the rails as simultaneously intersect-ing in both (opposite) directions if peripheral vi-sion permitted – in the manner of a Chameleon with lateral eye locations in which separate im-ages from both eyes are simultaneously merged into one by the brain—just as the human brain merges the images from two front-facing eyes. How is this possible? After all, the geometry of classical foreshortening (convergent) theory ap-plies to only a single direction of the central axis of the cone of vision. Th e author argues that it is possible because the retina of the eye is spherical in shape.Th e retina, as a part of the spherical orb of the eyeball, is a sector of a sphere. Th erefore, the im-age projected through the lens to the retina to the brain is the consequence of a projection onto a spherical (sector) surface-of-projection rather than onto a planar surface-of-projection, as is the case in both graphics and photography.To accommodate this phenomenon the paper proposes a spherical model as a surface of projec-tion rather than the standard planar model.Figs. 3 - 9 present a spherical surface-of-pro-jection as an analog of the retina, i.e., a ‘projec-tion sphere’ rather than a ‘projection plane’. Th e illustrations demonstrate that the step-by-step projection of spatially parallel lines onto the ret-ina-analog sphere-of-projection mimics human eyesight. Th e center of the sphere is analogous to the Focal Point of the retina. Th e resulting intersec-tions of the projected lines as they appear upon the “picture sphere” resolve the enigma of spatial-ly parallel lines appearing to intersect—in “both directions”.A brick wall serves as a model for spatially par-allel lines.[Artistic Note: Th e initial illustrations include strategically placed great circles to enhance the illusion of a sphere.]Fig. 3 features a brick wall with parallel hori-zontal and vertical edges. Th e wall serves as an object in nature an image of which is to be pro-jected onto the retina-analog spherical surface (P). By steps in the following illustrations, the edges are projected toward the center (S) of the “picture sphere”, namely, toward the analog of the Focal Point of the retina. Th e spherical sur-face-of-projection is analogous to the spherical (sector) shape of the retina.3 0  -  E n g i n e e r i n g  D e s i g n  G r a p h i c s  J o u r n a lv  o  l  u  m e    7 2    n  u  m b  e  r    3  FIGURE 3:  Sphericial Surface of Projection and Object WallFig. 4 shows the perspective projection of the horizontal top line of the wall to the Focal Point of the retina-analog,. [In life, the Focal Point of the retina is slightly in front of the retina—from where the image becomes inverted before con-tacting the retina and its subsequent transmission to the brain.] Th e intersecting projectors deﬁ ne a plane that intersects the sphere-of-projection in a great circle (G). [Note: For clarity, selected “artis-tic” great circles are occasionally omitted.]  FIGURE 4:  Projection of the Top Line of the WallFIGURE 5:  Projection of the Bottom Line of the WallFig. 5 similarly shows that the projection of the horizontal bottom line of the wall creates the great circle (H).  FIGURE 6:  Intersection of Great Circles of the WallFig. 6 shows that the two great circles G and H intersect at points A and B. (Take note that A and B do not lie on the outer circle of the illustra-tion, here artistically forming the sphere in the illustration.) A viewer at S (“Station Point”/Focal Point) looking in the direction of A sees all horizontal lines of the wall as appearing to con-verge at A; the same is true of point B. Th e points of intersection, A and B, are analogous to the two K e l s o  -   3 1A u t u m n  2 0 0 8points of intersection observed as one peers from between the rails down a railroad track simulta-neously “in both directions”.  FIGURE 7:  Projection of the Vertical Sides of the WallFig. 7 shows a similar determination of the two vanishing points, C and D, for the spatially vertical lines of the wall. Th e perspective image now has four vanishing points, A, B, C, and D. Two of the vanishing points appear at the “east-west” poles and two at the north-south poles. By logical extension, a perspective image of a single plane may have an unlimited number of vanish-ing points as deﬁ ned by an unlimited number of parallel straight lines thereon and their spatial directions. Note if the vertical lines of the wall represented the vertical corners of a tall building, a viewer at S looking upwards towards C will ob-serve the vertical corners of the building as con-verging toward that point. Fig. 8 shows that the horizon line, Z, is deter-mined by a plane deﬁ ned by the four vanishing points A, B, C, and D and is observed as a straight line from S, as are all great circles. Note  that the horizon line of the wall is determined by vanish-ing points; the horizon line does not determine the vanishing points. A view from S towards any point along the perimeter Z perceives that  point as located on the horizon line of the vertical wall. Th e horizon line would appear as horizontal if the wall were horizontal—and be more familiar to graphicians.FIGURE 8:  Hemisphere of Projection CONCLUSIONFrom the above the paper concludes that if, in graphical/photographic perspective projec-tion, it were possible to project an image onto a spherical surface and then to perfectly rectify the sphere into a planar surface the result would be a perspective projection image without distortion. However, this is not possible because it is impos-sible, geometrically, to perfectly rectify a sphere, e.g., the globe, into a plane, as especially cartog-raphy conﬁ rms.  Th erefore, all standard, perspec-tive projection images are necessarily distorted. 

Distortion in Perspective Projection

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Distortion in Perspective Projection

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Engineering Design Graphic Journal (ASEE - American Society for Engineering Education)