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Exocentric pointing in the visual field.

van Doorn A, Koenderink J, Wagemans J - Iperception (2013)

Bottom Line: Phenomenologically, such pointings show systematic deviations from veridicality of several degrees.The errors are very small in the vertical and horizontal directions, but appreciable in oblique directions.A general conclusion is that the visual field cannot be described in terms of one of the classical homogeneous spaces, or, alternatively, that the results from pointing involve mechanisms that come after geometry proper has been established.

View Article: PubMed Central - PubMed

Affiliation: Laboratory of Experimental Psychology, University of Leuven (KU Leuven), Tiensestraat 102 box 3711, B-3000 Leuven, Belgium; and Faculteit Sociale Wetenschappen, Psychologische Functieleer, Universiteit Utrecht, Heidelberglaan 2, 3584 CS Utrecht, The Netherlands; e-mail: andrea.vandoorn@telfort.nl.

ABSTRACT
"Exocentric pointing in the visual field" involves the setting of a pointer so as to visually point to a target, where both pointer and target are objects in the visual field. Phenomenologically, such pointings show systematic deviations from veridicality of several degrees. The errors are very small in the vertical and horizontal directions, but appreciable in oblique directions. The magnitude of the error is largely independent of the distance between pointer and target for stretches in the range 2-27°. A general conclusion is that the visual field cannot be described in terms of one of the classical homogeneous spaces, or, alternatively, that the results from pointing involve mechanisms that come after geometry proper has been established.

No MeSH data available.


Related in: MedlinePlus

The figure at left shows the case illustrated in Figure 7 (Lobachevsky's hyperbolic plane). The figure at the center shows a highly magnified view of the center. It looks like the simple Euclidean configuration of half-lines fanning out from a point, uniformly spreading over all directions. This occurs for any Riemannian (non-Euclidean) case. The figure at right shows an extreme case of non-Euclidean geometry. Here the geodesics “fanning out” from the point at center all start out in either a horizontal or vertical direction. There is no Riemannian space at all that would look like this. Thus the case at right is worse than a mere arbitrary warping of the Euclidean plane; it is even more “non-Euclidean” than that. Whether one would consider calling it a “plane” at all is a matter of choice.
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Figure 9: The figure at left shows the case illustrated in Figure 7 (Lobachevsky's hyperbolic plane). The figure at the center shows a highly magnified view of the center. It looks like the simple Euclidean configuration of half-lines fanning out from a point, uniformly spreading over all directions. This occurs for any Riemannian (non-Euclidean) case. The figure at right shows an extreme case of non-Euclidean geometry. Here the geodesics “fanning out” from the point at center all start out in either a horizontal or vertical direction. There is no Riemannian space at all that would look like this. Thus the case at right is worse than a mere arbitrary warping of the Euclidean plane; it is even more “non-Euclidean” than that. Whether one would consider calling it a “plane” at all is a matter of choice.

Mentions: This is a surprising conclusion with far-reaching implications. Here is why: The visual field is apparently unlike the Euclidean plane. Can it be described as a “geometry” at all? One would expect so given the fact that this is generally assumed in virtually every textbook, with the exceptions of non-generic cases (e.g., amblyopia or tarachopia; Hess, 1982). Here we might first consider the simplest “geometries,” which are the classical homogeneous spaces. These are spaces that appear the same from any point, and that admit of a continuous group of “movements,” or “congruences.” There are three qualitatively distinct contenders for the (non-degenerate) geometry of the plane (Coxeter, 1961, 1965).2 In each of these the infinitesimal neighborhood of any point is Euclidean; thus a unique geodesic passes through any point in any given direction.3 The Euclidean straight lines through the origin cannot be geodesics of some simple geometry, because then our results reveal the neighborhood of the origin to be non-projective (Coxeter, 1949).4,5 (See Figure 8.) But when the Euclidean line is not a geodesic itself, each of its points lies on a distinct geodesic (like the configuration in Figure 7), and points at different distances from the origin would be at different directions from the origin. However, this is contradicted by our empirical results. The conclusion is that the visual field cannot be described as one of the classical planar geometries at all. An extreme example of the kind of structure we mean here is illustrated in Figure 9. This is a very general, and perhaps unexpected finding.


Exocentric pointing in the visual field.

van Doorn A, Koenderink J, Wagemans J - Iperception (2013)

The figure at left shows the case illustrated in Figure 7 (Lobachevsky's hyperbolic plane). The figure at the center shows a highly magnified view of the center. It looks like the simple Euclidean configuration of half-lines fanning out from a point, uniformly spreading over all directions. This occurs for any Riemannian (non-Euclidean) case. The figure at right shows an extreme case of non-Euclidean geometry. Here the geodesics “fanning out” from the point at center all start out in either a horizontal or vertical direction. There is no Riemannian space at all that would look like this. Thus the case at right is worse than a mere arbitrary warping of the Euclidean plane; it is even more “non-Euclidean” than that. Whether one would consider calling it a “plane” at all is a matter of choice.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4129387&req=5

Figure 9: The figure at left shows the case illustrated in Figure 7 (Lobachevsky's hyperbolic plane). The figure at the center shows a highly magnified view of the center. It looks like the simple Euclidean configuration of half-lines fanning out from a point, uniformly spreading over all directions. This occurs for any Riemannian (non-Euclidean) case. The figure at right shows an extreme case of non-Euclidean geometry. Here the geodesics “fanning out” from the point at center all start out in either a horizontal or vertical direction. There is no Riemannian space at all that would look like this. Thus the case at right is worse than a mere arbitrary warping of the Euclidean plane; it is even more “non-Euclidean” than that. Whether one would consider calling it a “plane” at all is a matter of choice.
Mentions: This is a surprising conclusion with far-reaching implications. Here is why: The visual field is apparently unlike the Euclidean plane. Can it be described as a “geometry” at all? One would expect so given the fact that this is generally assumed in virtually every textbook, with the exceptions of non-generic cases (e.g., amblyopia or tarachopia; Hess, 1982). Here we might first consider the simplest “geometries,” which are the classical homogeneous spaces. These are spaces that appear the same from any point, and that admit of a continuous group of “movements,” or “congruences.” There are three qualitatively distinct contenders for the (non-degenerate) geometry of the plane (Coxeter, 1961, 1965).2 In each of these the infinitesimal neighborhood of any point is Euclidean; thus a unique geodesic passes through any point in any given direction.3 The Euclidean straight lines through the origin cannot be geodesics of some simple geometry, because then our results reveal the neighborhood of the origin to be non-projective (Coxeter, 1949).4,5 (See Figure 8.) But when the Euclidean line is not a geodesic itself, each of its points lies on a distinct geodesic (like the configuration in Figure 7), and points at different distances from the origin would be at different directions from the origin. However, this is contradicted by our empirical results. The conclusion is that the visual field cannot be described as one of the classical planar geometries at all. An extreme example of the kind of structure we mean here is illustrated in Figure 9. This is a very general, and perhaps unexpected finding.

Bottom Line: Phenomenologically, such pointings show systematic deviations from veridicality of several degrees.The errors are very small in the vertical and horizontal directions, but appreciable in oblique directions.A general conclusion is that the visual field cannot be described in terms of one of the classical homogeneous spaces, or, alternatively, that the results from pointing involve mechanisms that come after geometry proper has been established.

View Article: PubMed Central - PubMed

Affiliation: Laboratory of Experimental Psychology, University of Leuven (KU Leuven), Tiensestraat 102 box 3711, B-3000 Leuven, Belgium; and Faculteit Sociale Wetenschappen, Psychologische Functieleer, Universiteit Utrecht, Heidelberglaan 2, 3584 CS Utrecht, The Netherlands; e-mail: andrea.vandoorn@telfort.nl.

ABSTRACT
"Exocentric pointing in the visual field" involves the setting of a pointer so as to visually point to a target, where both pointer and target are objects in the visual field. Phenomenologically, such pointings show systematic deviations from veridicality of several degrees. The errors are very small in the vertical and horizontal directions, but appreciable in oblique directions. The magnitude of the error is largely independent of the distance between pointer and target for stretches in the range 2-27°. A general conclusion is that the visual field cannot be described in terms of one of the classical homogeneous spaces, or, alternatively, that the results from pointing involve mechanisms that come after geometry proper has been established.

No MeSH data available.


Related in: MedlinePlus