+New Treatise On Perspective

TREATISE ON PERSPECTIVE 1

A TREATISE ON PERSPECTIVE IN HUMAN VISION

A REVISION OF DIOPTRICS

Perspective foreshortening: a result, not a causeFocal Point: center of retina-sphere

Robert P Kelso SrProfessor of Engineering Graphics (Retired)Louisiana Tech University

1120 Cimarron CourtSan Marcos, Texas 78666, [email protected]

“If one’s vision is in perspective, it is verification that the focal point of one’s eye is located at the center of the retina-sphere”. – The author

to: marion, bob, julie & mandy

mailto:[email protected]


PREFACE:

he author demonstrates that the perspective illusion in vision is the result of a gnomonic projection to the retina, a concept predicated upon the focal point of the eye being positioned at the center of the

retina-sphere rather than on the retina proper. The author finds that only the center of the retina-sphere for the focal point location accommodates all the dioptrics of the eye. This location was first advocated in the 19th century by perhaps Sir David Brewster.

TIdentifying visual perspective as a dioptric gnomonic projection and the advocacy of a new system of dioptrics based on Brewster are original to this writer.

This, therefore, is a concomitant treatise: two dioptric concepts each of which supports the other and without both being true, neither being true; as a consequence, anyone who visually beholds ‘the world’ in perspective validates both concepts.

The first concept is entitled, On the Position of The Focal Point.

On the Position of The Focal Point contains two parts; part one contests the position of the focal point in the Standard Model (so called because it describes the dioptrics as presented by the current standard literature), and part two presents the efficacy of the author’s Alternate Standard Model (so called because it contains the new dioptrics advocated by the treatise).

The second concept is entitled: The Cause of Perspective in Vision. ________________________________________________________

The writer’s research into the history of optics leads to a provocative conclusion.

The focal point location within the eye has been mistakenly adopted over the centuries as concurrent with the retina. Although no scientific data supports this premise, it has been steadily passed on by circular citations. To further avoid this, doubtful readers are encouraged to support their argument with cited primal data, and avoid the recycled citation the ultimate origin of which is unknown.

________________________________________________________


NOTES:

Semantically, anterior and posterior, although generic are exclusively associated herein with the current ocular literature’s Standard Model; exterior and interior are similarly associated with the author’s Alternate Standard Model.

The writer construes from among various versions in the literature: 1) the geometric axis is the same as the optical axis; 2) the optical axis intersects the retina at the position of the nondescript posterior pole; and 3) the visual axis is the off-axis path connecting a fixation point in the field of view with the fovea. Hereinafter the axes’ point of intersection is demonstrated to be the focal point of the eye.

The writer’s deduced representation of the eye -- incorporating current literature selections from differing representations, many contradictory.

This treatise adopts the phrase, central trace, for two situations: 1) to describe the most nearly linear pathway connecting the reflected light from a given point in the field of view with the retina -- refraction considered – i.e., the treatment (implied) in the Standard Model literature and 2) for a modified treatment by the Alternate Standard Model. Each has its own connotation. In the former, the phrase describes the single ray pathway from among all the ambient rays from a point of light in the field of view. The same usage is true for The Alternate Standard Model except that a central trace completely replaces the anterior and posterior ambient rays with a single ray, which ray loses its linearity at the lens at which point it is refracted toward the advocated focal point at the center of the retina-sphere.

The illustrations are orthographic projections, i.e., parallel lines in

(ideated) actual space display as truly parallel in the computerized modeling space. Be reminded that the projections are naturally


inverted, and the final, beheld perception is then “corrected” by the brain. The illustrations are conceptual and not to scale.

ON THE POSITION OF THE FOCAL POINT

This section evaluates the erroneous dioptrics resulting from the traditional placement of the focal point on the retina surface. The viability of the evaluation relies upon The Law of Visual Directioni (The Law) as proposed by Sir David Brewsterii which places the focal point at the center of the retina sphere. To affect the comparison the writer assumes a radiating point light source within the field of view. The areas of evaluation are 1) the efficacy on vision of a Standard Model’s point source of ambient light, 2) parallel rays from a field of view, and 3) myopia/hyperopia.

The result of these comparisons: the dioptrics derived from The Law, i.e., a positioning of the focal point at the retina-sphere center, is consistent with elementary geometry whereas the position of the Standard Model i.e., at the retina, is not. Dioptrics in the current literature is found lacking in this aspect as described herein below.

THE STANDARD MODEL

FIGURE 1 Standard ModelTypical Dioptrics Trace from the Literature


FIGURE 2 Standard ModelDivergent/Convergent Cones;

Central Trace Tracks the Optical Axis

In Fig. 2, derived from Fig.1, ambient rays from point source P enter the pupil as a cone (anterior cone) of divergent rays. (Apex: source of light; base: the pupil.) These rays, upon exiting the lens, allegedly define a second cone (posterior cone) of convergent rays to the retina. (Base: the pupil. Apex: the alleged focal point on the retina.) The Standard Model’s central trace tracks the optical axis which is, per standard geometry, perpendicular to the retina. Note that the central trace alone produces the same visual result as all of the combined resultsiii of both the anterior and posterior cones. Ambient rays other than the central trace dissipate in the fluid of the eyeball as heat.


FIGURE 3 Standard Model

Dioptrics of a Trace Oblique to the Optical Axis

Point P, (Fig. 3) represents any point in peripheral vision, i.e., any of the countless points of reflected light in the field of view not-on-the-optical-axis. Per The Law, no peripheral rays as described by standard model dioptrics, including the central trace, produces vision. The rays do not access the ocular nervous system because the rays are not perpendicular to the retina.

Ray perpendicularity is required for vision because the photoreceptors are perpendicular to the retinaiv. For a ray to produce vision, it must transit the axis of a photoreceptor until it reaches the nervous systemv. For rays to be so perpendicular, per geometry, rays must first intersect at the center of the retina-sphere. As seen in Fig. 3 the rays of peripheral vision are not perpendicular to the retina and therefore do not produce an image; thus the standard concept is flawed.


PARALLEL TRACES

FIGURE 4 Standard ModelParallel Rays from the Field Of View

To analyze parallel traces the treatise first investigates light emitted from a single source at a great distance, say a star. The flawed principle of the Standard Model is the same in as Figs. 2 and 3 wherein an anterior cone of rays is formed which covers the pupil. However, the great distance compared to the size of the pupil renders the rays of the anterior cone virtually parallel. Nevertheless the central trace is produced and is also virtually parallel to the ambient rays of the anterior cone. If the central trace tracks the optical axis an accurate image of the source is produced on the retina because, by definition, the ray is perpendicular to the retina and parallel to a photoreceptor axis. In Fig. 5 the central trace and accompanying parallel ambient rays are not perpendicular to the retina therefore no vision occurs, thus the Standard Model also is flawed in the case of parallel rays. The Case of a Single Distant Source


The Case of Several Distant Sources

FIGURE 6 Standard ModelAnterior Rays Parallel to Optical Axis

FIGURE 7 Standard ModelNear Parallel Anterior Rays Oblique to Optical Axis

Fig. 6 illustrates parallel central traces from separate points, say, a select cluster of stars. It contains the comparable defect as Fig. 2, i.e., all the rays not tracking the optical axis are superfluous and may be dismissed. Fig. 7 contains the comparable defect as Fig. 3 and also may be dismissed.


Non-parallel rays from nearby sources may fill in the blank retinal areas, however Figs. 5, 6 and 7 address near-parallel traces, i.e., traces from a distance, the conclusion about which remains.


MYOPIA/HYPEROPIA


Myopia

The Standard Model posterior cone disputes the standard explanation of myopia/hyperopia. In myopia, as with all rays within the eye, the alleged posterior convergent cone proceeds unabated until it intersects the retina. By so doing, the alleged cone creates a filled circular image at the point at which it impinges upon the retina, i.e., myopia in Standard Model dioptrics causes a single point within the field of view to appear as a single, filled circle on the retina. See Fig. 9. Extrapolating this to each off-axis trace from the countless points of light comprising the field of view produces the same ‘circles’ defect, consequently, it follows that the entire field of view must appear chaotically blurred because of ‘circles’ replacing ‘points.’ As this is contrary to natural experience it may be dismissed. It will be seen herein after, that there is also blurring in the myopic/hyperopic Alternate Standard Model but from a different cause.

FIGURE 9 Standard ModelMyopia Representation


FIGURE 10Standard ModelHyperopia Representation

Figs. 9 and 10 reveal there should be no difference in blurred vision whether caused by myopia or hyperopia; both theoretically create indistinguishable types of circular images on the retina representing single points of light. Nothing prevents the ‘circles,’ due to either myopia or hyperopia, from being identical in size and location on the retina; therefore, in the Standard Model, there should appear no difference between the symptoms of myopia and hyperopia. This is contrary to the observed symptoms, therefore the Standard Model causes of myopia and hyperopia may be dismissed. SIDE-BAR

Unrelated to the difference between the Standard Model and the Alternate Standard Model is a persistent error in the standard literature regarding the angle between the optical axis and the visual axis. The angle is designated in the literature by the Greek letter, “α”. The error is that, given the parameters of the Standard Model, the two axes do not intersect. Furthermore “α” is usually shown in a right elevation view in which the optical axis is viewed perpendicularly, i.e., normally, while the visual axis is viewed obliquely. Fundamental to descriptive geometry is that both axes be viewed normally in the same view in order to show the true size of the angle between. (In the Alternate Standard Model all light rays, including that of the two axes, intersect at the retina-sphere center--thereby defining a plane of which it is possible to produce a normal view by elementary descriptive geometry protocols. Such will preclude the Standard error.

STANDARD MODEL


THE ALTERNATE STANDARD MODEL

Notevi

FIGURE 11From The Literature

Alternate Standard Model dioptrics is initiated by single central traces from each of the countless points of light in the field of view. This is in contrast to the dioptrics of the Standard Model which is initiated by divergent cones of ambient rays projected from such sources.

Two fundamental differences in the Alternate Standard Model from the Standard Model:

Exterior central traces from representative radiating points within the field of view replace without loss to vision the Standard Model’s anterior central traces. These exterior central traces collectively define an exterior conical geometry. (Fig. 12). Per The Lawvii, upon exiting the pupil, central traces refract to the center of the retina-sphere, namely the focal point.

FIGURE 12 Alternate Standard ModelsThe Cone of Vision


FIGURE 13Alternate Standard Model

The cone of vision encompasses all visible rays within an eye’s field of view. By definition the cone is composed of rays which trace toward the apex of the cone.

FIGURE 14Alternate Standard ModelDetail of Figure 13


FIGURE 15 Alternate Standard ModelsDetail of Figure 14

Per The Law, Brewster relocates the focal point to the center of the retina-sphere. This hypothesis plus elementary geometry mandate that visual rays intersect the retina perpendicularly, and concomitantly, if the rays intersect the retina perpendicularly then the rays must first intersect at the center of the retina-sphere, i.e., the focal point.

FIGURE 16Alternate Standard ModelsTraces Perpendicular to Retina


FIGURE 17Alternate Standard ModelPictorial

FIGURE 18Alternate Standard ModelA Complete Dioptrics Including Interior Cone of Vision

MYOPIA/HYPEROPIA


FIGURE 19Alternate Standard Model

If the pupil size remains stable there should be no difference between myopia and hyperopia images except size. The familiar relationship between image size and image acuity -- dependent on the distance at which an object is viewed – is beyond the scope of this treatise.

Fig.19 demonstrates that myopia/hyperopia in the Alternate Standard Model is due to a ray mal-refraction to a position either anterior or posterior to the ideal ocular focal point (center of the retina-sphere). This causes blindness because, per geometry, the rays will not transit the photoreceptors. If the rays entering the photoreceptors are only slightly off center, ≈± 1 micronviii, they may cause blurring rather than blindness because the rays may not transit the complete length of the photoreceptor but rather deflect into the surrounding photoreceptor tissues instead. The citation from thehumanmircle.blogspotix would seem a contradiction. However, the citation’s referenced “shielding effect” would seem at least only partial because of the evidence in Fig. 20x. The resulting meridianally smearing in the surrounding photoreceptors is the presumed cause of Alternate Standard Model blurring in myopic/hyperopic vision.


FIGURE 20 From the Standard LiteratureCause of blurring/blindness due to myopia/hyperopia in the Alternate

Standard Model. In the illustration on the right, the ray image is ‘smeared’ over the nervous system connected to the adjoining photoreceptors.

AMENDMENT TO THE LAW

The Law of Visual Direction relies on Brewster’s as yet unverified statement that the retina is nearly a perfect sphere. If not spherical, basic geometry reveals that Brewster’s statement may be amended without comprise to read, “The retina is spherical, or both, 1) the area of the retina perceiving the field of view is spherical, and 2) the orb is symmetrical about the geometric axis (optical axis).” See Fig. 21.

FIGURE 21Alternate Standard ModelPresumed Non-Spherical Orb

Q.E.D.


PERSPECTIVE ILLUSION IN VISION: THE CAUSE

erspective in vision is an optical illusion that defies Euclid: Parallel straight lines on a given plane appear to intersect at the “horizon” of the plane. Be reminded that a frequent rationale for the illusion, the

‘foreshortening factor’, is a result not a cause.PThe cause of the illusion is that straight lines, as seen in all normal sight, are a gnomonic projection through the center of the retina sphere. A gnomonic projection is a projection in which all straight lines in space, when projected toward the center of a sphere, create great circles of intersection with the sphere. The most familiar gnomonic projection is circles of longitude on the globe. From the center of the globe “vertical” north-south circles appear as straight lines passing through the north and south poles. The most prevalent lines seen on earth, e.g., roads, roof edges of buildings, railroad tracks, etc., being parallel to the plane of the earth (the curvature of the earth may be disregarded) are horizontal, therefore gnomonic projections of such to the retina have “horizontal” east and west poles (or an analog thereof) as opposed to “vertical” north-south poles.

FIGURE 22Gnomonic ProjectionsProjects a Straight Line as A Great Circle

The following is a series of illustrations of a gnomonic projection to the eye of straight railroad tracks which appear to merge. They constitute a validation that perspective in vision is a gnomonic projection.


FIGURE 23

Observer on a Railroad Platform

Fig. 23 shows an observer on a railroad platform. The location of the observer is irrelevant. He may stand in the middle of the rails. If the observer has either 180 º peripheral or bilateral vision, to him the rails appear to merge in opposite directions simultaneously. (See, Fig. 35.)

FIGURE 24Observer’s Cone of Vision


FIGURE 25Extended Interior Cone of Vision


FIGURE 26Observer with Extended Interior Cone of Vision

FIGURE 27Observer with Truncated Extended Interior Cone of Vision

FIGURE 28

Geometric Elements Reduced To Essentials


FIGURE 29Rail AB Projected To Retina

FIGURE 30

ABF Expanded To Intersect the Retina


Rays from points A and B project to the retina, through focal point -F-, thereby defining plane ABF. If sufficiently expanded within the orb, plane ABF (per gnomonic projection theory and as allowed here by computer software modeling precision) intersects the orb in a great circle of intersection. “A great circle [appears as] a straight line in a gnomonic projectionxi,” which is to say, great circles appear as straight lines when viewed from the center of the great circle – in this case, the ocular focal point.

FIGURE 31Rail CD Traced To Retina

FIGURE 32

CD Projected To Retina


i Brewster, Sir David, A treatise on optics, (168.) 2. On the law of visible direction. — When a ray of light falls upon the retina, and gives us vision of the point of an object from which it proceeds, it becomes an interesting question to determine in what direction the object will be seen, reckoning from the point where it falls upon the retina. In Figure 142., let F be a point of the retina on which the image of a point of a distant object is formed by means of the crystalline

lens, supposed to be at L L. Now, the rays which form the image of the point at F fall upon the retina in all possible directions from L F to L F, and we know that the point F is seen in the direction F C R. In the same manner, the points f f are seen somewhere in the directions /'S,/T. These lines F R,/' S,/T, which may be called the lines of visible direction, may either be those which pass through the centre C of the lens L L, or, in the case of the eye, through the centre of a lens equivalent to all the refractions employed in producing the image; or it may be the resultant of all the directions within the angles L F L, Lf L; or it may be a line perpendicular to the retina at F, f'f. In order to determine this point, let us look over the top of a card at the point of the object whose image is at F till the edge of the card is just about to hide it, or, what is the same thing, let us obstruct all the rays that pass through the pupil excepting the upper ones, R L, R C; we shall then find that the point whose image is at F, is seen in the same direction as when it was seen by all the rays L F, C F, L F. If we look beneath the card in a similar manner, so as to see the object by the lower rays, R L F, R C Fii The provenance of the “THE LAW” is uncertain.iii This may be validated experimentally by blocking a central trace as nearly as practicable in order to determine if the radiating point source remains visible on the retina as a result of the ambient convergent rays alone. Sheep’s eyes have been accessed from the rear to view an actual image and might serve to prove the concept. (See Optics and The Eye [Archive] - Physics Forums -- Physics Forums > Physics > General Physics > Optics and The Eye ... Ophthalmology students have been known to carefully scrape the back off a sheep's eyeball to see that image. -- www.physicsforums.com/archive/index.php/t-212864.html· Cached page.) If the point source is not visible upon blocking only the central ray then the anterior convergent cone of rays is ineffective except for the central trace. iv Per Brodal. The Central Nervous System, (p. 193): “As mentioned, the retina contains interneurons in addition to the photoreceptors, bipolars, and ganglion cells (Figure 7.4). The horizontal [Brodal employs the convention that the retina is “horizontal” - RPK] cells send their processes in the plane [spherical surface] of the retina ─ that is, perpendicular to the orientation of the photoreceptors and the bipolars (Figure 7.7).”v J.M. Enoch and V. Lakshminarayanan, Vision Optics and Instrumentation, ed. W.N. Charman (MacMillan Press, London, U.K., 1991), 280-308. “Because the photoreceptor inner and outer segments are short thin cylinders with a higher refractive index than the surrounding tissue, they act as small optical fibers and exhibit typical waveguide properties such that light entering along the fiber axis will be totally internally reflected, and light entering from peripheral angles can escape into the surrounding tissue. This optical property means that light entering on or near the receptor axis will pass through the entire length of the outer segment and therefore have an increased chance of isomerizing a photopigment molecule.vi This erroneous illustration of the Standard Model fortuitously describes the Alternate Standard Model. vii Brewster, Sir David, A treatise on optics, part iii. (p. 246). Brewster states, in order to be seen, “... the direction of the ray ... is always perpendicular to the retina; [and] ... as the interior eyeball is as nearly as possible a perfect sphere lines perpendicular to the surface of the retina must all pass through a single point, namely, the centre of its spherical surface.”

http://cc.bingj.com/cache.aspx?q=%22Optics+and+The+Eye+%22+sheep+scrape+student&d=4615591052183197&mkt=en-US&setlang=en-US&w=dd0629b3,dd4776d8

http://www.physicsforums.com/archive/index.php/t-212864.html


FIGURE 33Intersection of AB with CD at Focal Point

FIGURE 34Figure 33 as Seen Posterior to the Retina

To create a horizon-line on the retina which represents the horizon-line of the actual plane of the rails requires a cutting plane through the focal point and parallel to the plane of the geometry (here, the rails).

viii http://people.umass.edu/~phys125/P190Bsyllabus.html130 million photoreceptors in a human eye, each a couple of microns in diameterix http://thehumanmiracle.blogspot.com/ “Other pigments, such as melanin, are used to shield the photoreceptor cells from light leaking in from the sides. The opsin protein group evolved long before the last common ancestor of animals, and has continued to diversify since”.x F.W. Campbell, (1957). Depth of Field of the Human Eye. Optica Acta, 4, 157-164. A schematic diagram showing that obliquely incident rays pass through multiple photoreceptors and produce a meridional smearing of the image across several receptors. xi Steinhaus, H., Mathematical Snapshots, 3rd ed. (pp. 183 and 217)

http://thehumanmiracle.blogspot.com/

http://people.umass.edu/~phys125/P190Bsyllabus.html130


FIGURE 35

Cutting Plane Defining Ocular Horizon

The (parallel) rail images on the retina are, as with all gnomonic projections, concurrent -- here at “east-west poles” (or an analog thereof) of the earth’s “projected” visual horizon onto the orb. This “projection” occurs by virtue of a cutting plane parallel to the plane of the rails and through the focal point. The two visually concurrent points of the rail’s great circles are the visually perceived vanishing points of the two rails on the "projected” horizon. The precision of the computer graphics in Fig. 35 geometrically validates this phenomenon and confirms the premise. The justification of the earlier comment that a viewer with peripheral vision of 180º or bilateral vision will observe straight railroad tracks on an infinite plane to merge in opposite directions simultaneously is now obvious.

In order to fully appreciate the completed verification it is helpful to acknowledge that a plane which does not intersect a point centered on the retina-sphere does not create a great circle of intersection (a circle is created but not a great circle) – and the optical illusion will not exist. However, since the optical illusion is indeed known to exist it follows that the geometry of the foregoing reconfirms that the location of the ocular


focal point is the center of the retina-sphere and applies to all human vision.


VIEW TOWARD THE VANISHING POINT

FIGURE 36

The sequence in Fig. 36 shows a viewer gazing toward a vanishing point; orb with cone of vision (see Fig. 25); rails-plus-focal-point-expanded-planes as perceived through the cone of vision (see Fig. 33); creation of a visual horizon line by a cutting plane (see Fig. 35); resulting visual horizon line (see Fig. 35); hemisphere created using visual horizon line (see Fig. 35); continuation of sequence to reveal the image projected on the retina.


FIGURE 37

FIGURE 38


The retinal image produced by the viewer gazing toward a vanishing point. Note that the size of the field of view on the retina exactly maps

the size of the pupil.


Perspective in Human Vision: A More General Case.

The previous addresses parallel lines on a horizontal plane. This is the prevalent case but a special case nonetheless. Below is a more general case in which the plane of interest (roof) is oblique to the viewer.

FIGURE 39

FIGURE 40


FIGURE 41

FIGURE 42


FIGURE 43

FIGURE 44


FIGURE 45

FIGURE 46


FIGURE 47Sidelines projection

FIGURE 47


FIGURE 49

View Posterior to Retina

FIGURE 50Perspective Projection Analysis of Complete Roof


FIGURE 51Transparent Planes

FIGURE 52View Posterior to the Retina


FIGURE 53View Anterior To Pupil

FIGURE 54Worms-eye View Of Retina Hemisphere Showing Four-Point

Dioptrics Perspective Analysis of the Roof


FIGURE 55Eye Position Relative to Roof

FIGURE 56Detail of Figure 55

Q.E.D.

+New Treatise On Perspective

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