The mathematics

Each visual axis has a separate rotation in order to form the central location

Target 1 and target 2 are viewed ‘D1’ distance left, right of ‘F’ with the monocular view of the left, and right eye, than the fixation point. Target 1 is only viewed with the left eye because it is occluded from the view of the right eye and target 2 is only viewed with the right eye as it is occluded from the view of the left eye. In figure 5.16B, the right, and left eye view target 3, and target 4, farther than the fixation point. When ‘F’ is focused upon, target 1 and target 2, and target 3 and target 4, are viewed in the cortex eye the same D1, ‘D2’ distance (minus ‘D’ distance) left and right of ‘F’, nearer and farther than the fixation point. They are viewed in the same locations as they would if the occlusion were not present, in figure 5.16B.

Figure 5.16: Why each axis has a separate rotation.

The left and right eyes view target 1 and target 2 separately and not binocularly. They have moved the same ½ ‘D’ distance with the rotation of the visual axes of the left and right eyes. It proves that the visual axis of each eye rotates to a central location independently and it proves that each eye’s view has a separate, mathematical computation for the measurement of depth nearer to the fixation point. This diagram is also important in proving the mathematical computation of targets, located in the monocular areas occluded by the nose nearer than the fixation point, documented later in this chapter.

In figure 5.17, target 3 and target 4 are viewed ‘D2’ distance left, right of ‘F’ with the monocular view of the right, and left eye, farther than the fixation point. Target 3 is only viewed with the right eye as it is occluded from the view of the left eye, and likewise, target 4 is only viewed with the left eye because it is occluded from the view of the right eye. In figure 5.17A, target 1 and target 2 are viewed ‘D1’ distance left, right of ‘F’ by the right, and left eye, nearer than the fixation point. When ‘F’ is focused on in the cortex eye, the two sets of targets, target 1 and target 2, and target 3 and target 4, are viewed the same ‘D1’, ‘D2’ distance left, and right of ‘F’, minus ‘D’, respectively, nearer and farther than the fixation point.

Figure 5.17: Each axis has a separate rotation.

Target 3 and target 4 are viewed in the same locations if the occlusions are not present, in figure 5.17B. Target 3 and target 4 and are viewed separately, not binocularly, by the right and left eye. They move the same ½ ‘D’ distance with the rotation of the visual axes of each eye, and this proves that each eye’s view has a separate mathematical composition for the measurement of depth, both nearer and farther than the fixation point.

Disparity movement creates our visual perception of depth in our three-dimensional world. This is achieved primarily by the rotation of the visual axes of the right eye and the left eye, as they intersect and pass though the fixation point, when they come together as a single central axis in the cortex eye. The distance between these axes is the disparity, which equals the perceived movement between objects at their precise locations in depth along these axes, on either side of the retinal-section boundaries VL (the visual axis of the left eye nearer than the fixation point) and VR (the visual axis of the right eye farther than the fixation point). Unlike retinal disparity, the disparity created in SSSRD is the different angle that each separate, retinal area is viewed at and the movement of all these retinal-sections is created by the rotation of the visual axes in the cortex eye. This movement extends right into the distance for as far as the two eyes can see with clarity. The pivot on which the visual axes rotate is the fixation point. This rotation results in a disparity movement between all targets in right and left retinal-sections. The disparity movement is evident during every change in the fixation point. It becomes dramatically evident during monocular interchange whereby targets change from the monocular area of one eye to an opposite monocular area of the other eye.

As soon as the eyes converge on a fixation point, the visual field divides into six separate, opposite monocular sections; A and A1 of the right eye, B and B1 of the left eye, and two monocular sections of the left and right eyes, the latter two resulting from the occlusion of the nose. At the same moment fixation occurs, the two axes that intersect and pass through the fixation point rotate clockwise and anti-clockwise to form a single central axis. The pivot of this rotation is the fixation point. The central axis consists of the two entire visual axes, which includes the two retinal boundaries VL and VR. As illustrated, VL and VR are part of the visual axes of the left and right eyes nearer and farther than the fixation point, respectively.

There is no correspondence or matching problem in the single image, including in the overlapping area. Every target and part of the image is viewed separately by each eye, in opposite, separate, monocular areas, simultaneously in the cortex eye. Similarly, the same eye also separately views all targets in the area of the binocular single image that lies between the intersecting visual axes as they were before the overlap that occurs with the function of the cortex eye. Because of the rotation, all targets on opposite sides of the retinal boundaries VL and VR are perceived to be closer together in the single image (than they are observed by the monocular view of either eye) nearer and farther than the fixation point. The distance that they are perceived to be closer together is ‘D’ distance, which equals the rotation of the visual axes at their depth locations nearer and farther than the fixation point. This ‘D’ distance is the sum of two ½ ‘D’ distance movements of each axis. The only two retinal-sections where this does not occur are the two monocular areas farther than the fixation point (illustrated later in this chapter, using occlusion).

The left monocular area moves with the anti-clockwise rotation of the left visual axis and the right monocular area moves with the clockwise rotation of the right visual axis. This in turn results in targets in these two areas moving outwards, farther than the fixation point, which is the opposite directional movement to that of all the other targets in the binocular field nearer than the fixation point. The near side of the fixation point, the monocular and binocular areas are viewed by the same eye as they adjoin the retinal-sections of section B and section A in the binocular field, which are both viewed by the left and right eyes, respectively. As a result, all targets move inwards with the rotation of the visual axes nearer than the fixation point. In short, all retinal-sections in the binocular field and monocular areas, including occluded zones in these areas, move ½ ‘D’ distance with the rotation of the visual axis of the eye that they are viewed with.

The rotation of the visual axes in the cortex eye creates a very precise movement change in the already assembled dominant retinal-sections. This movement change creates a change in position of all targets in a mathematical structure by which depth and distance judgement is coded by the brain, in an already assembled stereo single image. As illustrated, this movement change in targets resulting from the rotation of the visual axes in the cortex eye is not restricted to the binocular field and it occurs in the monocular areas. This means that depth and distance judgements are measured in the monocular areas in exactly the same way as in the binocular areas and this proves that targets in the monocular areas are linked to the rotation of each individual axis in the same way as targets in the binocular field.

The eyes converge to different degrees in order to focus on a near or far target. By understanding SSSRD and how these four opposite, separate and monocular retinal areas come together in the cortex eye to form BSV, we can logically understand and prove the mathematical process by natural observation. All the dominant snapshot images of both eyes representing the capacity of one single eye are precisely assembled in the visual cortex before the function of the cortex eye occurs. Integral to this assembly are the precise original locations of the entire two visual axes when all the separate snapshots of the retinal-sections of both eyes were taken. The spines of these retinal-sections are the visual axes as their movement in the rotation of the axes in the cortex eye dictates the movement of all retinal-section divisions, creating the mathematical structure by which depth perception is measured.

·    The assembly of snapshots viewed at different angles is a stereo image.

·    The function of the cortex eye results in the two visual axes rotating ½ ‘D’ distance in clockwise and anti-clockwise directions to become a straight central axis. The pivot of their rotation is the fixation point. The rotation is a very precise and accurate movement. Each of the axes moves exactly the same distance, which is ½ ‘D’ distance in opposite directions.

 ·    All targets in the dominant retinal-section snapshot areas of the right eye move ½ ‘D’ distance relative to their depth locations, anti-clockwise with the visual axis of the left eye. All targets in the dominant retinal-section snapshot areas of the left eye move ½ ‘D’ distance relative to their depth locations, anti-clockwise with the visual axis of the right eye.

·    The equal rotation of each axis to a single central axis not only means that the convergent angle of each eye is accurately measured, but the ½ ‘D’ distance movement of every target in each sectional area of each eye is accurately recorded during every fixation.

·    The clockwise and anti-clockwise rotation in the cortex eye during every fixation, results in continuous motion in an already created stereo image. This continuous motion is always occurring with the slightest movement of the eye, even when the head is in a static position. The movement of the eye creates a new mathematically structured movement in the visual field.

·    Occluded targets move in opposite ½ ‘D’ directions to all other targets in the retinal-section that they are occluded in. The opposite ½ ‘D’ movements arise because they are attached to the opposite visual axis, but they have the same structured, mathematically computed movement in the visual field.

·    The convergent angle of each eye is equal, nearer and farther than the fixation point in the cortex eye, as opposite angles are equal. Consequently, the same mathematical computation exists inversely nearer and farther than the fixation point, when the visual axes intersect and pass through it.

All of these proven observations create a logical understanding as to how the stereo-image is formed, how the cortex eye functions, how occlusions occur, how transparency occurs, how visual direction is judged, how optic flow is created and, most important of all, how the brain can mathematically compute our depth perception. As a result of my discovery of SSSRD in coordination with the cortex eye, a precise mathematical structure by which direct, visual sensory perception is created, for understanding instant depth, distance, textures and angular movement. This sensory perception is judged by the fast changing formations of textures, surfaces and objects, and their perceived movements relative to each other utilising the incredibly fast functional speed of the cortex eye. This sensory perception is also greatly enhanced with the constant change in the image resulting from the constant movement of retinal-sections with every change in the fixation point.

Not only do targets change from a retinal-section of one eye to an opposite retinal-section of the other eye, but the vantage angle of view of the targets also changes. The function of the cortex eye during every fixation creates perpetual, mathematically computed judgements of depths and distances. The visual system must work in exactly the same way for every animal with front-loaded eyes, regardless of how far their eyes are apart or how large or small their binocular overlap may be. The movement in distance is always a very consistent movement. The perceived movement is represented by ½ ‘D’ distance at the depth location of each target nearer or farther than the fixation point. This perceived movement is represented in the entire visual field in a single cortex eye, which represents each eye separately. Thus, the movement is varied in direction, but consistent in computing the depths and distances for all targets at their depth locations nearer and farther than the fixation point. This is again the case because the same mathematical formula exists for all targets viewed by each eye separately in the single image.

Figure 5.18: Visual axes passing through the fixation point.

In the cortex eye, the left and right eyes have two separate, opposite rotational movements. Figure 5.18A illustrates the two visual axes as they intersect and pass through the fixation point. Figure 5.18B illustrates that they are two separate identities, even though they are interlinked. That is, the visual axis rotates all the retinal-sections of that eye. The two axes rotate on pivot ‘F’ and form a central axis, but each axis carries with it all the retinal-sections of that eye, creating opposite directional movements between all these sectional areas.

Each visual axis rotates to form a single axis at every angle of view and the distance of the rotation at every depth location, as well as the closing angle at ‘F’, are accurately measured. Each axis moves by a precise ½ ‘D’ distance. Therefore, every angle of convergence for each eye is accurately measured in a fixation, and it is the same angle for each eye. As opposite angles are equal, the angle of convergence is mirrored by the same angle of convergence farther than the fixation point. That is, the convergent angles of the right eye in sections A and A1 are equal, and the convergent angles of the left eye in sections B and B1 are equal, nearer and farther than the fixation point.

All targets are viewed at right angles to the visual axis of each eye in the cortex eye and all targets move with the rotation of each visual axis to a central location, forming a central axis. This has the effect of each visual axis creating a right-angle triangle nearer and farther than the fixation point during each fixation. The right angle triangle, created by each visual axis nearer than the fixation point, inversely mirrors the right angle triangle created by the same visual axis farther than the fixation point. In the cortex eye, each eye computes it own depth judgement for every target in the retinal-sections of the right and left eye. In every fixation, the same mathematical, computational formula measures the depth of all targets and surfaces at their individual depth locations nearer and farther than the fixation point, throughout the entire visual field. The mathematical formula and the ½ ‘D’ distance movement of all targets at their depth location in the retinal-sections of each eye in each fixation are illustrated in the following figures.

In figure 5.19, the two eyes focus on ‘F’ and the distances of target 1 and target 2 (located nearer than ‘F’), and target 3 and target 4 (located further than ‘F’) are unknown. When the function of the cortex eye occurs, the two visual axes rotate clockwise and anti-clockwise to become a central axis. For the two axes to become one central axis, the convergent angles of the two eyes on ‘F’ have to close to zero. This means the movement of the two axes to a central location and the closing of the angle of convergence and that the opposite angle is registered in the brain.

Figure 5.19: Depth measurements of targets nearer and farther than the fixation point.

The convergent angle of each eye is accurately registered during each fixation. The ½ ‘D’ distance rotation, combined with the convergent angle of each eye, creates a mathematical process by which the brain can compute depth in three-dimensional space. This diagram illustrates the structure of this mathematical process in the binocular field, nearer and farther than the fixation point. In figure 5.19A, the convergent angle of target 1 is angle A and the converging angle of target 2 is angle B. Target 1 has a rotational movement of ½ ‘D’ and this supplies a mathematical computation to the brain. The perpendicular distance of target 1 to the fixation point equals [½ ‘D’ ÷ tan ∠ bº] and the distance between target 2 and the fixation point is equal to [½ ‘D’ ÷ tan ∠ aº].

For targets farther than the fixation point, the mathematical computation is inversed. That is the distance measurements are from the focal plane not the fixation point. Target 3 has a rotational movement of ½ ‘D’, this supplies a mathematical computation to the brain that the distance of target 3 from the focal plane equals [½ ‘D’ ÷ tan ∠ a1º] and the distance of target 4 from the fixation point equals [½ ‘D’ ÷ tan ∠ b1º]. As all opposite angles are equal, [∠ bº = ∠ b1º] and [∠ aº = ∠ a1º]. The following figure 5.19A, illustrates how the visual system, knowing the converging angle and the ½ ‘D’ rotation distance of each target, is enabled to compute much more information other than the perpendicular distance to and from the fixation point regarding these same targets.

Figure 5.19A: The vantage angle view of a target by each eye is computed.

The four targets above are at different depth locations nearer and farther than the fixation point. Target 1 and target 4 rotate anti-clockwise, ½ ‘D’ distance with the visual axis of the left eye, while ttarget 2 and target 3 rotate clockwise with the rotation of the visual axis of the right eye. This rotational movement creates the mathematical computation of a right angle at the depth location of each target, the apex of each triangle being the convergence of each eye at the fixation point. All the right angle triangles are different, but the rotation of the axes supplies the visual system with accurate measurements of all the angles and the sides of each triangle.

The ½ ‘D’ rotational movement supplies the dimensions of the base of each triangle at its depth location, nearer and farther than the fixation point. The same rotational movement supplies the converging angle of each eye. The formula [½ ‘D’ ÷ tan ∠ º] is used for each converging angle, and supplies the perpendicular distance of all the targets nearer, and inversely farther than, the fixation point. In turn, this computation supplies the visual system with the vantage angle distance, ‘C’, at which each target is viewed at its depth location. It also supplies the distance of the slope (hypotenuse) of each target at its depth location (the square of the other two sides equals the square of the hypotenuse).

Our perception of the surfaces and textures of all targets have hundreds and thousands of variations in every fixation. The same mathematical computation exists for every target, texture, and surface in that split second when dominant, separate, retinal-section images are transmitted to the visual cortex, and the function of the cortex eye occurs. We will also observe later in this chapter, that the same mathematical computation also exists for all targets in the two monocular retinal-sections and for all targets in occlusion zones in all retinal-sectioned areas.