Visual outdoor space perception

Abstract

Visual space perception has been a topic of sustained research since the nineteenth century. Much of this research into the geometry of visual space, however, required observers to make judgments about spatial relationships between isolated points in total darkness. While a sizeable number of previous investigations have now explored visual space perception in outdoor natural environments, nearly all of the previous investigations evaluating the curvature of visual space have utilized only small numbers of observers. In the current experiment, a large number (30) of observers adjusted triangular configurations of markers in an outdoor field until they appeared either as equilateral or right triangles in depth. There was a wide range of outcomes, such that the observers' judgments were consistent with elliptic, Euclidean, and hyperbolic geometry. There is thus no single consistent relationship between physical space and perceived space. Furthermore, the geometry of visual space frequently changes as the size of spatial configurations is varied-for many observers, judgments for small configurations are consistent with elliptic or Euclidean geometry while judgments for large configurations are frequently consistent with hyperbolic geometry.

Fig. 1

Illustrations of possible outcomes of the task described by Blank. The observer views a triangular configuration defined by points A, B, and C. The observer first estimates the midpoints of sides AB and BC. These midpoints are referred to by points D and E. The distance d indicates the distance between the perceived midpoints. The observer then marks a point F that is located at a perceived distance d from A along the path from A to C. The observer then marks a point G that is located at a perceived distance d from C along the path from C to A. If the observers’ judgments are consistent with Euclidean geometry (top), then the adjusted points F and G should coincide, both being located at a point halfway between A and C. If the observers’ judgments are consistent with hyperbolic geometry (lower left), then the adjusted points F and G will not coincide and F will be located closer to A, while G will be located closer to C. If the observers’ judgments are consistent with elliptic geometry (lower right), then the adjusted points F and G will not coincide and G will be located closer to A, while F will be located closer to C.

Fig. 2

A photograph of the 89 m × 24 m grassy field where the experiment was conducted. This photograph was taken by the first author (J.F.N.) using an Apple iPhone XR digital camera.

Fig. 3

Depictions of equilateral triangles on curved surfaces. Notice on the left that an equilateral triangle on a hyperbolic surface (curved like a horse saddle) has vertex angles less than 60 degrees, while equilateral triangles on elliptic surfaces (see right) possess vertex angles greater than 60 degrees. In Euclidean geometry (where space is not curved) equilateral triangles always have vertex angles of 60 degrees. The degree of deviation from 60 degrees indicates the magnitude of the curvature of space. This original drawing was made by the second author Maria Carmichael.

Fig. 4

Overall (mean) results for the equilateral triangle task (left) and isosceles right triangle task (right); see text and Blank. The vertex angles at the observers’ location are plotted as a function of the triangle size in meters. If the judgments are consistent with Euclidean geometry, the angles at the observers’ location are 60 and 45 degrees for the equilateral and right triangle tasks, respectively. The dashed lines in each plot indicate the observer vertex angles consistent with Euclidean geometry. The error bars indicate ± 1 SE.

Fig. 5

Results for all 30 individual observers for the equilateral triangle task. The vertex angles at the observers’ location are plotted as a function of the triangle size in meters. If the judgments are consistent with Euclidean geometry, the angles at the observers’ location are 60 degrees. The dashed lines indicate the observer vertex angles consistent with Euclidean geometry. The observers who exhibit the typical pattern (highest vertex angles at the smaller stimulus sizes and whose judgments are consistent with hyperbolic geometry for the largest stimulus size) are shown on the left, while the remaining observers’ (who exhibit other response patterns) judgments are shown on the right.

Fig. 6

Results for all 30 individual observers for the isosceles right triangle task. The vertex angles at the observers’ location are plotted as a function of the triangle size in meters. If the judgments are consistent with Euclidean geometry, the angles at the observers’ location are 45 degrees. The dashed lines indicate the observer vertex angles consistent with Euclidean geometry. The observers who exhibit the typical pattern (highest vertex angles at the smaller stimulus sizes and smaller vertex angles for the larger stimulus sizes) are shown on the left, while the remaining observers’ (who exhibit other response patterns) judgments are shown on the right.

Fig. 7

Individual results (left) and overall results (right) for the isosceles right triangle task. The angles at the left triangle vertex are plotted as a function of the triangle size in meters. In this task the observers were asked to create triangles so that (1) the left and opposite sides of the triangle appeared equal in length, and (2) the left triangle vertex appeared to be a right angle (i.e., 90 degrees). One can readily see that while some individual observers created left vertex angles close to or larger than 90 degrees, that most of the observers produced left vertex angles smaller than 90 degrees (e.g., see overall results on the right). The error bars included in the overall results indicate ± 1 SE.