Perception of Perspective Angles

Abstract

We perceive perspective angles, that is, angles that have an orientation in depth, differently from what they are in physical space. Extreme examples are angles between rails of a railway line or between lane dividers of a long and straight road. In this study, subjects judged perspective angles between bars lying on the floor of the laboratory. Perspective angles were also estimated from pictures taken from the same point of view. Converging and diverging angles were judged to test three models of visual space. Four subjects evaluated the perspective angles by matching them to nonperspective angles, that is, angles between the legs of a compass oriented in the frontal plane. All subjects judged both converging and diverging angles larger than the physical angle and smaller than the angles in the proximal stimuli. A model of shallow visual space describes the results. According to the model, lines parallel to visual lines, vanishing at infinity in physical space, converge to visual lines in visual space. The perceived shape of perspective angles is incompatible with the perceived length and width of the bars. The results have significance for models of visual perception and practical implications for driving and flying in poor visibility conditions.

Keywords: models; perspective angles; visual space.

Figure 1.

Geometries of spaces. The squares represent a large section of 2-D space in front of the head (gray half discs). Arrows indicate the viewing direction. (a) Physical space. Lines indicate visual directions (orange) and parallel lines (gray) that vanish at infinite distance. (b) Shallow visual space. Visual directions of (a) are unchanged but its parallel lines are rotated toward the visual directions. (c) Converged visual space. Visual directions and parallel lines of (a) are rotated toward the viewing direction. (d) Curved visual space. Crosses indicate rotated directions at several locations. Converging (red) and diverging (blue) angles in physical space (a) are larger in a shallow visual space (b), smaller in a converged visual space (c), and of equal size in a curved visual space (d).

Figure 2.

Stimuli. Pictures of converging bars (left figures) were taken from the three different eye heights used in the experiments. Right figures show similar pictures for diverging bars.

Figure 3.

Stimulus geometry. Boxes show stimulus geometries for physical (a, b) and depicted (c, d) bars viewed from one of the three eye and camera heights (e). (a) The subjects viewed (E represents the position of the cyclopean eye) physical bars (red) converging at A on the floor. Angle LA’R (blue) indicates the proximal stimulus of the bars representing the retinal stimulus and the stimulus on the picture taken by the camera. The plane of the proximal stimulus is determined by the positions L and R of the near ends of the bars and an orientation perpendicular to EA, the visual direction of A. (b) Stimulus geometry for a subject viewing diverging bars. (c) The subject viewed the proximal stimulus LA’R (blue) on a frontal screen. (d) Stimulus geometry for the subject viewing pictures of diverging bars. For reasons of clarity, the dimensions of bars and eye height are not to scale.

Figure 4.

Matched angles of converging bars. Panels show means (±1 SD) of angles matched by subjects A, B, C, and D. Judgments were made for physical (dots) and depicted (squares) bars. Colored horizontal lines indicate proximal angles at eye and camera heights of 0.75 m (blue), 1.15 m (green), and 1.65 m (red). Proximal angles of physical (darker lines) and depicted (lighter lines) bars show differences due to small errors in the camera positions. Gray lines indicate the angle between the bars on the floor.

Figure 5.

Matched angles of diverging bars. Panels show means (±1 SD) of angles matched by subjects A, B, C, and D. Judgments were made for physical (dots) and depicted (squares) bars. Colored horizontal lines indicate proximal angles at eye and camera heights of 0.75 m (blue), 1.15 m (green), and 1.65 m (red). Proximal angles of physical (darker lines) and depicted (lighter lines) bars may show differences due to small errors in the camera positions. Gray lines indicate the angle between the bars on the floor.

Figure 6.

Weights of the proximal angle in the matched angles. Panels show mean weights (+1 SD) for angles matched by subjects A, B, C, and D. Judgments were made of angles between physical (darker colored bars) and depicted (lighter colored bars) bars.