A kinematic cue for active haptic shape perception

BR A IN RE S E A RCH 1 2 67 ( 20 0 9 ) 2 5 –3 6

a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m

w w w. e l s e v i e r. c o m / l o c a t e / b r a i n r e s

Research Report

A kinematic cue for active haptic shape perception Abram F.J. Sanders⁎, Astrid M.L. Kappers Physics of Man, Helmholtz Institute, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands

A R T I C LE I N FO

AB S T R A C T

Article history:

This study quantitatively investigates the actual movements that observers make when

Accepted 23 February 2009

exploring a shape. It addresses the question of how the kinematics of these movements

Available online 4 March 2009

relate to and may affect perceived shape. It is one of the first studies to do so for real stimuli and for relatively unconstrained exploration. Observers discriminated the curvature of

Keywords:

circularly shaped strips. We identify a kinematic cue for a single finger stroking across

Haptic perception

circular strips under conditions of slip. This cue consists of two terms that are related to the

Active touch

shift of the skin contact surface across the fingertip and the rotation angle of the finger. The

Shape

rotation angle of the finger is found to increase linearly with the curvature of the stimulus.

Exploratory movement

Observers rotated their finger less on a concave curvature by a constant amount, while at

Curvature

the same time they overestimated the radius of the concave strips compared to the convex

Psychophysics

ones. We show that responses were related to kinematic properties of the actual movements and we consider several mechanisms that could explain this finding. © 2009 Elsevier B.V. All rights reserved.

1.

Introduction

One approach to the study of haptic shape perception of real objects is to look at the geometric stimulus properties that determine performance. Gordon and Morison (1982) provide an early study addressing this question. Pont et al. (1998, 1999) and Louw et al. (2000, 2002a,b) offer a more thorough investigation into the geometric cues relevant in haptic shape perception. The general idea behind these studies is that detection and discrimination thresholds expressed in terms of the effective geometric cue should be independent of shape and size. This study takes a different approach. We address the problem of haptic shape perception by looking at how a shape is actually explored by the fingers and hands. Shape cues that are related to these exploratory movements can be identified. By quantitatively analyzing exploratory movements and relating movement characteristics to observers' responses,

we can get indications if and to what extent perceived shape may possibly be affected by the way that the shape is explored. One could argue that the strength of approaching the problem of haptic shape perception by investigating relevant geometric cues lies in the fact that no reference is made as to how the fingers actually explore the shape. Discrimination performance is explained entirely in terms of stimulus properties; any findings will thus have a high level of generalizability. However, an explanation purely in terms of physical stimulus characteristics cannot be the whole story, since perceptual abilities are also tied to the characteristics of the sensory apparatus. For example, a number of researchers have noted a critical interaction between the nature of movements of the hands exploring a haptic stimulus and the stimulus property to be perceived (e.g., Davidson, 1976; Gibson, 1962; Klatzky and Lederman, 2003; for an overview, see Appelle, 1991). Note that the same holds for eye movements and apprehension of visual stimuli (e.g., Yarbus, 1967).

⁎ Corresponding author. E-mail address: [email protected] (A.F.J. Sanders). 0006-8993/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.brainres.2009.02.038

26

BR A IN RE S EA RCH 1 2 67 ( 20 0 9 ) 2 5 –36

Insights into mechanisms at work in haptic shape perception may be gained from studies using haptic interfaces (e.g., Drewing and Ernst, 2006; Henriques and Soechting, 2003). Although valuable in identifying possible shape cues, a fundamental drawback of this approach is that the set of shape cues is reduced compared to real stimuli. In addition, cues are defined a priori because the haptic device obviously has to be manufactured and programmed. Consequently, shape cues delivered through a haptic device might not be the same as the cues that observers use when freely exploring real shapes. The question then remains as to how exactly observers explore real objects and what shape cues they are using in that case. The difficulty of the problem lies in measuring movements for natural exploration of real objects. In this research, we asked observers to judge the radii of circularly shaped concave and convex strips. The possible quantitative relation between characteristics of exploratory movements and an observer's shape percept for relatively unconstrained exploration of real objects will be addressed. As such, this is one of the first studies to do so (cf. Sanders and Kappers, 2008; Wong, 1977). First, we briefly discuss studies of geometric stimulus properties. We mainly focus on active dynamic touch (i.e., observers actively moving their fingers across the surface of an object), as this manner of touch is investigated in the present study. Taken together, the two studies by Pont et al. (1998, 1999) show that discrimination of circularly shaped strips is first and foremost based on a comparison of slope differences. A strip is judged more curved if the slope difference between the two outermost points is larger. Discrimination performance proved to be independent of the tilt of the strip (Pont et al., 1998). Moreover, observers hardly correct for differences in scanning length (Pont et al. 1999). The slope difference over the surface depends not only on the radius of the strip, but also on the length of the scanned surface. Pont et al. (1999) found that a strip with a larger scanning length is perceived as more curved than a strip of the same radius but with a smaller scanning length. Observers made only minor corrections for these scanning length differences. Slope difference over the surface represents a first-order geometrical shape parameter. The zeroth-order and secondorder geometrical structures correspond to height differences over the surface and local curvature, respectively

(curvature is the local change of the slope). Experiments on static discrimination (Pont et al., 1999) showed that both geometric cues do not contribute markedly to discrimination of circular strips. Louw et al. (2000, 2002a,b) provide an extension to more generic shapes. The shapes used in these studies were strips with Gaussian height profiles and strips with second-order derivatives of Gaussian profiles (Mexican hat-like shapes). These stimuli have varying curvature, whereas the circular strips used by Pont et al. (1998, 1999) have constant curvature. Results from the three studies by Louw et al. (2000, 2002a,b) point out that, for both detection and discrimination, the effective cue is the maximum slope, even when different shapes are being discriminated. However, this time a minor but unquestionable role is played by another geometric cue, the maximum local curvature. A starting point for incorporating properties of the hand and hand movements into our understanding of haptic shape perception is offered by Hayward (2008). The author gives a theoretical description of a number of possible cues related to the mechanics of contact between finger and circular shapes. For example, kinematic cues involve the displacement of the skin contact surface between finger and object when the finger moves over the surface. For small scanning lengths or non-stationary objects, the finger may roll on the object's surface. However, in case of stationary objects and scanning lengths considerably larger than the fingertip's width (as in the present study), slip between finger and object must occur. We can then identify two extreme cases (Fig. 1). First, if the finger keeps a constant orientation while moving over the strip, the skin contact surface shifts across the fingertip (Fig. 1A). The shift s is related to the radius of the object (Hayward, 2008): 1 s = ; R rL

ð1Þ

where R is the object's radius, L is the scanning length, and r is the effective finger's radius. It is defined to be positive for concave surfaces and negative for convex surfaces. If the magnitude of the shift s is larger, then the radius R of the surface must be smaller, given fixed scanning lengths. A study by Dostmohamed and Hayward (2005) supports the idea that the shift of contact may indeed be a powerful source of information used by observers.

Fig. 1 – Illustration of the two terms of the kinematic cue: (A) contact shift, and (B) rotation angle of the finger. Arrows indicating the orientation of the finger are drawn as vectors perpendicular to the nail. Note that radii used in the experiment are much larger than depicted.

BR A IN RE S E A RCH 1 2 67 ( 20 0 9 ) 2 5 –3 6

Second, if the contact point is kept stationary on the fingertip, then the finger will rotate along with the surface normal (Fig. 1B). In that case, the rotation angle of the finger θ is the same as the slope difference ϕ, which is related to the radius of the surface: 1 / = : R L

ð2Þ

If the finger rotates in the same direction as the surface normal, like in Fig. 1B, the rotation angle of the finger θ is defined to be positive for concave surfaces and negative for convex surfaces. This mode of exploration may be especially suited for larger objects (cf. Klatzky and Lederman, 2003, p.156: contour following). It was not described by Hayward (2008), probably because the author focused on small shapes having low curvatures. The two exploration modes described above span the range of all possible finger movements when there is slip between the exploring finger and the stationary object. For free exploration, the rotation angle of the finger (θ) will typically be between zero rotation (finger maintaining constant orientation) and full rotation along with the surface normal (θ being equal to the slope difference). For any exploration mode, the radius of the surface can be observed from the sum of two terms, which we will call Contact Shift (CS) and Finger Rotation (FR): 1 sðhÞ h = CS + FR = + : R rL L

ð3Þ

The shift of contact (s) depends on the rotation angle: it is shortened by an amount rθ compared to the shift for θ equal to zero (Eq. (1)).
 L h : ð4Þ sðhÞ = r R Thus, we have identified a kinematic cue for haptic shape perception under conditions of slip. This cue is the sum of two terms CS and FR. Observers can judge the radius of the surface by observing the shift of contact (s), the amount of rotation of the finger (θ), and the scanning length (L), and combining these variables according to Eq. (3). Moreover, note that this kinematic cue is invariant with respect to the amount of rotation of the finger. Interestingly, the shift of contact is observed from cutaneous input at the contact between finger and surface, whereas the amount of finger rotation is based on proprioception. Finally, note that the slope difference over the surface can be sensed from both the contact shift and the rotation angle of the finger, so that this geometric cue is in agreement with either term. A recent study by Van der Horst and Kappers (2008) describes an experiment that might enable us to investigate the kinematic cue described above. Participants had to compare a convex with a concave circularly shaped strip and decide which of the two had the larger radius. They found biases indicating that a convex strip systematically appeared flatter than a concave strip of the same radius (Van der Horst and Kappers, 2008, Experiment 1: weakly curved stimuli). The crucial observation here is that the geometry of the convex strip is the same as that of the concave strip, apart from a sign. Thus, it is not possible to

27

explain biases in terms of geometric cues: we may have to look at the way in which the strips are actually being explored by the finger. Unfortunately, Van der Horst and Kappers (2008) did not record movement profiles of the index finger. In addition, the lengths over which participants scanned the strips were left free, so there might have been confounding effects of systematic differences in scanning lengths between convex and concave stimuli (Pont et al., 1999). In this paper, we present the same curvature-magnitude discrimination experiment on concave and convex strips as described by Van der Horst and Kappers (2008). However, we fixated the scanning length by using movement stops, such that the scanning length was the same for every stimulus. Thus, observers compared concave and convex circular strips and decided which of the two had the larger radius. We tested a single reference curvature magnitude of 2 m− 1. Fig. 2 shows the stimuli used in the present study. We recorded movement profiles of the tracing index finger, expecting to find curvature biases also in the case of fixed scanning length. If we hypothesize that observers use the kinematic cue described above to discriminate concave and convex curvature magnitudes, two conditions are needed for any curvature biases to occur. As a first condition, observers do not properly take both terms of this cue into account. For veridical performance, observers need to observe both the contact shift (s) and the rotation angle of their index finger (θ) and add them according to Eq. (3). If curvature perception proves to be biased, observers might have ignored one of the two terms (CS or FR) and have only based their curvature percept on the other term. Alternatively, observers could still have observed both terms but have taken one term or both terms only partially into account. We suppose that in first-order approximation one of the terms is dominant, which means that this term dominates the curvature percept and that effects of the other term are secondary or negligible. This seems to be a reasonable supposition in view of the fact that contact shift and finger rotation are observed through different subsystems (cutaneous vs proprioceptive). As a second condition, observers stroke their index finger differently across concave and convex strips. A particular

Fig. 2 – Convex and concave reference stimuli along with their dimensions. Convex and concave stimuli are presented one after the other; they are positioned approximately in the participant's frontoparallel plane. Surfaces are touched with the finger pad of the right index finger.

28

BR A IN RE S EA RCH 1 2 67 ( 20 0 9 ) 2 5 –36

concave and convex strip are judged as equally curved if values of the effective cue are the same for both strips. For example, if CS is the dominant term, two strips will be perceived as having the same curvature when values for CS are the same for the two strips. However, it is important to understand that equal values of CS do not necessarily imply that physical curvatures of the concave and convex strips have been the same (generally, this is not the case). When observers stroke their finger across the strips, there may be systematic differences in movement patterns between concave and convex curvatures due to, for example, biomechanical constraints. Because the contact shift s depends on the rotation angle θ (Eq. (4)), it is possible that values of CS are the same for a concave and a convex strip of different curvature magnitudes. As a result, curvature biases could then arise, whereas in terms of CS there would be no bias. A similar reasoning applies to the case of FR being the dominant term. To summarize, if observers move their finger in the same way, or more precisely, if the dependence of rotation angle of the index finger on curvature is the same for the two types of strips, there would be no curvature biases, even when observers use just one of the two terms of the kinematic cue. Moreover, note that the sum of CS and FR (Eq. (3)) is invariant with respect to the manner in which the finger sweeps across concave and convex strips, as we have already mentioned above. In other words, both conditions are needed for curvature biases to occur. Finally, we point out that under this hypothesis we assume that perception of kinematic properties of arm movements is the same for the two types of curvatures (this could represent another source of curvature biases). Thus, participants are able to observe kinematic properties of exploratory movements and they do so in the same way for concave and convex curvatures. Given these considerations, we proceeded as follows. From the movement data, we extracted the rotation angle of the finger. For every trial, we computed values for CS and FR according to Eqs. (3) and (4). Next, we fitted psychometric curves to the data as a function of physical curvature C (geometric cue) and each of the two kinematic terms, CS and FR. We analyzed goodness-of-fit for each curve to assess whether observers' responses were related with movement characteristics. As the independent variable for each psychometric curve, we took the differences ΔC, ΔCS, and ΔFR between the concave and convex curvatures that were discriminated in a particular trial.

Fig. 3 – Rotation angle per unit scanning length as a function of curvature. (A) A typical data set for one of the participants. (B) Schematic illustration of the two observations that the slope of the regression line is less than unity and the y-intercept is negative. For both strips, the finger follows the curvature of the strip only partially (nail normal vectors are not perpendicular to the surface) and the amount of rotation is less for the concave strip compared to the convex strip.

Note that the vertical distance between a data point and the dashed line represents the shift of the skin contact surface (term CS in Eq. (3)). Three observations can be made. First, rotation angle per unit scanning length clearly correlated with the curvature of the stimulus. Second, the slope of the regression line is between zero rotation and full rotation. Third, there is a negative y-intercept, indicating that the finger systematically rotated less on a concave curvature by a constant amount. The last two observations are schematically illustrated in Fig. 3B. These observations resemble the pattern of results averaged over all participants. All R2-values were highly significant (ps < 0.001). The mean y-intercept was −0.31 and significantly different from zero, with t(11) = 3.94, p = 0.002. The mean slope was 0.37 and significantly different from zero, with t(11) = 12.25, p < 0.001, as well as significantly different from one, with t(11) = 20.90, p < 0.001. Thus, the rotation angle of the finger was only 37% of the slope difference over the surface: the finger followed the curvature of the stimulus only partially.

Psychometric curves as a function of ΔC, ΔCS, and ΔFR

2.

Results

2.2.

2.1.

Rotation angles as a function of curvature

As a next step in our analysis, we fitted psychometric curves as a function of curvature differences (ΔC) and differences in observed values for each of the two terms of the kinematic cue (ΔCS and ΔFR). First, we computed values for CS and FR from the movement data. Second, we computed values of ΔCS and ΔFR for every psychometric trial and fitted these to response fractions (see Experimental procedures). In every psychometric trial, observers compared two stimuli. If we assume that they discriminated curvature magnitudes by observing and comparing movement characteristics for the two strips in a particular trial, it is essential that we analyze

First, we extracted rotation angles of the finger (θ) from the movement data (see Experimental procedures). For every participant, we then performed a linear regression on curvature and rotation angle per unit scanning length (θ/L; term FR in Eq. (3)). Fig. 3A shows a representative example of the dependence of rotation angle per unit scanning length on curvature. The dashed line represents the case of the index finger perfectly following the curvature of the strip: the rotation angle (θ) would then equal the slope difference (ϕ).

BR A IN RE S E A RCH 1 2 67 ( 20 0 9 ) 2 5 –3 6

differences in observed values of CS and FR on a trial-to-trial basis. Fig. 4 shows a representative example of a psychometric curve from one observer fitted as a function of each of the three independent variables (data points indicated in black). Note that independent variables (ΔCS and ΔFR) form a continuum, because they were extracted from the movement data. The so-called Point of Subjective Equality (indicated as μ) corresponds to a response score of 50% and represents the combination of concave and convex stimuli that were judged equal when expressed in terms of the respective variable. Goodness-of-fit was assessed by computing deviances for every curve and comparing these against bootstrap-simulated distributions (see Experimental procedures). Deviances for all experimentally obtained curves, except for one, fell within the 95% confidence intervals. In other words, responses were related to each of the three independent variables and their relationships were well described by the psychometric function. To better appreciate the distribution of data points for the right two curves, we have binned the data (see Experimental procedures); binned data sets are indicated in gray. Zigzag bands denote the 95% confidence bands for the respective psychometric curves (see Experimental procedures). For this particular observer, all observed response fractions fell within the corresponding 95% confidence intervals. The pattern of results was similar for every other curve obtained in this study: only 2 out of 324 data points (12 observers × 3 variables × 9 bins) were outside the respective confidence bands. This finding corroborates conclusions from the analysis of deviances. Fig. 5 shows Points of Subjective Equality (PSE) averaged over all twelve observers. The average PSE in terms of curvature difference (ΔC) was 0.48 m− 1. This was significantly different from zero, with t(11) = 3.24, p = 0.008. The positive bias indicates that a concave strip must be curved more than a convex strip in order to be perceived as equal: a concave curvature appeared flatter than the same convex curvature. The average PSE in terms of contact shift (ΔCS) was 1.15 m− 1, which also differed significantly from zero, with t(11) = 4.83, p < 0.001. In other words, the shift of skin contact was larger for the

29

Fig. 5 – Points of Subjective Equality. Average PSEs expressed in terms of curvature differences (ΔC) and differences in contact shift and finger rotation (ΔCS and ΔFR). Error bars represent standard errors of the mean. PSE-ΔC and PSE-ΔCS differed significantly from zero, ps < 0.01.

concave strip in case of the two curvatures that were subjectively equal. However, the average PSE in terms of finger rotation (ΔFR) was −0.21 m− 1. This was not significantly different from zero, with t(11) = 1.04, p = 0.319. The non-significant bias indicates that concave and convex finger rotations were the same for the two strips that the observers judged equally curved. Finally, PSEs in terms of curvature difference (ΔC) correlated significantly with PSEs in terms of contact shift (ΔCS), with N = 12, r = 0.68, p = 0.016, as well as with PSEs in terms of finger rotation (ΔFR), with N = 12, r = 0.63, p = 0.028. Furthermore, PSEs in terms of contact shift (ΔCS) did not correlate with PSEs in terms of finger rotation (ΔFR), with N = 12, r = 0.06, p = 0.85, so that their effects on PSEs in terms of curvature difference (ΔC) were independent. Slopes of separate regression lines fitted to PSE-ΔC as a function of PSE-ΔCS and PSE-ΔFR were 1.08 ± 0.37 and 0.87 ± 0.34, respectively.

2.3.

Other variables

Observers will never move their index finger in exactly the same way twice over a certain stimulus. For a fixed curvature,

Fig. 4 – Psychometric curves. Representative examples of psychometric curves fitted as a function of curvature differences (ΔC) and differences in observed values for each of the two terms of the kinematic cue (ΔCS and ΔFR). All curves are from the same observer, and each consists of 144 2AFC trials. The Points of Subjective Equality are indicated by μ; the average threshold σ expressed in physical curvature (ΔC) in this study was 0.81 m− 1 (SD = 0.25 m− 1). We have also indicated deviance values for each curve. Finally, gray data points represent binned data sets and zigzag bands are the 95% confidence bands for the fitted psychometric functions.

30

BR A IN RE S EA RCH 1 2 67 ( 20 0 9 ) 2 5 –36

there will always be a certain amount of variation in movement profiles. If observers base their curvature percept on movement characteristics of the sweeping index finger, one would expect that for a fixed curvature, responses also vary systematically within the – small – range of observed values for the respective movement parameter. Therefore, we split psychometric trials for each curvature difference into two groups that corresponded to the two response alternatives and we analyzed whether ΔFR differed between the two categories. We grouped the data for all observers; average values are shown in Fig. 6. Squares lie above boxes in eight out of nine cases. Note that the vertical distance between a data point and the y = x-line (not drawn) would represent ΔCS, so that an analysis in terms of ΔCS is the mirror version of the present analysis. Error bars are relatively large for squares at negative curvature differences because these averages are based on relatively small numbers of data points: if the convex stimulus was much more curved than the concave one, the observer judged the convex stimulus to be more flat only incidentally. To exclude effects of curvature, we did a multiple regression on the pooled data set (N = 12 observers × 144 trials), including predictors ΔC and Response. We tested whether the y-intercepts of the regression lines of ΔFR on ΔC were significantly different for the two response categories: the regression equation was ΔFR = a + b ΔC + c Response. (Response is 0 for “Concave”-responses and 1 for “Convex”-responses.) The model was significant (R = 0.44, p < 0.001); all parameters, including parameter c, made significant contributions to the model (a = −0.65, p < 0.001; b = 0.30, p < 0.001; and c = 0.10, p = 0.034). Furthermore, we tested whether there were any systematic differences in movement speed between concave and convex stimuli. To exclude effects of curvature, we only analyzed trials in which both the concave and convex strips had reference curvature (N = 16 trials per observer). We computed average movement speeds for the concave and convex reference stimulus for every observer. There was considerable variation between observers: average movement speeds could differ by more than a factor of three (Mdn = 22.2 cm/s, Min = 12.5 cm/s, Max = 46.4 cm/s). Pairedsamples t tests indicated that concave speeds were slightly higher than convex speeds (t(15) = 5.74, p < 0.001) for only one observer. For all other observers, speed differences were non-significant (ps N 0.141).

Fig. 7 – Dependence of rotation angle per unit scanning length on sweep number. To exclude effects of curvature, we analyzed only those trials in which both strips had reference curvature.

Finally, we tested whether the amount of finger rotation systematically varied over the course of the four sweeps. We only analyzed trials in which both stimuli had reference curvature and grouped the data for all observers (N = 12 observers × 16 trials). Fig. 7 shows average rotation angles for every sweep and both the concave and convex reference curvatures. Recall that we found that observers systematically rotated their finger less on a concave curvature. Fig. 7 shows that this effect may be more pronounced for the last three sweeps compared to the first one, as the difference in absolute values of rotation angles increases in magnitude. Indeed, a one-way ANOVA with a repeated-measures design of values of ΔFR for each of the four sweeps showed a significant main effect, with F(3,573) = 72.33, p < 0.001. Pairedsamples t tests with Bonferroni corrections showed that the first sweep differed significantly from all subsequent sweeps (ps < 0.001); pairwise comparisons for the last three sweeps were all non-significant. We remark that absolute values of concave and convex rotation angles were significantly different already in the first sweep, as shown by a pairedsamples t test, with t(191) = 4.07, p < 0.001. As a final note, we mention that we again fitted psychometric curves as a function of ΔCS and ΔFR, this time taking either the first sweep or the last three sweeps into account. In both cases, the pattern of results was similar.

3.

Fig. 6 – Multiple regression analysis of ΔFR for the two response categories. For a fixed curvature difference (ΔC), ΔFR was significantly larger for the “Convex”-response category.

Discussion

We found biases indicating that a concave curvature appeared flatter than a convex curvature of the same radius. Scanning lengths were fixated through the use of movement stops, so that the two types of strips were exactly the same in terms of geometric cues, apart from a sign (Pont et al., 1998, 1999). Therefore, finding these biases calls for an analysis of finger movements across the stimuli. From recordings of the location and orientation of the index finger tracing the stimulus, we could deduce that there were systematic differences in movement patterns between concave and convex curvatures. Regressions of rotation angles on curvature showed that

BR A IN RE S E A RCH 1 2 67 ( 20 0 9 ) 2 5 –3 6

observers on average rotated less on a concave curvature by a constant amount (negative y-intercepts). As a next step, we fitted psychometric curves as a function of differences in observed values for two kinematic variables: the shift of the skin contact surface (CS) and the rotation angle of the finger (FR). Analyses of goodness-of-fit of individual psychometric curves showed that response fractions were related to these variables. Moreover, for fixed curvature differences (ΔC) we found that ΔFR was significantly higher for the “Convex”-response category (Fig. 6). In conclusion, these findings show that observers' responses were related to kinematic properties.

3.1.

Mechanisms underlying curvature perception

This relationship between observers' responses and movement data may then be interpreted in the following two ways. The first explanation states that perceived curvature causally depends on kinematic properties, while the second states that kinematic properties follow from perceived curvature.

3.1.1. Perceived curvature based on observed kinematic properties The first explanation corresponds to the hypothesis put forward in the Introduction. We argued that a possible mechanism for haptic shape perception is based on observing kinematic cues. Kinematic cues relate movement of the hand and displacement of the skin contact surface to the shape of the object. For the case of slip occurring between finger and object, we have identified a kinematic cue, consisting of two terms that are related to the shift of the skin contact surface across the fingertip and the rotation angle of the finger (CS and FR, respectively). If observers indeed use this cue to discriminate concave and convex curvature magnitudes, we argued that curvature biases would be caused by observers taking only one of the terms into account (the dominant term). The negative y-intercepts indicating that observers systematically rotated their finger less on a concave curvature by a constant amount (cf. Fig. 3) would then already suggest that FR is the dominant term, as negative y-intercepts are incommensurate with CS being the dominant term. A more sophisticated analysis to identify the dominant term is to determine for which of the two terms the independent variable is zero at the Point of Subjective Equality, as two curvatures are assumed to be judged equal if values of the effective cue are the same. The psychometric curve would thus have a vanishing bias in case of the dominant term and, conversely, a non-zero bias in case of the non-dominant term. The results indicate that biases expressed in terms of FR disappear, whereas biases in terms of CS remain significant (Fig. 5). Thus, FR would be the primary effective cue for curvature discrimination in our experiment. In that case, the curvature biases we found are due to observers failing to account for the shift of the skin contact surface across the fingertip (CS). The significant correlation between PSEs in terms of curvature difference (ΔC) and PSEs in terms of contact shift (ΔCS) supports this conclusion: observers who had a large curvature bias had a large bias in terms of CS as well. If observers indeed judge curvature by closely observing finger rotation, we would not only expect PSEs in terms of

31

finger rotation (ΔFR) to be zero, but also that they do not correlate with curvature biases. However, PSEs in terms of curvature difference (ΔC) did correlate with PSEs in terms of finger rotation (ΔFR). Thus, although the latter biases did not differ significantly from zero, part of the variation in curvature biases could be explained by the small variation around zero of biases in terms of FR. Note that effects of biases in terms of CS and FR on curvature biases were independent and that both correlations were equally strong. Moreover, although the slope for the regression line fitted to PSE-ΔC as a function of PSE-ΔFR was slightly smaller, slopes of separate regression lines did not differ significantly. We don't know yet how to explain the correlation between PSEs in terms of curvature difference (ΔC) and PSEs in terms of finger rotation (ΔFR). It may be suggested, for example, that observers to some extent do make small corrections for the asymmetries in finger rotation between concave and convex curvatures, or that other – veridical – curvature cues are playing a role too. However, more research is needed to settle this issue. If finger rotation is indeed the primary effective cue for curvature discrimination in our experiment, an important question that has to be answered then is why so? And why do observers fail to account for the shift of skin contact across the fingertip? Clearly, when tested in isolation using a haptic device, the contact shift can be a powerful source of information (Dostmohamed and Hayward, 2005). However, scanning lengths were a factor of five smaller than in our experiment. We suggest that threshold levels for cutaneous and proprioceptive input are critical here. Depending on the size and the curvature of the object, and possibly also depending on whether or not the object has constant curvature, one or the other source of stimulation might be more salient in terms of being above threshold. For example, if an observer has to detect a change in curvature for small stimuli, the corresponding contact shift on the fingertip's skin might be easier to detect than a change of orientation signaled by muscle spindles and joint receptors in the arm. Instead of adding the two terms together, the more salient term might take over and dominate an observer's curvature percept.

3.1.2.

Movement patterns adapted to perceived curvature

Other sources of curvature information that are not considered here include local skin deformation patterns (cf. Hayward, 2008). The experimental difficulty lies in measuring local skin deformation for the human finger under normal conditions (e.g., Pawluk and Howe, 1999). At present, one would have to divert to haptic devices (e.g., Provancher et al., 2005; Wang and Hayward, 2006) and simulations of fingertip deformations (e.g., Serina et al., 1998) to understand the exact role of skin deformations in shape perception. However, the studies by Louw et al. (2000, 2002a,b), which showed that the geometric cue of local curvature played a minor but significant role, suggest that local skin deformation patterns may indeed contribute significantly to haptic shape perception of real stimuli. As such, they may also cause biases. Clearly, local skin deformation patterns are different for concave and convex surfaces of the same radius due to the convex shape of the fingertip. A second explanation for the present results could then be as follows: observers use cues other than kinematic properties

32

BR A IN RE S EA RCH 1 2 67 ( 20 0 9 ) 2 5 –36

(notably cues related to local skin deformation patterns) to perceive curvature, overestimate concave curvatures according to observed values for these particular cues, and adapt movement patterns to perceived curvature. Thus, kinematic properties (notably finger rotation) would follow from perceived curvature instead of the other way around. Since we essentially provide a correlative analysis, we cannot exclude this possibility. We analyzed finger rotation over the course of the four sweeps (Fig. 7) and found that it changed significantly between the first and second sweep, after which it remained constant. The change between the first and second sweep is in the direction of the observed overall systematic difference between concave and convex rotation angles. This finding supports the second explanation that movement patterns are adapted to perceived curvature. However, the finding that finger rotation changed significantly between the first and second sweep does not necessarily disprove the first explanation. Observers have to remove and replace their hand for every new stimulus presentation. They may need the first sweep to get an idea of the spatial layout of the strip and use the subsequent sweeps to observe kinematic properties more closely in order to judge the strip's curvature. Moreover, note that the systematic difference between concave and convex rotation angles was already present in the first sweep, in which adaptation, if any, has probably not yet occurred. Furthermore, the restriction to four sweeps for exploring the stimuli was based on pilot experiments. We observed that this was about the minimum number of sweeps that observers needed to perform the task satisfactorily. Although this observation is based on introspective reports from the participants, it may provide another indication that adaptation is not very likely to have played a role in our experiment. If observers were indeed adapting movement patterns to perceived curvature, a smaller number of sweeps would have been sufficient for them to do the discrimination task. Finally, although curvature cues related to local skin deformation patterns may contribute to curvature discrimination, Pont et al. (1999) showed that they are substantially weaker compared to slope difference over the surface: thresholds for discrimination from flat of circular strips were much higher for a single finger touching the stimulus statically (only skin deformation) compared to stroking the stimulus with a scanning length similar to the scanning length in the present study. Based on threshold values provided in Pont et al. (1999), none of the stimuli used in the present study can be reliably discriminated from flat using static contact with a single finger. It is not very likely that cues that contribute only marginally to discrimination performance in terms of discrimination thresholds are strong enough to cause significant biases at the same time.

3.2.

Other remarks

We were surprised to find curvature biases opposite to curvature biases found in Van der Horst and Kappers (2008). In their study, convex curvatures were judged flatter. Possibly, biases in Van der Horst and Kappers (2008) are the compound effects of biases as found in the present experiment and biases

due to systematic differences in scanning length (convex scanning lengths being shorter and as a consequence, physical slope difference being smaller) and biases. In that case, the latter bias must have been larger in magnitude than the former. However, it should also be noted that there were slight differences in setup, the most notable one being observers resting their forearm on a table. If the biases that we found in the present experiments are due to systematic differences in movement patterns, different movement constraints related to the setup might have induced slight differences in movements of the index finger. We addressed the problem of haptic shape perception by investigating a curvature-magnitude discrimination task. We should, of course, be cautious in generalizing results from this particular task to curvature perception and shape perception in general. Van der Horst and Kappers (2008) found that thresholds for concave–convex discrimination were approximately 70% higher than concave–concave or convex–convex discrimination thresholds. This suggests that there are additional cues involved in the latter two tasks that may not play a role in concave–convex discrimination. In other words, cues important in curvature-magnitude discrimination may be less prominent in other tasks. Note that the finding by Van der Horst and Kappers (2008) is in agreement with both interpretations of the relationship between curvature perception and movement data discussed above, as the additional cues involved in concave–concave and convex–convex discrimination could be related to kinematic properties as well as to local skin deformation.

3.3.

Conclusion

In this study, we focused on circular strips. However, the emphasis on surfaces with constant curvature does not mean a loss of generalizability. In mathematical terms, smooth surfaces can be locally approximated by doubly curved surfaces that are either egg-like or saddle-like (e.g., Hilbert and Cohn-Vossen, 1932; Porteous, 2001). Any shape can thus be rendered as an ordered array of these elementary surface patches. For example, the case of the kinematic cue for circular strips may be easily extended to shape perception in general. The observer would be sensing instantaneous values of the kinematic cue while moving the finger across the object's surface. The contact shift term (CS) is expressed as the ratio of contact velocity and scanning velocity and the finger rotation term (FR) as the ratio of angular velocity and scanning velocity. Further research is needed to investigate to what extent results presented in this study generalize to perception of more generic shapes. The present study provides a quantitative investigation into how observers explore real shapes by touch and how the manner of exploration relates to and may affect perceived shape. In a curvature-magnitude discrimination experiment, observers compared concave and convex strips and then decided which felt more flat. We constrained exploration as little as possible. When sweeping over the circular strips, the finger partially followed the curvature of the surfaces. In addition, we found that the rotation angle increased linearly with the curvature of the stimulus and that responses were related to the kinematic properties of the finger movements.

BR A IN RE S E A RCH 1 2 67 ( 20 0 9 ) 2 5 –3 6

We considered two explanations that could account for this relationship. However, more research is needed to further elucidate the interaction between properties of exploratory movements and an observer's shape percept.

4.

Experimental procedures

4.1.

Participants

The participants consisted of eleven males and one female in their 20 s and early 30 s. They were all right-handed. Participants were naïve as to the aims and design of the experiment. Informed consent was obtained from each participant. They were paid for their efforts.

4.2.

Stimuli and setup

Stimuli were cylindrically curved strips made of PVC (see Fig. 2). The same strips were used by Van der Horst and Kappers (2008). For circular strips, local curvature is constant and equal to the reciprocal of the strip's radius. Curvatures for both the concave and convex strips ranged from 0.4 m− 1 to 3.6 m− 1 in steps of 0.4 m− 1. The base-to-maximum (convex) or base-tominimum (concave) height was 3 cm for all strips. The heights of the end points varied slightly with the curvature of the stimulus. We placed cylindrical aluminum stops (height 7 mm) on the curved surfaces such that the distance between the stops measured along the surface was always 15 cm, irrespective of the stimulus' curvature. The scanning path was positioned symmetrically on the surface. The movement stops themselves did not convey any information about the curvature of the stimulus. Upon stimulus presentation, curved strips were put in a thin cardboard placeholder fixed on a table (height 75 cm), behind which the participant was comfortably seated. Stimuli were oriented to be in the participant's frontoparallel plane, with their midpoints approximately in the sagittal plane passing through the right shoulder. Participants touched the stimuli with the finger pad of the index finger of their right hand (volar surface of distal phalanx). Thus, when tracing the stimuli, participants moved their index finger from left to right and vice versa. The distance between the participant and the table's edge was adjusted according to their liking. Generally, when touching the stimuli, the angle between upper arm and forearm was more than 90°, with zero degrees meaning that the arm was completely flexed. Participants could move their arms freely when tracing the stimuli, but they did not rest their elbow on any support or touch the table's edge. Talcum powder was used to reduce friction between skin and stimulus surface.

4.3.

Design and procedure

4.3.1.

Measuring psychometric functions

We measured psychometric functions according to the twoalternative forced-choice (2AFC) method using constant stimuli to determine which concave and convex curvature were perceived as equal. Each trial proceeded as follows. Participants were presented with a concave or a convex

33

curvature. They placed their right index finger on the curved surface against the left movement stop. After a signal from the experimenter, they traced the stimulus four times, going back and forth twice. When they lifted their hand, the experimenter quickly placed the second stimulus, having an opposite curvature, in front of them. The participants positioned their finger against the second curvature's left stop and, after a signal from the experimenter, traced the next stimulus four times. They then had to decide which of the two stimuli felt more flat. Participants were instructed to make smooth movements, to touch the stops when moving across the stimuli, and to keep their finger perpendicular to the stimulus (parallel to the z-axis, Fig. 2) as much as possible. The curvature difference ΔC between the concave and convex curvature (Cv–Cx) served as the independent variable. Curvature differences sampled were ± 1.6, ±1.2, ±0.8, ±0.4 and 0 m− 1 at a reference curvature magnitude of 2 m− 1. Each curvature difference corresponded to four possible combinations. For instance, a curvature difference of + 1.6 m− 1 corresponded to the following four combinations: (1) Cv— 3.6 m− 1 presented first and Cx—2 m− 1 presented second (Cv– Cx = + 1.6 m− 1); (2) Cx—2 m− 1 first and Cv—3.6 m− 1 second (presentation order reversed); (3) Cv—2 m− 1 first and Cx— 0.4 m− 1 second (reference curvature changed to concave); or (4) Cx—0.4 m− 1 first and Cv—2 m− 1 second (presentation order again reversed). The two curvatures were presented in both time orders to avoid response biases. Furthermore, the reference curvature magnitude of 2 m− 1 could be both concave or convex to prevent any learning effects: if, say, the reference curvature had always been concave and comparisons convex, observers might have noticed this and started using information from previous trials in making their judgment. The set containing the four possible stimulus combinations for each of the nine curvature differences was presented four times, amounting to a total of 144 trials per psychometric curve. Trials were independently randomized within each of the four blocks. A psychometric curve was measured in two sessions on separate days, with each session containing half of the trials. Every session started with a few practice trials. Participants were blindfolded throughout the experiment. A single session lasted about 45 min.

4.3.2.

Measuring location and orientation of the index finger

Movement profiles of the index finger were recorded with an Optotrak Certus system (Northern Digital Inc., Canada). Three position markers were placed on the nail of the index finger, enabling us to measure not only the finger's location but also its orientation. In addition, position markers were fixed to the setup such that we could retrieve coordinates of the stimulus' curved surface. Three-dimensional spatial coordinates of the position markers were read at a frequency of 100 Hz. The vector perpendicular to the plane defined by the three markers on the nail represented the finger's orientation. The orientation of this plane corresponds to the orientation of the nail only in rough approximation. This is not a problem, since in the end we are only interested in the change of orientation. Finally, the location of the index finger was defined as the average location of the three position markers.

34

BR A IN RE S EA RCH 1 2 67 ( 20 0 9 ) 2 5 –36

4.4.

Data analysis

4.4.1.

Rotation angle of the finger

We extracted two main parameters for analysis. First, we projected the finger's location on the space curve passing through the stimulus' curved edge (i.e., the point on the stimulus with the shortest distance to the finger's location). A location parameter specifying the position of the index finger was defined as the distance from the projection to the midpoint of the stimulus (measured along the curved edge). Second, the vector perpendicular to the nail was projected on the xy-plane, and the angle between the projected vector and the horizontal was used to parameterize the finger's orientation. Fig. 8A shows typical orientation–location profiles for a concave and a convex reference curvature. The orientation of the finger with respect to the horizontal is given as a function of the location on the stimulus (cf. Fig. 3B). The participant starts at the left end point and goes back and forth over the stimulus twice. At the left and right end points, there is a sudden change in the finger's orientation, indicating that there is a more or less instantaneous tilt of the finger at movement onset due to friction between the skin and the surface (Fig. 8B). Note that the change in orientation when moving across the stimulus is less for the concave curvature than for the convex one. Furthermore, the full range of the location parameter is less than the distance between the movement stops (15 cm). This is because the effective scanning length is shortened by the width of the fingertip. Next, to exclude any effects due to friction or due to the movement stops, we extracted those parts from the orientation–location profiles that the finger was really sweeping across the stimulus. We proceeded as follows. First, we

computed velocity profiles by taking the three-point derivative of location–time profiles (Press et al., 2002, Chapter 5). Sweeps were defined as those parts of the movement profiles for which the speed reached a certain threshold. Values for speed thresholds were a fixed fraction of the peak velocity for the respective trials. For all observers except one, a fraction of one quarter of the peak velocity yielded reliable sweep extractions; for one observer, this was a fraction of one third. We ended up with four sweeps for each trial (black data points in Fig. 8A). For each sweep, we calculated the rotation angle by performing a linear regression and then calculating the orientation difference according to the regression equation. Thus, for each sweep, we have a location range (scanning length) and a change of orientation over this range. The mean scanning length and mean rotation angle for the four sweeps were taken to represent the scanning length (L) and rotation angle (θ) for each particular trial. We also performed a full analysis by defining rotation angles as the orientation difference between the first and the last recordings for each sweep. This method yielded the same pattern as the overall results.

4.4.2.

Points of Subjective Equality

We used cumulative Gaussian distributions as psychometric functions:
 1 1 xl PFðxÞ = + erf pffiffiffiffiffiffi ; ð5Þ 2 2 2r where erf is the Error Function (Gautschi, 1972). The shape of this psychometric function is determined by two parameters: μ, which represents the location of the curve, and σ, which indicates its shallowness (Fig. 4). They are interpreted as

Fig. 8 – Extracting rotation angle of the finger. (A) Typical orientation–location profiles for concave and convex reference curvature. Black data points correspond to extracted sweeps (i.e., those parts of the movement profiles for which movement speed reached a certain threshold). (B) Schematic illustration of the sudden change in nail orientation at movement onset or when movement direction changes.

BR A IN RE S E A RCH 1 2 67 ( 20 0 9 ) 2 5 –3 6

follows: μ is the point at which the response score is 50% and is assumed to represent the observer's Point of Subjective Equality (PSE); σ is the difference between the 50% point and the 84% point and is taken as a measure of the discrimination threshold. Psychometric curves were fitted to the data by maximizing a likelihood function, assuming that responses were generated by Bernoulli processes. Goodness-of-fit was assessed by computing log-likelihood ratios, or deviances, and comparing these against bootstrapsimulated distributions (parametric bootstrap: N = 10,000). The deviance is a goodness-of-fit measure similar to the r2 statistic for linear relationships. If the experimentally obtained deviance fell between the 2.5th and 97.5th percentile of the deviance distribution, the fit was accepted. For computational details, we refer the reader to Wichmann and Hill (2001a,b). Psychometric curves were fitted as a function of curvature differences (ΔC) and as a function of differences in observed values for each of the two terms of the kinematic cue (ΔCS and ΔFR). Independent variables ΔCS and ΔFR were defined in the same way as the curvature difference ΔC: concave–convex. We computed CS and FR according to Eqs. (3) and (4). The assumption underlying Eq. (4) is that the finger can be regarded as effectively circular with some radius r over the range of contact shifts observed. This seems reasonable, since contact shifts were small compared to the perimeter of the index finger. Both the mean rotation angle and the mean scanning length (θ and L, respectively) were taken into account on a trial-to-trial basis. Instead of nine discrete values resulting from sixteen repetitions, the psychometric function is now sampled at 144 points in the case of independent variables ΔCS and ΔFR, because values were extracted from movement data. Response fractions for each of these 144 points are either zero or one.1 We have also binned data sets for the psychometric curves as a function of ΔCS and ΔFR. Note, however, that psychometric functions were fitted to the original data sets, indicated by the black data points in Fig. 4, as parameters from these fits are more accurate. Each bin contained 16 data points and was located at the median value of the independent variable. Finally, we have constructed 95% confidence bands for the fitted psychometric functions (zigzag bands in Fig. 4). Each point on the psychometric curve corresponds to a binomial experiment with parameters p, given by the value of the psychometric function at that point, and n, given by the number of data points in the respective bin (n = 16). Confidence intervals were computed as the intervals between the 2.5th and 97.5th percentile for the respective binomial distributions. Confidence bands are zigzagged because response fractions take on discrete values.

1 It is perfectly valid to fit the psychometric curve on binary data. Take, for example, the psychometric curve as a function of the curvature difference in Fig. 4 and consider the response fraction at 1.6 m− 1. This point is the result of 14 “Convex”responses and two “Concave”-responses and its coordinates are therefore (1.6; 0.875). However, we could as well replace this single point with 14 points having coordinates (1.6; 1) and two points having coordinates (1.6; 0). If we replace all other points with their binary counterparts and fit the curve again, this time as a function of 144 data points, we obtain the same fit parameters.

4.5.

35

Error analysis location parameter

Recall that the location parameter is defined as the perpendicular projection of the average location of the three nail markers on the stimulus' surface. This is the only feasible way to parameterize the location of the finger. However, the projection might not correspond precisely to the actual location of the finger (i.e., the contact point between finger and surface). Consequently, small errors in the measured scanning length might be introduced. If the finger fully rotates with the surface's curvature (Fig. 1B), then the scanning length is measured correctly. However, if the finger does not fully rotate or maintains a constant orientation, then a systematic error in scanning length between the two types of stimuli occurs. Considering Fig. 1A, if we draw a line through the center of the circular strip and the midpoint of the nail, the intersection of this line with the strip's surface is the location as we measure it. For convex stimuli, the measured scanning length is slightly smaller than the actual scanning length; for concave stimuli, the measured scanning length is slightly larger. These systematic errors are in the same direction as the reported effects: the results show that the rotation angle per unit scanning length was smaller for concave stimuli. The difference between the measured and actual scanning lengths depends on the scanning length itself as well as on the radius of the surface and the effective radius of the finger. The latter we do not know. To estimate the magnitude of this effect, we measured the difference in scanning length between zero rotation and full rotation for three observers. We asked them to place their index finger against the left and right end stops on both the concave and convex reference curvatures. They oriented their finger either horizontally or parallel to the surface. Overall, the average difference in scanning length between zero and full rotation was 4.5 mm (SD = 1.5 mm). This corresponds to 3.5% of the overall average scanning length (130 mm). Next, we fitted the psychometric curves again, adding 4.5 mm to all convex scanning lengths and subtracting this amount from all concave lengths. The pattern of results remained the same: the average PSEs in terms of contact shift (ΔCS) and finger rotation (ΔFR) were 1.09 m− 1 and −0.17 m− 1, respectively. Note that this is a conservative error estimate. Since participants rotated with the curvature of the surface for 37%, the systematic errors in convex and concave scanning lengths were maximally 63% of 4.5 mm.

Acknowledgments The authors would like to thank Rob van Beers, Vincent Hayward, and Maarten Wijntjes for their helpful comments. This research was supported by the Netherlands Organization for Scientific Research (NWO).

REFERENCES

Appelle, S., 1991. Haptic perception of form: activity and stimulus attributes. In: Heller, M.A., Schiff, W. (Eds.), The Psychology of Touch. Lawrence Erlbaum Associates, New Jersey, pp. 169–188.

36

BR A IN RE S EA RCH 1 2 67 ( 20 0 9 ) 2 5 –36

Davidson, P.W., 1976. Haptic perception. J. Pediatr. Psychol. 1, 21–25. Dostmohamed, H., Hayward, V., 2005. Trajectory of contact region on the fingerpad gives the illusion of haptic shape. Exp. Brain Res. 164, 387–394. Drewing, K., Ernst, M.O., 2006. Integration of force and position cues for shape perception through active touch. Brain Res. 1078, 92–100. Gautschi, W., 1972. Error Function and Fresnel integrals In: Abramowitz, M., Stegun, I.A. (Eds.), Handbook of Mathematical Functions. Dover, New York, pp. 295–330. Gibson, J.J., 1962. Observations on active touch. Psychol. Rev. 69, 477–491. Gordon, I.E., Morison, V., 1982. The haptic perception of curvature. Percept. Psychophys. 31, 446–450. Hayward, V., 2008. Haptic shape cues, invariants, priors, and interface design. In: Grunwald, M., Srinivasan, M. (Eds.), Human Haptic Perception: Basics and Applications. Birkhäuser Verlag. Henriques, D.Y.P., Soechting, J.F., 2003. Bias and sensitivity in the haptic perception of geometry. Exp. Brain Res. 150, 95–108. Hilbert, D., Cohn-Vossen, S., 1932. Anschauliche Geometrie. Springer, Berlin. Klatzky, R.L., Lederman, S.J., 2003. Touch. In Handbook of Psychology: Vol. 4. Experimental Psychology, I.B. Weiner, ed.-in-chief, A.F. Healy and R.W. Proctor, vol. eds. John Wiley and Sons, New York, pp. 147–176. Louw, S., Kappers, A.M.L., Koenderink, J.J., 2000. Haptic detection thresholds of Gaussian profiles over the whole range of scales. Exp. Brain Res. 132, 369–374. Louw, S., Kappers, A.M.L., Koenderink, J.J., 2002a. Active haptic detection and discrimination of shape. Percept. Psychophys. 64, 1108–1119. Louw, S., Kappers, A.M.L., Koenderink, J.J., 2002b. Haptic discrimination of stimuli varying in amplitude and width. Exp. Brain Res. 146, 32–37. Pawluk, D.T.V., Howe, R.D., 1999. Dynamic contact of the human fingerpad against a flat surface. J. Biomech. Eng. 121, 605–611.

Pont, S.C., Kappers, A.M.L., Koenderink, J.J., 1998. The influence of stimulus tilt on haptic curvature matching and discrimination by dynamic touch. Perception 27, 869–880. Pont, S.C., Kappers, A.M.L., Koenderink, J.J., 1999. Similar mechanisms underlie curvature comparison by static and dynamic touch. Percept. Psychophys. 61, 874–894. Porteous, I.R., 2001. Geometric Differentiation: For the Intelligence of Curves and Surfaces, Second Edition. Cambridge University Press, Cambridge. Provancher, W.R., Cutkosky, M.R., Kuchenbecker, K.J., Niemeyer, G., 2005. Contact location display for haptic perception of curvature and object motion. Int. J. Robot. Res. 24, 691–702. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 2002. Numerical Recipes in C++: The Art of Scientific Computing, Second Edition. Cambridge University Press, Cambridge. Sanders, A.F.J., Kappers, A.M.L., 2008. Curvature affects haptic length perception. Acta Psychol. 129, 340–351. Serina, E.R., Mockensturm, E., Mote Jr., C.D., Rempel, D., 1998. A structural model of the forced compression of the fingertip pulp. J. Biomech. 31, 639–646. Van der Horst, B.J., Kappers, A.M.L., 2008. Haptic curvature comparison of convex and concave shapes. Perception 37, 1137–1151. Wang, Q., Hayward, V., 2006. Compact, portable, modular, high-performance, distributed tactile transducer device based on lateral skin deformation. Proceedings of the 14th Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems, IEEE VR, pp. 67–72. Wong, T.S., 1977. Dynamic properties of radial and tangential movements as determinants of the haptic horizontal–vertical illusion with an L figure. J. Exp. Psychol. Hum. Percept. Perform. 3, 151–164. Wichmann, F.A., Hill, N.J., 2001a. The psychometric function: I. Fitting, sampling, and goodness of fit. Percept. Psychophys. 63, 1293–1313. Wichmann, F.A., Hill, N.J., 2001b. The psychometric function: II. Bootstrap-based confidence intervals and sampling. Percept. Psychophys. 63, 1314–1329. Yarbus, A.L., 1967. Eye Movements and Vision. Plenum Press, New York.