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ON SOVE PECULIARITIES OF PHANTOM PLANES.
We have already shown how the figures of a regularly figured plane, like a tessellated pavement or a regularly patterned carpet or papered wall, may be combined either by crossing the eyes or (if the figures be not too large) by looking beyond the plane of the figures, as in the stereoscope, so as to make phantoms, which are nearer or farther off, and the figures smaller or larger than reality, according to the degree of ocular convergence, and therefore the apparent distance of the phantom. In Fig. 48, page 134, we have represented these phantoms as planes parallel to the real plane; but if closely observed they are seen to be neither perfect planes nor parallel to the real plane. The phenomena now about to be described have, some of them, not been heretofore noticed, and none of them have been satisfactorily explained.
1. The Phantom Plane not parallel to the Real
Plane. Experiment 1.—I sit in a chair in the middle of a tessellated floor and direct the eyes on the floor at an angle of 45°. By ocular convergence I now combine successively the figures of the floor, stopping a little at each combination until the phantom image clears. These phantom floors are distinctly perceived to be not horizontal, as they ought to be by geometric construction, but inclined, dipping away from the observer at higher and higher angle, as by greater convergence the phantom floor comes nearer and nearer. I am sure that by extreme convergence I can make the phantom slope at an angle of 300-40°.
In addition to the slope of the plane, and closely connected therewith, another phenomenon is perceived. The figures change their shape, becoming elongated in the direction of the slope and in proportion to the angle of the slope. If, for example, the figures are regular squares, looked at in the direction of their diagonals they become greatly elongated rhombs, or if circles they become long ellipses.
Explanation. Principles.—We have already seen (pages 140 and 141) that a slender rod held horizontally in the median plane, but a little below the horizontal plane passing through the two eyes and looked at with both eyes, is seen thus or thus
according as we look at the nearer end or the farther end or the middle point. In this case-i. e., when the rod is horizontal—the angle between the two images is small.
If now the nearer end of the rod be lifted so as to bring it nearly in coincidence with the plane of sight, the angle between the images will become greater, but the vertical length of the projection less, until when the rod is in the plane of sight, the angle becomes 180° and the two images are in the same horizontal straight linè. Of course, to the binocular observer it does not seem like a horizontal straight line, because he introduces the element of depth of space by binocular perspective. To him it would look like a V or an X looked at end on.
Application of Principles in Explanation of the Slope.—The figures of a tessellated plane of course lie in parallel lines. We will suppose these lines to run from the observer. By geometric or monocular perspective such lines converge to a vanishing point on the horizon. Leaving out the figures of the pattern, Fig. 82 represents the projection of such parallels as seen with one eye. As seen with two eyes, of course,
there are two images of all these lines crossing one another at small angle, as shown above. Let us fix the mind on the
middle one, a, alone. In natural vision its two images would cross at small angle at the point of sight. But in making the phantom plane, 6 b or c cor d d or e e, etc., are brought together in the middle, combined and viewed as the middle line. But it is evident that the angle of perspective convergence, and therefore the angle of crossing one another when they come together, is greater and greater, as by greater convergence they are brought from greater distance right and left.
In other words, the perspective angle is added to the binocular angle and all is credited to the binocular angle, be cause viewed as a middle line which ought not to have any perspective angle at all. But we have already seen that the crossing of binocular images of lines at higher angle means a nearer approach to the plane of sight -a looking more end on. In other words, it means a lifting of the nearer end of the plane. Therefore as more and more separated lines of figures, b b, c c, d d, e e, etc., are brought forward and united in front, and the angle of crossing of the lines becomes greater, the slope of the phantom plane becomes greater, until, if we could bring together lines from an infinite distance, the phantom plane would coincide with the plane of sight—i. e., would slope 45°.
Elongation of the Figúres.—This follows as an obvious and necessary consequence of the slope. For since the angle subtended by the plane at the eye, or the retinal image, remains the same, the length of the plane and all its figures must seem greater in proportion to the degree of slope, precisely as shadows cast on a plane are longer in proportion as the angle of the light to the plane becomes less.
Experiment 2.-In experimenting with the floor, the observer's body prevents viewing the phantom in the contrary direction. Therefore we take next a vertical wall, such as a regularly patterned wall-papering or a coarse wire-netting. The windows of the basement of one of the university buildings are protected by a coarse wire-netting with lozenge-shaped meshes about 21 inches in their shorter or horizontal diameter. Standing before this and combining by extreme convergence, on looking upward the phantom slopes away upward; looking downward, it slopes away downward. So that sweeping the plane of sight upward and downward alternately the phantom plane seems to dip away up and down from an anticline, or arch. The explanation of this is, of course, the same as that already given in the case of the floor.
By careful experiment it is found that the top of the arch is not on a level with the eyes, but a little above, making with the horizontal an angle of about 70. This is the result of the rotation of the eyes on the visual axes in convergence, already demonstrated on pages 199–212, and is a beautiful proof of such rotation.
2. The Phantom not a Plane. Experiment 3.—Instead of looking obliquely on the experimental plane we next look perpendicularly on it, or, more accurately, 7° inclined upward. By extreme convergence in this position the phantom plane is seen to slope away on either side so as to form a sort of saddle. Similarly, on looking upward or downward, the sloping plane is not a perfect plane, but bulged a little along the middle line. Sweeping the point of sight about in all directions, all these effects are combined and the phantom surface slopes away in all directions, forming a mound.
Explanation of the Transverse Arching. It will be remembered that impressions on non-corresponding points produce double images; moreover, that when the non-corresponding points are nearer together than corresponding points or central spots, the double images are homonymous, and when farther apart than central spots they are heteronymous; and still further, that homonymously double images mean that the object which produces them is farther away than the point of sight, while heteronymously double images indicate