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By employing basic principles of physics andeuclidean geometry, we will attempt to provide a

rational explanation of binocular vision and thehoropter. The concept of the horopter encompasses anumber of corresponding retinal points that project todefinite single points in space within the field of sin-gle binocular vision. Nonetheless, there is a finite areaaround this horopter wherein fusion is possible and iscalled the Panum fusional area. Objects that lie with-in the Panum fusional area can produce single binoc-ular vision with stereoscopic vision.1

The above presentation appears intuitive and sim-ple enough. Nonetheless, the explanations on whichthese observations are based have been anything butsimple and clear. In the 1600s, Aguilonius postulatedthe existence of a surface on which single visionoccurred and termed this locus of points “thehoropter.”2 In the 1800s, Vieth3 and Müller4 producedadditional work on the horopter, resulting in the nowwell-known Vieth-Müller circle. This circle has beenused extensively as the basis for binocular vision andthe explanation for the horopter as well as the Panumfusional area. Essentially, the Vieth-Müller circle is anarbitrary construct wherein a circle is created onwhich the circumference intersects the fixation pointof vision as well as an arbitrary nodal point containedinside each eye. However, much controversy hasexisted over the integrity of the Vieth-Müller circle.Wright5 states: “Physiologic experiments have shownthat the Vieth-Müller circle is not accurate, and thatthe circle of corresponding points is actually anellipse, which is termed the empirical horopter.”Insight into a reason why empirical data display anelliptical horopter has been limited.

Richard J. Fugo & Dawn M. DelCampo

The Ellipseof Vision

The contention of this article is that the horopter, as well as the

Panum fusional area, is more precisely explained as an ellipse that

employs corresponding points on each retina as geometric foci. This

analysis, using principles of physics and euclidean geometry, pro-

duces the theory of the “ellipse of vision.”

Reprints:Fugo Eye Institute, 100 W. Fornance St., Norristown, PA 19404.

A B S T R A C T

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The contention of our article is that the horopter, aswell as the Panum fusional area, is more preciselyexplained as an ellipse that employs correspondingpoints on each retina as geometric foci from which theellipse of vision may be constructed.

Early thinkers following the teachings of Aristotleand his contemporaries described an earth-centereduniverse with other celestial bodies orbiting the earthin circles. The work of Copernicus and Galileodescribed a celestial sky in which celestial bodiesrevolved around the sun in a circular pattern.Johannes Kepler took the astronomical data collectedby the Danish nobleman Tycho Brahe, but Keplercould not mathematically justify that the planets cir-cled the sun in a true circular fashion. It is said thatKepler struggled to the point of near nervous exhaus-tion to justify the “obvious” circular paths of the plan-ets around the sun. Nonetheless, he could notmathematically justify the orbit of Mars. The orbit ofMars is more elliptical than most planets and did notconcede to any mathematical “fudge factors,” whichKepler employed as he struggled with his “war onMars.” When his analysis was done, Kepler exclaimedin a scientific manuscript, “Oh, ridiculous me!” andconceded that the path of planets was elliptical andnot a circle. In fact, Kepler’s First Law of Astrophysicsis known as “The Law of Ellipses.”6 Similarly, whenNobel prize winner Niels Bohr proposed a novel struc-ture of the atom, he described a circular path of elec-trons around the nucleus. To simplify a verycomplicated topic, the concept of particle perturbationmakes circular electron orbits nearly impossible, justas it makes circular planetary orbits nearly impossi-ble.7 Likewise, the 3-dimensional (3D) structure of theretina represents a perturbed ellipse, rather than a cir-cle. This fact is key to the ellipsoid nature of the fieldof vision and the continuous array of retinal focalpoints. These points represent focal points for con-struction of the immense array of ellipses that gener-ate the ellipsoid of the field of vision.

The ancient Greeks were fascinated by conic sec-tions (a plane that intersects the body of a cone at dif-ferent angles). They had developed the field of conicsections to a level that is beyond the grasp of many oftoday’s scientific elite. Such conic sections can pro-duce circles, ellipses, parabolas, and hyperbolas. Asimple definition of an ellipse would be a plane curvesuch that the sum of the distances from any point onthe curve to 2 fixed points within the curve is a con-stant (Fig 1). The 2 fixed points within the curve arecalled the focal points or nodal points of the ellipse. InFigure 2, F′ and F are the 2 focal points of the ellipse.If F′P and FP were a continuous piece of string thathad a pen or pointer holding the string taut at P, thenrotating the pen 360° will generate the circumferenceof an ellipse. An interesting corollary of this charac-teristic of an ellipse is based on an analysis when theinterior surface of the ellipse becomes reflective tolight rays. If the inner surface of the ellipse acts as amirror reflective to light, any light rays striking the

inner surface of the ellipse emanating from the light-bulb located at 1 focal point would always reflect backto the second focal point of the ellipse (Fig 3). Theexplanation for this phenomenon is based on thegeometry of a tangent line to the precise spot that thelight ray strikes the circumference of the ellipse, thusacting as a reflective mirror. Actually, this simpleexperiment based in physics helps define the conceptof the horopter and the concept of binocular stereo-scopic vision from the horopter. In this case, thehoropter is a portion of the circumference of an ellipsewherein focal point pairs represent correspondingpoints on each retina, such as the fovea of each retina.A line projecting perpendicular from each fovea inter-sects at a point in space. This point is 1 location on theellipse of vision. In actuality, there is a continuousarray of corresponding nodal points on each retinathat serve to generate a continuous array of ellipsesthat exist within the Panum space. Viewed in 3 dimen-sions, this serves to create a perturbation of the sur-

Fig 1.—An ellipse is a curve formed around focal points F′ and F where the sum ofF′P plus FP is a constant.

Fig 2.—Ends of a string are tacked at focus points F′ and F. The string is pulled tautwith the end of a feather pen at point P. While keeping the string taut, we use the pento trace 360° around the tacks. This circumference forms an ellipse.

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face of an ellipsoid shell that exists within a 3DPanum space.

Another basic premise of geometry is the conceptof increasing or decreasing the total length of F′P +FP (Fig 1). In other words, if F′P and FP were a con-tinuous piece of string that had a pen holding thestring taught at P, what would happen if the totallength of the string were increased or decreased? Theanswer is shown in Figure 4. If we decrease the totallength of the string, the effective area of the ellipse isdecreased. However, if we increase the total length ofthe string, the total area of the ellipse is increased.This basic principle of geometry explains the effect ofabduction and adduction of the eyes with regard totheir effect on the horopter of vision. If we adduct theglobes, the geometric lines emanating from thefoveas will intersect nearer to the face and therebyform an elliptical horopter that is closer to the eyesand that possesses a smaller area within the confines

of the ellipse. On the other hand, if we abduct theglobes, the geometric lines emanating from eachglobe would intersect in space at a distance much far-ther away from the globes. Thus, an ellipse of visionwould be formed that is much farther away from theface and that contains a greater area within the con-fines of the ellipse.

The arrangement of corresponding retinal points inhumans favors stability in the geometric configura-tion of the ellipse of vision. Since temporal retinalpoints in 1 eye correspond to nasal retinal points inthe second eye, this histologic arrangement serves tostabilize the distance between the retinal focal pointsin 1 eye to those in the other eye. This stabilization ofdistances between retinal focal points responsible forthe ellipse of vision is quite important. Because achange in focal point separation alters the shapeeccentricity of the ellipse geometry, an increase infocal point separation would produce a more elongat-ed ellipse. Bringing the focal points closer together,however, would produce a more circular ellipse.

Extending our argument to a 3D view of thehoropter, we may best explain this entity by a qua-dratic structure known as an ellipsoid (Fig 5). Theimage from the ellipsoid horopter is projected onto theretina. An ellipsoid may be generated by turning theellipse 180° around its longest axis, thereby forming astructure similar in appearance to a watermelon. Sim-ilarly, the inner surface of the retina is itself a sectionof a perturbed ellipsoid, thereby providing an idealsurface on which to project the 3D image of visionfrom the ellipsoid horopter. In actuality, there is a con-tinuous array of corresponding nodal points on eachretina that serve to generate a continuous array ofellipses that exist within the Panum space. Viewed in3 dimensions, this serves to create a perturbation ofthe surface of an ellipsoid shell that exists within a 3DPanum space. Obviously, using all correspondingpoints on the retinas will create a spacial array of

Fig 4.—Original ellipse is the middle ellipse designated by points F′, F, and P. If a per-son’s eyes are abducted, the line of sight of each eye is rotated outward and createsan ellipse with a larger area, as designated by points F′, F, and Q. If a person’s eyesare adducted, the line of sight of each eye is rotated inward and creates an ellipsewith a smaller area as designated by points F′, F, and q.

Fig 5.—Ellipsoid. X, Y, and Z graph coordinates are shown. Straddling the Y axis,ellipse a is drawn along the X axis and ellipse c is drawn along the Z axis. Imaginerotating either ellipse a or c for an excursion of 180° around the Y axis. Point o definesthe center of the ellipsoid. Ellipse b is used to assist the reader to perceive depth inthe diagram.

Fig 3.—Question: What happens if a point of light, shown here as a lightbulb, isplaced at 1 focal point of an ellipse (point F) and the inner surface of the ellipse ismade into a reflective surface? Answer: All light rays originating from the lightbulbwill reflect off the inner surface of the ellipse and will reflect back onto the other focalpoint (F′).

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ellipses of vision. Accordingly, field of vision exhibitsa complex elliptical character.

As discussed, the Panum fusional space is based ona continuous spacial array of ellipses of vision. It isalso based on the fact that the retina structure has anintegrating character; for example, the fovea is not adiscrete, ideal point but represents an area whereinvision with only half a fovea still gives 20/20 visualacuity. It is also based on relative velocities in thevisual pathways. For example, the Pulfrich phenome-non provides an illusion of depth from a 2-dimension-al swinging pendulum. This may be seen in a normaleye by placing a red filter over 1 eye and a green filterover the other eye. This may also be seen in optic neu-ritis, wherein the diseased eye has an abnormal opticnerve signal velocity. As a matter of fact, electronicengineers have employed a similar concept for yearsand have used the oscilloscope to create Lissajous fig-ures, which mimic the Pulfrich phenomenon.8 Herein,2 sine waves are fed into the oscilloscope, 1 into thehorizontal electrode and the other into the verticalelectrode. If there is no phase angle between the 2 sinewaves, a straight line is created. If 1 sine wave arrivesat the oscilloscope slower than the other, the image onthe scope demonstrates a degree of depth such that a

45° phase difference (difference in signal velocity) cre-ates an ellipse on the scope screen, whereas a 90°phase difference creates a circle on the scope screen.Refractive issues such as chromatic aberration andmonochromatic aberration also contribute to thePanum area. Central nervous system processing andoculomotor activity likewise contribute to this space.

Our objective was to provide a discussion of visionthat may be supported by physics and euclideangeometry. The ellipse of vision accomplishes this goal.

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Menschen für Vorlesungen. Koblenz, Germany: Hölscher; 1840.5. Wright KW. Textbook of Ophthalmology. Baltimore, Md: Williams

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