**Objects are often identified by the shape of their contours. In this study, visual search tasks were used to reveal a visual dimension critical to the analysis of the shape of a boundary-defined area. Points of maximum curvature on closed paths are important for shape coding and it was shown here that target patterns are readily identified among distractors if the angle subtended by adjacent curvature maxima at the target pattern's center differs from that created in the distractors. A search asymmetry, indicated by a difference in performance in the visual search task when the roles of target and distractor patterns are reversed, was found when the critical subtended angle was only present in one of the patterns. Performance for patterns with the same subtended angle but differing local orientation and curvature was poor, demonstrating insensitivity to differences in these local features of the patterns. These results imply that the discrimination of objects by the shape of their boundaries relies on the relative positions of their curvature maxima rather than the local properties of the boundary from which these positions are derived.**

^{2}ambient luminance) observers viewed a Sony G520 CRT monitor from a distance of 65.5 cm, constrained by the position of a chin rest. At this distance, each pixel of the monitor subtended 2 minutes (2′) of visual angle. The screen comprised 768 × 1,024 pixels refreshed at 100 Hz. Visual stimuli were presented to the screen from the frame buffer of a Cambridge Research Systems (CRS) ViSaGe visual stimulus generator (Cambridge Research Systems, Rochester, Kent, UK). Observers made their responses using a CRS CB6 button box (Cambridge Research Systems).

^{2}). Luminance calibration was performed using a CRS Optical and associated software.

*R*(

*θ*) is the radius at an angle

*θ*relative to the positive

*x*-axis,

*R*is the unmodulated radius,

_{0}*A*is the amplitude of modulation,

*ω*is the frequency of modulation, and

*ϕ*is the phase of modulation of the pattern (controlling the orientation of the pattern in the frontoparallel plane—the RF3 pattern illustrated in Figure 1 is in zero phase). An

*R*value of 60′ was used for both RF patterns. Different integer frequencies of modulation (

_{0}*ω*) create patterns with different characteristic shapes. The RF patterns were given amplitudes of modulation, defined as a proportion of the unmodulated radius, of 1/(1 +

*ω*

^{2}), where

*ω*is the frequency of modulation (cycles of modulation in 360°). At these amplitudes the patterns have a curvature, a rate of change of orientation with

*θ*, of zero at the minima of the modulation of radius and therefore resemble familiar geometrical shapes with flat sides but smoothed corners (Dickinson et al., 2012). An RF4 pattern, for example, resembles a square but with rounded corners (the RF4 pattern is not depicted but contains the same local features as the RF3 and differs in periodicity of those local features). Curvature defined as the rate of change in orientation with

*θ*will differ slightly from curvature defined as the rate of change of orientation with path length. However, curvature discontinuities require abrupt changes in orientation and therefore the curvature discontinuities are constrained to particular points defined in polar coordinates. Similarly, the points of maximum curvature occur at points where the tangent to the path is perpendicular to the local radius, so RF patterns are characterized by the existence of points of maximum curvature and their relative positions. RF patterns with differing frequency have differing configurations of points of maximum curvature. The curvature around the pattern is, however, continuous, so RF patterns do not contain curvature discontinuities. Because the points of maximum curvature repeat periodically around the patterns, the RF patterns also contain a shape cue defined by the relative positions of the adjacent points of maximum curvature (Dickinson et al., 2013). This cue can be described in a manner that is invariant to the orientation of the pattern, as the angle subtended at the center of the pattern by adjacent points of maximum curvature. For RF3 patterns this angle is 120° and for RF4 patterns 90°.

*ω*= 3. To control for the range of local orientation differences from a circle, the amplitude,

*A,*was double that used for the RF3 patterns (the frequency of the modulation is halved resulting in the magnitude of the gradient of the function being reduced by half and so a doubling of amplitude is applied to compensate). This results in the patterns having the same histogram of local orientation difference from circular, precluding the use of orientation statistics to differentiate the patterns. So that the RRF3 patterns approximated the RF3 patterns in size, a value of 54′ was used for

*R*for the RRF3 patterns. Rectification of the sinusoidal modulation creates curvature discontinuities at the zero crossings of the sinusoidal modulator. In contrast to the K&T patterns, these curvature discontinuities are also orientation discontinuities. These rectified RF patterns, therefore, have curvature discontinuities that transition from continuous curvature to effectively infinite curvature and then back again at their points of maximum curvature. They also contain the same shape cue as the RF3 pattern. That is, the angle between adjacent points of maximum curvature is 120°.

_{0}- A pairing of K&T patterns and ellipses: This pattern pairing allows us to verify the search asymmetry reported by Kristjánsson and Tse (2001) for these patterns (see the top row of Figure 2). Both patterns contain points of maximum curvature (indicated by black triangles) but only the K&T stimulus contains curvature discontinuities (white triangles). An impression of the expected result can be obtained from inspecting the example arrays. The K&T pattern in (a) is readily identified among ellipses but the ellipse in (b) is hard to find among K&T patterns.
- A pairing of RF3 patterns with circles: This pairing allows us to ensure that a comparable pattern of results may be obtained from a pair of patterns that are not differentiated by the existence of curvature discontinuities in one but not the other (see the second row of Figure 2). The RF3 pattern contains points of maximum curvature but the circle does not. Again, inspection suggests that the RF3 pattern in (c) is readily identified but the circle in (d) is hard to find.
- A pairing of RF3s and RF4s (see the third row of Figure 2): The patterns of this pairing vary the number of cycles of sinusoidal modulation used to complete the pattern. Curvature around both patterns is continuous but the RF3 patterns contain three points of maximum curvature and the RF4 patterns four. As well as differing in the number of points of maximum curvature, the patterns also differ substantially in the angles subtended at the center of the patterns by adjacent points of maximum curvature. This pattern pair allows us to verify that patterns with differing frequencies of modulation are rapidly discriminated. Inspection suggests the RF3 and RF4 are readily identified in (e) and (f), respectively.
- A pairing of RF3s and RRF3s: This pattern pairing examines whether the existence of curvature discontinuities (or indeed, orientation discontinuities - points of infinite curvature) is sufficient for supporting rapid identification of RRF3s among RF3 patterns (see the bottom row of Figure 2). RRF3 patterns have points of maximum curvature that are coincident with curvature discontinuities, while RF3 patterns have similarly distributed points of maximum curvature that are not curvature discontinuities. Inspection suggests that both the RF3 and RRF3 patterns are hard to find in (g) and (h), respectively.

*t*-test results demonstrating that performance in the forward and reverse conditions do not differ significantly for the RF3/RF4 and RF3/RRF3 pattern pairs but do differ for the K&T/ellipse and RF3/circle pattern pairs.

*t*-tests comparing the forward and reverse condition gradients.

*t*-tests are similar to those reported in Table 1, which compared gradients of functions fitted to response times for all SSs. Again, the gradients for the K&T/ellipse and RF3/circle pattern pairs are asymmetrical across the forward and reverse conditions. The RF3/RF4 pattern pair gradients are, however, also shown to be asymmetrical for target-absent trials and RF3/RRF3 pattern pair gradients for target-present trials. These results are inconsistent across present and absent trials and the ratios between forward and reverse conditions are small, as can be seen in Figure 5.

- A pairing of RF3 patterns with ellipses: This pairing has the potential to provide supporting evidence that the RF3 pattern contains a shape cue that allows for its rapid identification among patterns that do not contain the same shape cue. Both patterns contain points of maximum curvature but on the basis of the results of Experiment 1, we expect that the angle of 180° subtended by the two points of maximum curvature in the ellipse will not constitute a usable shape cue. The angle of 120° subtended by adjacent points of maximum curvature in the RF3 pattern is, however, expected to characterize its shape and allow it to be rapidly identified among ellipses.
- A pairing of ellipses with circles: Making the assumption that the ellipse is devoid of the postulated shape cue, this pairing is expected to show that the existence of points of maximum curvature on a pattern will allow it to be rapidly identified among patterns that do not contain points of maximum curvature (circles) but not vice versa.
- A pairing of the K&T pattern with a circle: This pairing is expected to show an asymmetry in performance. The K&T pattern contains points of maximum curvature, curvature discontinuities, and also the shape cue while the circle is devoid of all of these features.
- A pairing of RF3 patterns with K&T patterns: This pairing provides a strong test of whether both of these patterns contain a shape cue by which they can be discriminated, allowing each to be rapidly identified among examples of the other. The RF3 pattern has an angle subtended by adjacent points of maximum curvature, at the center of gravity of the pattern, of 120°, and the K&T pattern of 156°. The K&T pattern, of course, also contains curvature discontinuities while the RF3 pattern does not. In Experiment 1, the difficulty of identifying RRF3 patterns among RF3 patterns was taken as evidence that a curvature discontinuity does not allow the RRF3 patterns to be rapidly identified. As pointed out in the General methods section, however, the curvature discontinuities on the RRF3 patterns differ from those on the K&T patterns. The curvature discontinuities on the RRF3 patterns occur at points of infinite curvature (orientation discontinuities) so the magnitude of the discontinuity is larger for the RRF pattern but the curvature on either side of the orientation discontinuity is the same. This pattern pairing, thus allows us to rigorously test whether the curvature discontinuity on a K&T pattern is more salient than the shape cue that distinguishes an RF3 pattern.

*t*-tests comparing the forward and reverse condition gradients.

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