## Abstract

The *two-thirds power law*, postulating an inverse local relation between the velocity and cubed root of curvature of planar trajectories, is a long-established simplifying principle of human hand movements. In perception, the motion of a dot along a planar elliptical path appears most uniform for speed profiles closer to those predicted by the power law than to constant Euclidean speed, a kinetic-visual illusion. Mathematically, complying with this law is equivalent to moving at constant planar equi-affine speed, while unconstrained three-dimensional drawing movements generally follow constant spatial equi-affine speed. Here we test the generalization of this illusion to visual perception of spatial motion for a dot moving along five differently shaped paths, using stereoscopic projection. The movements appeared most uniform for speed profiles closer to constant spatial equi-affine speed than to constant Euclidean speed, with path torsion (i.e., local deviation from planarity) directly affecting the speed profiles perceived as most uniform, as predicted for constant spatial equi-affine speed. This demonstrates the dominance of equi-affine geometry in spatial motion perception. However, constant equi-affine speed did not fully account for the variability among the speed profiles selected as most uniform for different shapes. Moreover, in a followup experiment, we found that viewing distance affected the speed profile reported as most uniform for the extensively studied planar elliptical motion paths. These findings provide evidence for the critical role of equi-affine geometry in spatial motion perception and contribute to the mounting evidence for the role of non-Euclidean geometries in motion perception and production.

- motion invariants
- motion perception
- non-Euclidean geometry
- relation between motion perception and production
- constant equi-affine speed

the interaction between mechanisms of perception and action is pivotal in understanding the workings of the brain. One facet of this interaction is the exquisite sensitivity of the visual system to certain forms of biological motion (Johansson 1973, 1977; Viviani and Stucchi 1992; Blakemore and Decety 2001), which is often understood from the evolutionary point of view as required for survival.^{1} One prominent invariant of biological motion, as it pertains to human movement production, is the *two-thirds power law*:
(1)

The name “two-thirds power law” stems from the originally used equivalent formulation of this kinematic constraint in terms of angular speed, *A*, where *A* = ακ^{2/3} (Lacquaniti et al. 1983). In *Eq. 1*, the hand movement speed is:
(2)
and the curvature of the movement path of the hand is:
(3)
(Oprea 1997; O'Neill 1997), where for any function of time, ρ(*t*), the first and second time derivatives are ρ̇ and , respectively (i.e., and ). For the power law of *Eq. 1*, the exponent β is roughly −1/3, and the velocity (or speed) gain factor α is roughly constant for simple elliptical trajectories. For more complex shapes the velocity gain factor is piecewise constant and the exponent β may deviate from −1/3 (Viviani and Cenzato 1985; Viviani 1986). This power law is a long-established model of trajectory formation in the plane, allowing the derivation of local trajectory kinematics from local path geometry (up to the constant coefficient α).

#### Two-thirds power law and constant equi-affine speed.

The power law in *Eq. 1* describes various types of movement invoking different effectors and utilizing various muscle groups across the body. It is applicable to drawing movements and to hand movements under isometric force conditions (Massey et al. 1992). Further, smooth pursuit eye movements (deSperati and Viviani 1997) and tongue movements during speech kinematics (Tasko and Westbury 2004; Perrier and Fuchs 2008) obey the power law. Locomotive trajectories also comply with the power law (Vieilledent et al. 2001; Hicheur et al. 2005, 2007; Pham et al. 2007), as well as proprioceptive information during arm and leg movements (Viviani et al. 1997; Albert et al. 2005). Compliance with the power law further seems to evolve with age in children (Sciaky et al. 1987; Viviani and Schneider 1991), while adults find it extremely difficult to produce trajectories that significantly violate this law (Viviani 2002). Motion according to the power law is perceived as more uniform than movement at constant speed (Viviani and Stucchi 1992; Levit-Binnun et al. 2006; Dayan et al. 2007), and human movement prediction is facilitated by trajectories conforming to the power law (Kandel et al. 2000).

The ubiquity of the two-thirds power law as a movement invariant motivated previous attempts to discover underlying optimization principles that could account for this law. These included the minimum-jerk model (Flash and Hogan 1985) that favors smoothness maximization (Wann et al. 1988; Viviani and Flash 1995; Todorov and Jordan 1998; Richardson and Flash 2002) and the minimum-variance principle (Harris and Wolpert 1998). More recently, it was hypothesized that the power law may stem in part from correlated noise in the motor system (Maoz et al. 2006). Others have suggested that it may result from more peripheral smoothness effects due to muscle mechanics (Gribble and Ostry 1996) or to nonlinearities involved in mapping from joint to hand coordinates and to the smooth oscillatory nature of joint rotations (Schaal and Sternad 2001; Dounskaia 2007). However, recordings from single neurons in monkey motor and premotor cortexes have shown that population-vector coding obeys the two-thirds power law during execution of motion (Schwartz 1994; Schwartz and Moran 1999, 2000, Polyakov 2001). Moreover, Polyakov et al. (2009b) found neurons in primary motor cortex that encode equi-affine, rather than Euclidean, speed. A more recent functional MRI (fMRI) study that focused on human perception of visual motion also reflected this constraint; the response to a visual display of a cloud of dots moving along elliptical trajectories while obeying this law of motion was much stronger and more widespread than to other types of visual motion (Dayan et al. 2007). Another fMRI study also showed that human movement complying with the two-thirds power law preferentially activates frontal brain regions, including areas within the left premotor cortex associated with biological motion perception (Casile et al. 2010). These studies have indicated that the power law is reflected in both motion production and motion perception processing suggesting that its origin does not lies in low-level biomechanical effects.

Importantly, when planar movement obeys the two-thirds power law, this is equivalent to moving at a constant equi-affine speed (Flash and Handzel 1996, 2007; Pollick and Sapiro 1997; Handzel and Flash 1999).^{2} Formally, for a planar trajectory [*x*(*t*), *y*(*t*)], equi-affine speed is defined as:
(4)
which, geometrically, corresponds to the cubed root of the signed area of the parallelogram created by the instantaneous first and second time derivatives of position vectors along the trajectory. Algebraic manipulations of *Eq. 2*–*4* then yield:
Comparing this expression with the power law (*Eq. 1*), we see that motion with constant equi-affine speed (i.e., with *v*_{es} = const) is equivalent to motion obeying the two-thirds power law.

Within the equi-affine framework, moving at a constant equi-affine speed is the simplest possible kind of motion. Hence, if the brain tends to generate movement within this framework, the simplest kind of motion would be that which obeys the two-thirds power law. Indeed, analysis of recordings from the monkey brain during planar scribbling movements suggests that equi-affine speed is represented in the firings rates of motor-cortex neurons (Polyakov 2006). The range of phenomena described by the power law may thus result from the central nervous system using constant equi-affine speed for planning and controlling movement.

This interpretation suggests that the power law may be more than some empirically observed and possibly accidental motion invariant. More recently Bennequin et al. (2009) have developed a theory suggesting that movement invariants, including the two-thirds power law and the isochrony principle (Viviani and Flash 1995) can be successfully accounted for by a theory of movement timing based on geometrical invariance and that movement duration and kinematics arise from cooperation among Euclidian, equi-affine and full affine geometries. In the present study, however, we focus on motion perception in three dimensions (3D) following the extension of a model based on the constancy of equi-affine speed.

#### Constant equi-affine speed in 3D.

We have recently shown that constant equi-affine speed generalizes to 3D and that 3D movement at constant equi-affine speed entails a power law different from the two-thirds power law. Moreover, general self-paced unconstrained scribbling and shape-tracing hand movements in 3D both seem to be produced at roughly constant spatial (3D) equi-affine speed (Pollick et al. 1997, 2009; Maoz et al. 2009).

Following a rationale similar to that used to prove the equivalence of the two-thirds power law to moving at planar constant equi-affine speed, we derived the formula for motion at constant spatial equi-affine speed (see Pollick et al. 2009). Geometrically, spatial equi-affine transformations preserve the volume (rather than the area) enclosed by a 3D shape. Hence we define spatial equi-affine speed at any point on a curve in terms of the volume of the parallelepiped defined by the first, second, and third derivatives of the position vectors at that point. More formally, we define:
(5)
where *r*(*t*) = [*x*(*t*), *y*(*t*), *z*(*t*)] and “|*u*, *v*, *w*|” denotes the scalar triple-product of any vectors *u*, *v*, *w* ; |*u*, *v*, *w*| = *u*·(*v* × *w*). Here the “·” operator denotes the dot-product between two vectors and “×” denotes the cross-product.

We now need to introduce torsion, which intuitively is the local deviation from planarity^{3}:
(6)
Here, for any vector *u*, ‖*u*‖ is its vector-norm or magnitude. Algebraic manipulations of *Eqs. 2*, *3*, *5*, and *6* yield:
Therefore, motion at a constant equi-affine speed entails a new power law (Pollick et al. 2009):
(7)
which we name the *curvature-torsion power law*. Here β is roughly −1/3 and γ is approximately −1/6. This law suggests that spatial movement speed (*v*) is inversely related to curvature (κ) and, to a lesser extent, to torsion (τ).

#### From movement production to movement perception.

The two-thirds power law seems to generalize from motion production to motion perception in humans. For example, Viviani and Stucchi (1989, 1992) have demonstrated a visual illusion carrying the imprint of this power law. Subjects viewed a moving spot on a computer screen, which repetitively traced out elliptical paths of different eccentricities, or, on different trials, continuous random scribbling-like paths. The movements followed the functional relationship between speed and curvature described by *Eq. 1*. The subjects could control the exponent of that functional relationship, thus altering the velocity profile, and were instructed to select the velocity profile for which they perceived the movement as most uniform. For β = 0, the tangential speed was independent of the curvature and was thus constant throughout the path. If the subjects had judged motion uniformity according to this parameter, they should have selected motion conforming to β = 0. Conversely, the value β = −1/3 denoted motion exactly according to the two-thirds power law. The results of Viviani and Stucchi (1992) suggested that the movements perceived by all subjects as having uniform speed were actually closer to obeying the two-thirds power law (β = −1/3) than to having a constant Euclidian speed (β = 0).

They found that the β selected by the subjects became increasingly closer to −1/3 as the eccentricity of their ellipses increased; the selected β ranged from −0.17 for elliptical motion with eccentricity 0.52 to −0.22 for an eccentricity of 0.96, the mean β-value being −0.18. The β-values chosen by the subjects for the scribbling trajectories were even closer to −1/3 than for the ellipses, with an overall average value of −0.35. Note that, under the conditions selected, the Euclidean tangential speed of the elliptical motion selected as being most uniform varied by up to 200% along the path (Viviani and Stucchi 1992).

More recently, Levit-Binnun et al. (2006) examined the effects of shape (eccentricity) and size (perimeter) of the elliptical trajectories as well as of motion duration and eye fixation on the β-values selected. The ellipse eccentricity and the average tracing speed significantly affected subjects' decisions. Fixation also appeared to play a role for more eccentric ellipses. Ellipse perimeter had little effect on the subjects' decisions even though perimeter changes should have had the same effect on the average speed as changes in the tracing duration. Angular speed and equi-affine speed were examined using simulations to determine whether subjects based their decision of movement uniformity on speed variables other than tangential speed. The results demonstrated the clear sensitivity of the motion perception system to variations in speed along an elliptical trajectory in a manner dependent on the path curvature. Furthermore, the movement appeared to be judged as more uniform when there were minimal differences in angular, tangential, and perhaps also in equi-affine speeds along the trajectory.

The power law is also involved in perceptual anticipation of human movement. Kandel et al. (2000) used a moving dot to reproduce the generation of a handwritten letter, the middle letter excerpted from a cursively written trigram. Subjects were asked to predict which of two possible letters would come next, a task they accomplished well. Prediction accuracy dropped markedly when the kinematics of the letter trajectories were manipulated by shifting the β-value of the two-thirds power law away from -1/3 (leaving the path unchanged). Flach et al. (2004) extended these results to general curvilinear motions. When visual motion vanishes abruptly, the localization is misjudged to be in the anticipated direction of motion (e.g., Freyd and Finke 1984, the representational momentum theory). Flach et al. (2004) demonstrated that this anticipation error is specifically influenced by the power law.

Finally, it has been suggested that non-Euclidean affine geometry plays an important role in human vision. A key transformation in vision is projection. Allowing the physical space to deform in a manner equivalent to a projection of one plane on another (with uniform scaling in an arbitrary direction) results in affine geometry (Smeets et al. 2009). The importance of affine geometry was demonstrated in the tracking of planar trajectories and in extracting structure from motion (Koenderink and van Doorn 1991; Pollick 1997; Todd et al. 1998). Psychophysical studies provided evidence that the judgments of Euclidean properties are considerably less accurate than those of affine properties (Todd and Bressan 1990). More recently, it was also argued that successful interaction with the environment does not depend on accurate metric knowledge and that inaccurate estimates of metric properties do not prevent, for example, accurate grasping (Domini and Caudek 2013). Affine geometry has also been mathematically analyzed and applied in computer and human vision (Lamdan et al. 1988; Faugeras 1993, 1995; Shashua and Navab 1994; Sato and Cipolla 1997; Munich and Perona 1999; Olver et al. 1999). One advantage of the affine representation is that different viewing directions of an object, to first order of approximation, are locally affine invariant (Lindeberg 2013). What is more, it was shown that, at least in monkey inferior temporal cortex, there are populations of single neurons that have invariant representations over different views of familiar objects (Logothetis et al. 1996; Booth and Rolls 1998), potentially assisting in object recognition independently of the viewing direction (Rothganger et al. 2003; Obdrzalek and Matas 2002). Therefore, processing the motion path of an object under an affine framework would render it more invariant to viewing direction.

Equi-affine geometry has a more restricted application in vision than general affine geometry. Yet, if constant equi-affine speed is one of the factors determining when planar motion appears uniform, perhaps motion at constant spatial equi-affine speed (i.e., according to the curvature-torsion power law) is a deciding factor in judging whether spatial motion appears to be more uniform. It has also been argued that affine properties of a shape are invariant to the relative orientation of the eye and the plane of hand movements, for example, when drawing (Pollick and Sapiro 1997). Thus an affine perceptual encoding might simplify the process of drawing the same shape despite large changes in the relative orientation of the eyes and the hand on the drawing plane. Furthermore, similarity between motion production and perception with respect to the dependence of speed on trajectory curvature could be of importance in many different behaviors and tasks requiring visuo-to-motor mapping, especially when the motion is far away from the eye. Some examples are imitation, motor learning from observation, action recognition, social interaction, and so on.

In summary, the ubiquity of the two-thirds power law in planar motion production has been abundantly demonstrated. Furthermore, planar motion according to this power law is perceived as most uniform (Viviani and Stucchi 1989, 1992). The equivalence of the power law to planar motion with constant equi-affine speed has led to its successful generalization to 3D motion generation (Pollick et al. 2009; Maoz et al. 2009). Here we first and foremost tested whether constant equi-affine speed for 3D visual motion is also perceived as more uniform. To answer this question we adapted the experimental setup of Viviani and Stucchi (1992) and Levit-Binnun et al. (2006) to the perception of spatial motion using a virtual reality setup with stereoscopic vision capabilities. The subjects could control the exponent of the power law and had to decide which speed profile seemed most uniform to them. To better compare our results in the first experiment with the previous results of Viviani and Stucchi (1992) and Levit-Binnun et al. (2006), we decided to additionally test whether the viewing distance affects the speed profiles that would be perceived as most uniform for the better-studied planar elliptical motion paths.^{4} For this we used visual stimuli similar to those of Viviani and Stucchi (1992) and Levit-Binnun et al. (2006) but varied the distance of the subjects from the screen.

## MATERIALS AND METHODS

#### Participants.

Seven subjects participated in *experiment 1*, with another seven taking part in *experiment 2*. Subjects' ages ranged from 27 to 39 yr. All but *subjects A.B.* and *R.F.* were right handed. We found no systematic effect of handedness on the results. The Ethical Committee of the Weizmann Institute of Science approved both experiments. All subjects gave their informed consent before their participation in the study. All had normal or corrected-to-normal vision.

#### Apparatus.

The experiments were run in a quiet, dimly lit room. In *experiment 1*, an Apple Macintosh Quad G5 computer presented the stimuli and recorded the subjects' responses. The computer was connected to a high-end Christie Digital Mirage 4000 DLP stereoscopic projector, which back-projected the image at 4,000 lumens in a polarized manner onto a 2,690 × 2,150 mm screen. Frame rate was 96 Hz (48 Hz for each eye), and screen resolution was set to 1,280 × 1,024 pixels in 32 bit color. The 3D virtual environment was created and maintained in real time with the OpenGL Utility Toolkit (GLUT) over GNU C++. We rendered the scene with perspective projection and used a single white-light source infinitely high above the center of the floor (20% ambient, 100% diffuse and specular, and 0% emission), making the red ball drop its shadow directly vertically (Fig. 1, *A*–*E*). The ball was maximally specular and highly shiny to clearly reflect the light. The blue floor was neither specular nor shiny. The camera was set at a viewing angle 20° above the plane that the horizontal plane ellipse traversed (Fig. 1*B*). This graphical environment enabled us to accurately and rapidly create convincingly realistic 3D scenes (Fig. 1, *A*–*E*). We made every effort to keep the scene realistic in terms of lighting, shadows, stereoscopic projection, the correct point of view for the subjects, etc. The subjects, wearing polarized glasses, were seated on a chair optimally placed in the room for viewing the stereo image (∼6,250 mm from the screen, with their eyes opposite the center of the screen). Subjects chose their most comfortable sitting position. They responded via the computer keyboard.

The second experiment was run on an LG Flatron L1950H LCD computer screen connected to a Lenovo Thinkpad T61p computer. Screen dimensions were 378 × 302 mm, and resolution was set to 800 × 600 pixels for fast and clear rendering. A small white spot was rendered over a dark gray background without any shadow or lighting effects (Fig. 1*F*). The distance of the subjects' eyes from the screen was carefully controlled using a chin rest and head brace. Each subject ran the experiment three times with their eyes at a distance of 410, 878, and 1,580 mm from the center of the screen. The screen was set exactly perpendicular to each subject's line of sight (up to the accuracy of a professional spirit level).

#### Task.

In both experiments, once subjects were seated, they were given a sheet of instructions to read. Those instructed them to choose the form of motion resulting in the least absolute variation of perceived speed along the trajectory, i.e., the most uniform speed. They were informed that there was always a unique correct solution and that they controlled the form of motion using the left and right arrow keys of the keyboard in front of them. When they reached the motion that appeared most uniform to them, they were to press the <Space> bar. After reading the instructions they could approach the experimenter for further clarification if needed.

#### Experimental trials and stimuli.

In *experiment 1*, the subjects were presented with five types of paths, which were smoothly and continuously traversed by a small red ball until the subject intervened. The paths are depicted in Fig. 1, *A*–*E*, with the experimental (*x*, *y*, *z*) coordinate system in Fig. 1*A*. The paths were a frontal-plane ellipse (FPE; major:minor axes 26:17, eccentricity 0.76), horizontal-plane ellipse (HPE, same major: minor axis ratio and eccentricity), bent ellipse (BE; *x*:*y*:*z* extension proportions 26:17:9), left-to-right spiral helix (SH; ratio of *x* extension to final SH diameter was 26:17), and slightly bent figure-eight (FE; *x*:*y*:*z* extension proportions 26:17:2; see Table A5 for the mathematical formulas of the different figural forms).^{5}

The subjects were given trial runs of the experiment on the FPE path until they were convinced that they understood the task and knew how to operate the keyboard. This usually took no more than half a minute. Only when the subjects indicated that they fully understood the operation of the keyboard and the progression of the experimentation did the experiment begin. The FPE paths were presented first, then the HPE paths, followed by BE, SH, and FE.

The first two paths (FPE and HPE) were planar and thus had zero torsion throughout. Hence, they were run with only the power law of *Eq. 1*. The FPE path was one of the paths used by Viviani and Stucchi (1992) and Levit-Binnun et al. (2006) (although there the ellipse was slanted at 45°). It was included here mainly to test whether our 3D virtual reality experimental setup produced the same results as those obtained on a planar computer screen. The HPE was included to test the effect of the spatial orientation of the elliptical path.

The remaining paths contained some torsion and were presented to the subjects in different trials with either the *coupled curvature-torsion power law*:
(8)
or with the *torsion-only power law*:
(9)
The first power law formed an experimental condition where the subjects had to seek the most uniform speed when controlling (rather than κ alone). The second power law fixed the curvature exponent and gave the subjects control only over the torsion exponent.

Each path was run once with six different initial β-values: −1/2, −1/3, −1/6, 0, 1/6, and 1/2. These initial β-values are termed β_{0}. A random order of presentation of the initial β-values and power laws for each path was created using a pseudorandom generator and the constraint that consecutive initial β-values could not appear consecutively for the same path. The planar-elliptical paths thus appeared 6 times each and the other paths appeared 12 times each (6 β_{0} × 2 power laws). A complete experiment for each subject therefore consisted of 48 trials.

In *experiment 2*, subjects were presented with a small white spot that traversed only one shape, the FPE used in *experiment 1* without any lighting effects and without casting a shadow (Fig. 1*F*). The subjects were given trial runs and the experiment commenced similarly to *experiment 1*. As the path was planar, it was run with only the power law of *Eq. 1*, using the same initial β-values (β_{0}) as in *experiment 1*. We aimed to repeat the experimental stimuli of Viviani and Stucchi (1992) and Levit-Binnun et al. (2006) (although with the elliptical path not slanted at 45°) to test the effect of viewing distance on the speed profiles chosen as most uniform. All subjects viewed the motion from distances of 410, 878, and 1,580 mm. The middle distance, 878 mm, replicated the viewing angle of the subjects in *experiment 1* (average of 12.15°; range of 12.05–12.24°). The first distance, 410 mm, was about half this middle distance and was close to the ranges of the viewing distances reported by Viviani and Stucchi (1992) and Levit-Binnun et al. (2006) (350–450 and 300–400 mm, respectively). The third distance, 1,580 mm, was found in a pilot experiment to be the farthest that subjects could sit from the screen and still perform well. The experiment was run using 6 different β_{0} at 3 different distances from the screen, resulting in 18 trials per subject.

#### Within-trial experimental procedure.

In each trial the red ball (in *experiment 1*) or white spot (in *experiment 2*) traversed a single path. All paths were defined by 1,000 quintuplets of (*x*, *y*, *z*, κ, τ) that were accurately analytically precomputed in Matlab (Mathwork) and saved to a file. This file was read into the realtime management program before the experiment began. The *x*, *y*, and *z* were the spatial coordinates of the points on the path, κ was the curvature, and τ was the torsion at each point.

The ball/spot was always initially placed as in Fig. 1, *A*–*E*, for *experiment 1* and as in Fig. 1*F* for *experiment 2*. The initial curvature and torsion were used to compute the initial speed, *v*_{0}, from *Eqs. 1*, *8*, or *9*, depending on the path and power law used in that trial. Every time the scene was refreshed (∼150 times per second, i.e., faster than the screen refresh rate) the computer clock was queried, and the duration from the previous screen-refresh, Δ*t*, was computed. This Δ*t*, together with the speed at the previous scene-refresh, *v*_{t−Δt}, enabled us to compute the distance traveled on the path and thus the new position on the path. The new curvature and torsion were now computed by linear interpolation from the neighboring quintuplets, and from them the new speed was computed, based on the appropriate power law, and so on.

This continuous repetitive progression on the path continued uninterrupted until the subject pressed the left or right arrow key, which decreased or increased the β-values, respectively. Alternatively, the subject could press the <Space> bar, which signaled that the subject had decided that the motion with the current β-value was the most uniform and she or he was ready to move on to the next trial. We designate this final β-value value as β_{f}. After the <Space> bar was pressed, the program paused for half a second between different trials of the same path and 2.5 s when a new path was loaded. Motion following a press of the left or right arrow key, during a change of β-values in the same trial, was smooth and without a pause. Possible β-values were as follows: −2/3, −1/2, −5/12, −1/3, −1/4, −1/6, −1/12, 0, 1/12, 1/6, 1/4, and 1/2. For a β-value of −2/3 pressing the left-arrow key had no effect; consequent pressing the right-arrow key immediately moved the β to −1/2. The same, in the opposite direction, was true for β = 1/2. However, the subjects were not notified when reaching the extreme β-values and thus could not know where they were on the β-scale (nor were they informed how many steps the β-scale included). The α-values in the power laws were updated every time the β-value was changed to keep the average speed fixed at 500 mm/s. When debriefed after the experiments, subjects reported smooth transitions between trajectories when pressing the left/right arrow keys.

#### Data recording and analysis.

The empirical information of each trial was saved in a file. This included the type of power-law tested and all α- and β-values along the trial together with the time stamp describing when these values went into effect on the screen. This resulted in an accurate, timed record of all the user inputs for each trial.

We were especially interested in the β_{f} (the final β-value of each trial). There were 42 such β_{f} values for each power law over each shape in *experiment 1* or distance in *experiment 2* (6 β_{f} × 7 subjects). On rare occasions a subject pressed the <Space> bar twice consecutively or otherwise ended up with clear outlier values for β_{f}. Outliers were defined as any β_{f}-value more than two standard deviations away from mean value. In *experiment 1*, across all subjects and shapes, there were 4 outliers in 84 β_{f} samples for the two-thirds power law, 7 in 126 β_{f} samples for the coupled curvature-torsion power law, and 4 in 126 for the torsion-only power law (5%, 6% and 3%, respectively). Over all power laws, 15 of the 336 β_{f} values (<5% of the data) were designated outliers. In *experiment 2*, 5 of the overall 126 values (42 values per distance × 3 distances; <4%) were designated as outliers. All outliers were removed from the analysis.

Analysis for statistically significant difference in the distribution of the different β_{f} over the various β_{0} generally used Kruskal-Wallis nonparametric one-way ANOVA and multiple comparison test of means with Tukey's honestly significant difference criterion at the 0.05 significance level for all pairs of means. Comparisons of β_{f} -distributions with specific mean values, like −1/3,−1/6, and 0, were performed using a one-tailed *t*-test at the 0.05 significance level.

## RESULTS

#### Experiment 1.

Here we tested which exponents would result in speed profiles that appeared most uniform to the subjects for the power-law in *Eq. 1* on FPE and HPE and for the power-laws in *Eqs. 8* and *9* on BE, SH, and FE (Fig. 1, *A*–*E*).

The initial exponent value of each trial, β_{0}, had little or no effect on β_{f}, the final exponent chosen by the subjects for the speed profile perceived as the most uniform. For the two-thirds power law, significant differences (at *P* = 0.05) were found for just 1 of 30 pairs tested, β_{0} = −1/2 & β_{0} = 0 for FPE, which is within what would be expected by chance (*P* = 0.79, binomial test). For the coupled curvature-torsion power law, significant differences were found between two of 28 pairs tested, β_{0} = −1/2 & β_{0} = 1/6 and β_{0} = −1/2 & β_{0} = 1/2, which is again not significant (*P* = 0.41, binomial test). No pairs were significantly different for the FE and for the torsion-only power law. Overall, only 3 of 86 pairs of the β_{f}-distributions tested were significantly different over the β_{0} (*P* = 0.81, binomial test). This accords with previous results for the two-thirds power law on planar elliptical paths (Viviani and Stucchi 1992). We therefore pooled the β_{f} results across all β_{0}.

A histogram of β_{f}-values over all subjects and β_{0} for the two-thirds power law (*Eq. 1*) for FPE and HPE is given in Fig. 2, *A* and *B*, respectively. The means ± SD of those β_{f} distributions are −0.32 ± 0.08 and −0.31 ± 0.12, respectively (see Table A1 for a subject-by-subject breakdown). The β_{f}-distributions for FPE and HPE are significantly negative (*P* = 2·10^{−25} and *P* = 10^{−19}, paired one-tailed *t*-test, respectively) yet not significantly different from −1/3 (*P* = 0.21 and *P* = 0.23, paired *t*-test, respectively), nor are they significantly different from one another (Kruskal-Wallis at *P* = 0.05).

Figure 3 depicts histograms of β_{0}-values over all subjects and β_{0} for the coupled curvature-torsion power law (*Eq. 8*) for BE, SH, and FE, with means ± SD values of −0.11 ± 0.12, −0.20 ± 0.14, and −0.22 ± 0.08, respectively (see Table A2 for a subject-by-subject breakdown). The β_{f}-distributions for BE, SH, and FE are all significantly negative (*P* = 5·10^{−7}, *P* = 4·10^{−12}, and *P* = 2·10^{−19}, paired one-tailed *t*-test, respectively). Comparing between shapes, the β_{f}-distribution for BE is significantly different from those of SH and FE, while the latter are not significantly different from each other (Kruskal-Wallis at *P* = 0.05).

Figure 4 depicts histograms of β_{f}-values over all subjects and β_{0} for the torsion-only power law (*Eq. 9*) for BE, SH and FE, with means ± SD values of −0.05 ± 0.04, −0.06 ± 0.11, and −0.16 ± 0.15, respectively (Table A3 for a subject-by-subject breakdown). These mean β_{f} values for BE, SH, and FE are all significantly negative (*P* = 5·10^{−11}, *P* = 0.002, and *P* = 5·10^{−24}, paired one-tailed *t*-test, respectively). The means of the distributions for BE and SH are significantly different from −1/6 (*P* = 4·10^{−21} and *P* = 2·10^{−7}, paired *t*-test, respectively), while the mean of FE distribution is not significantly different from −1/6 (*P* = 0.17, paired *t*-test). Comparing between shapes, the β_{f}-distribution for FE is significantly different from those of BE and SH, with the latter two not significantly different from each other (Kruskal-Wallis at *P* = 0.05).

#### Experiment 2.

The means of the β_{f}-distributions we found for the two planar paths, FPE and HPE, in *experiment 1* above were very close to −1/3. Yet, previous literature that investigated planar elliptical motion found that the exponent values perceived as most uniform ranged from −0.15 to −0.25 (Viviani and Stucchi 1992; Levit-Binnun et al. 2006). We hypothesized that this difference stems from the difference in the viewing distance (in the depth direction) or viewing angle between their experimental condition and ours. We therefore simplified the stimulus in our experiment to a small, white spot moving on a regular computer screen with no lighting, shadowing, or stereoscopic effects (Fig. 1*F*, similarly to that used in previous studies) and tested the effect of viewing distance on the β_{f}-distributions.

Figure 5 depicts histograms of β_{f}-values over all subjects and β_{0} for the distances of 410, 878, and 1,580 mm, with means ± SD values of −0.18 ± 0.09, −0.22 ± 0.11, and −0.23 ± 0.10, respectively (see Table A4 for a subject-by-subject breakdown), all significantly negative (*P* = 6·10^{−16}, *P* = 3·10^{−16}, and *P* = 8·10^{−17}, paired one-tailed *t*-test, respectively). In accordance with the means, the histograms for 410, 878, and 1,580 mm are left skewed, roughly centered and right skewed, respectively. Moreover, a linear regression through the β_{f}-data over the three distances (Fig. 5, in grey) results in a line with a significantly positive slope (*P* = 0.047), suggesting that the upward trend in β_{f}-values over the distances is significant.

## DISCUSSION

In *experiment 1* a ball, which was projected on a screen using a stereoscopic projector, with perspective, lighting, and shadowing effects, smoothly traversed one of several paths in 3D space. In *experiment 2* a small, white spot smoothly traversed a simple, planar elliptic path on a computer screen, and subjects viewed it from three different distances from the screen. In both experiments, the subjects controlled the power-law relation between the moving spot's instantaneous speed and the local geometry of the path and were instructed to find the value that resulted in the most uniform speed along the path, the exponent β_{f}.

#### The kinetic-visual illusion of uniform speed in 3D.

The β_{f}-values for FPE and HPE suggest that the stereoscopic display was appropriate for investigating a 3D version of the kinetic-visual illusion reported by Viviani and Stucchi (1992) and Levit-Binnun et al. (2006) for planar motion. If anything, the β_{f} we found were more negative and closer to −1/3 than those in the earlier studies (see below), suggesting a stronger effect of the curvature on the speed profile that was perceived as most uniform. The β_{f}-values of the coupled curvature-torsion power law (*Eq. 8*) were significantly negative for all 3D shapes (BE, SH, and FE). As β_{f} = 0 is compatible with constant Euclidean speed, the significantly negative values (Fig. 3) suggest that such constant tangential Euclidean speed did not appear most uniform to our subjects. Rather, for all shapes, the speed profile they selected as most uniform was locally dependent on geometry (i.e., on , where κ and τ are the path curvature and torsion, respectively).

The difference between planar and spatial motion at constant equi-affine speed is the dependence of the spatial speed on torsion in addition to curvature in the latter (compare *Eq. 1* with *Eq. 7*). We examined torsion specifically using the torsion-only power law (*Eq. 9*), where the curvature exponent was fixed and the subjects controlled only the torsion exponent. There we would expect the chosen exponent, β_{f}, to distribute rather uniformly on all possible values if torsion played no part in determining the most uniform-looking motion under our experimental conditions, and we would expect β_{f} = 0 if torsion only disrupted the illusion of uniform speed. However, the β_{f}-distributions were not uniform and were also significantly negative for all shapes (Fig. 4).

Both and torsion alone affected the speed profile the subjects selected as most uniform. Therefore, we conclude that the local-geometric measures of curvature and torsion of the movement path appear to affect the chosen speed profile in 3D just as the local-geometric measure of curvature affects the selected planar speed profile (Viviani and Stucchi 1992; Levit-Binnun et al. 2006). An effect of the local geometry of the path on the local speed is in line with motion at constant equi-affine speed (*Eq. 7*).

However, constant equi-affine speed appears not to explain all the variability in the speed profiles selected as most uniform. The local-geometric factor, , had a significantly greater impact on shapes SH and FE than on BE, and torsion alone had a significantly greater influence on FE than on BE and SH. These differences indicate that the global geometry of the path might play a role together with the very local geometry to determine the most apparently uniform speed profile over that path. We found a similar effect for 3D motion production, where the curvature-torsion power-law explained much, but not all, of the variance in the speed profile. There too the global geometry of the path played a role in determining the local relation between speed and geometry for each shape (Maoz et al. 2009).

#### Planar elliptical paths and the kinetic-visual illusion.

The kinetic-visual illusion was discovered for planar, elliptical motion paths (Viviani and Stucchi 1992). In the original study and in a later and more extensive one (Levit-Binnun et al. 2006), the power law-exponents chosen as most uniform were significantly influenced by the eccentricity of the elliptical paths; the higher the eccentricity (i.e., the more elongated the ellipse,) the more negative the β_{f}. In both of those studies the mean β_{f}-values ranged from roughly −0.15 to −0.25. The β_{f} of elliptical paths with an eccentricity similar to our FPE and HPE and traced at a comparable average speed and without eye fixation were significantly negative, averaging around −0.17. In contrast, the mean β_{f}-values for FPE and HPE in *experiment 1* were −0.32 and −0.31, respectively, and hence while significantly negative were not significantly different from −1/3 nor significantly different one from the other.

These results are interesting for three reasons. First, it suggests that something in our experimental conditions resulted in a stronger kinetic-visual illusion than that obtained using the simpler, planar visual displays of earlier studies. Second, our subjects favored statistically indistinguishable speed profiles as most uniform for two very differently oriented elliptical paths, frontal (FPE) and horizontal (HPE), despite the various depth cues used in *experiment 1* (cf. Figs. 1, *A* and *B*). Third, on average, our subjects chose as most uniform a speed profile in which the Euclidean speed at the least curved point was >1.5 times faster than that at the point of greatest curvature.

Examining the different exponents chosen as most uniform for our ellipses and those investigated in earlier literature (Viviani and Stucchi 1992; Levit-Binnun et al. 2006), we now focus on the FPE of *experiment 1*, as it was the most similar path to those investigated in this earlier literature. There are three prominent differences between our experimental conditions and those of the earlier literature. First, our stimulus was more natural and realistic, viewed with a stereoscopic display together with perspective projection, lighting, and shadow casting, all enhancing the three-dimensionality of the scene. Second, our subjects were seated 6.25 m from a roughly 3.5-m screen (diagonal measure), while subjects in earlier studies sat ∼30 cm away from roughly 40-cm screens. The viewing angle was therefore considerably different. Moreover, the larger distance and screen size in *experiment 1* rendered any body movement negligible in altering our subjects' viewing angles. In contrast, in the earlier studies subjects' head position varied by up to 15 cm across a distance of ∼30 cm from the 40-cm screen.

In pilot studies (not shown) we removed the various depth cues from the visual stimulus traversing the FPE, while maintaining the viewing distance and viewing angles of *experiment 1*. The speed profiles selected by the subjects as most uniform remained virtually the same. Therefore, we hypothesized that the viewing distance or viewing angle appear to be the major factors leading to the difference between our results and those of Viviani and Stucchi (1992) and Levit-Binnun et al. (2006) for the FPE. *Experiment 2* was designed to test this hypothesis as well as to assess whether the main factor contributing to the difference was the viewing distance or the viewing angle.

In *experiment 1*, the FPE was viewed at a distance of 6,250 mm from a 2,690 × 2,150 mm screen. In *experiment 2*, where we showed subjects, who sat at different distances from the screen, a small, white spot traversing a planar, elliptical path, one of the viewing distances was 878 mm from a 378 × 302 mm screen. This was specifically chosen to have the same horizontal and vertical viewing angles as *experiment 1* (24.3 and 19.5°, respectively). Yet, the distributions of the β_{f} in the two experiments were significantly different (Wilcoxon rank sum test, *P* = 10^{−4}), with mean β_{f} for FPE in *experiment 1* and mean β_{f} for the viewing distance of 0.878 in *experiment 2*. It therefore seems more likely that the value of the two-thirds power law exponent, corresponding to the speed profile perceived as most uniform, depends on the distance of the viewer from the stimulus rather than on the viewing angle.

In *experiment 2*, at a distance of 470 mm, which was closer to the distances used in previous studies, the β_{f}-distribution we found was −0.18 ± 0.09, well within the range reported there (Viviani and Stucchi 1992, Levit-Binnun et al. 2006). The systematic differences between the β_{f}-distributions for the three viewing distances in *experiment 2* (Fig. 5) are also noteworthy. The increasing distance of the computer screen from the retina should have had a similar effect to uniformly scaling the ellipse size on the retina. However, while previous work showed a very small effect of the perimeter size of such elliptical paths on the speed profile chosen as most uniform (Levit-Binnun et al. 2006), changing the distance between the subjects and the screen in *experiment 2* did significantly affect the distribution of selected speed profiles. As the distance from the screen increased, the mean β_{f} decreased from −0.18 to −0.23 (Fig. 5), coming closer to the β_{f}-values found in *experiment 1* for the FPE and HPE. A potential explanation for the differences between the β_{f}-values in *experiments 1* and *2* has to do with the greater distance from the screen in *experiment 1* than even the greatest distance in *experiment 2*. This greater distance in *experiment 1* may have resulted in a parallel projection of the stimulus image on the retina, while *experiment 2* afforded only polar projections for the different distances.

It could be claimed that the differences between the β_{f}-results in *experiments 1* and *2* are attributable to differences in the screen resolution. Let us therefore examine how different the visual angles of one pixel in the two experiments are. The size of a pixel in *experiment 1* was about 2.1 × 2.1 mm, while that in *experiment 2* was 0.5 × 0.5 mm, resulting in a visual angle of 0.019° in *experiment 1* and 0.07, 0.033, and 0.018° for the distances of 410, 878, and 1,580 mm, respectively, in *experiment 2*. Therefore, the visual angle of a single pixel for the 1,580 mm distance in *experiment 2* was similar to that of *experiment 1*. However, the β_{f}-results were different, −0.32 vs. −0.23, on average, providing evidence against a major contribution of the screen resolution to the difference between the results in *experiments 1* and *2*.

Another possibility is that the sensitivity of the visual system to noise in the experimental conditions that prevailed in *experiments 1* and *2* contributed to the dissimilarity between the results of the two experiments. However, it was shown that noisy-looking scribbling-like trajectories are perceived as most uniform for β_{f} rather close to −1/3 (Viviani and Stucchi 1992) and that, at least for motion production, both white and correlated noise actually similarly contribute to the two-thirds power law (Maoz et al. 2006). Hence, it is hard to conceive of a noise type that would shift β_{f} closer to −1/3 for *experiment 1* and not for *experiment 2*. Last, it is also unlikely that the perspective projection, which we used to render the stimulus in *experiment 1* but was not used in *experiment 2*, may have contributed to the different results between *experiments 1* and *2*. This is because the perspective projection had relatively little effect on the FPE in *experiment 1*, because it lay on a frontal plane.

#### Our results and the stereo-kinetic effect.

The long-known stereo-kinetic effect (or kinetic depth-effect) occurs when a planar object rotates, and with it a shadow or an outline changes in a manner compatible with a rotation of a 3D object. This evokes an overall sensation of depth, i.e., of a 3D object rotating, with the 3D interpretation overriding the planar one (Fisichelli 1946; Wallach and O'Connell 1953). Under appropriate conditions, even a single dot can evoke this effect (Mefferd and Wieland 1967a,b; Ullman 1979; Braunstein 1962; von Hofsten 1974). This effect might be claimed to play a role in our subjects' velocity judgment: presented with a spot moving along an ellipse, subjects may perceive it as a planar projection of a spot moving on a slanted circle. If the spot were to traverse this slanted circle at constant Euclidean tangential speed, its elliptical projection would progress at exactly constant equi-affine speed (see Viviani and Stucchi 1992). Could our subjects have selected the roughly constant equi-affine speed profiles as most uniform on ellipses because they interpreted these stimuli as planar projections of underlying 3D stimuli moving at a constant Euclidian velocity?

Perceiving an ellipse as a circle slanted in 3D would have given rise to different flatness cues (e.g., accommodation, vergence, pixel size, etc.; Young et al. 1993) than those perceived by our subjects, rendering this interpretation unlikely. Also, Viviani and Stucchi (1992) already provided evidence against this claim, suggesting that this effect did not explain why an additional type of trajectories on which they tested their subjects, i.e., scribbling trajectories, were perceived as most uniformly traversed for β_{f} ≈ −1/3. The 3D setup we used here provides further evidence against the interpretation of our results as possibly stemming from the stereo-kinetic effect. Our stimuli included a more realistic 3D sphere than the stimuli of Viviani and Stucchi (1992) and Levit-Binnun et al. (2006), with lighting, shadows and perspective congruent with 3D motion paths. Moreover, as shown in Fig. 1, in our virtual reality setup a sphere moving along the FPE would look very different from one moving on a slanted plane. The shadow would have traversed an elliptical path instead of the linear one it followed, the lighting would be different, and the size of the ball, which is constant for frontal-plane movement, would vary along a slanted plane, increasing when the ball was closer and decreasing when it was farther away. The interpretation of the FPE stimulus of Fig. 1*A* as a projection of a circle on some slanted plane is therefore unlikely. The same is true for all other paths used in *experiment 1*. In *experiment 2*, the stereo-kinetic effect could not explain why different velocity profiles were selected for different distances from the screen, so it is once again unlikely as an alternative explanation of the results.

#### Geometric invariance and perception of uniform motion.

It is insightful to view the invocation of the stereo-kinetic effect, as a potential explanation for our results and those of the earlier literature, as part of a wider attempt to uncover the functional geometry used by the central nervous system for motion perception and production. The stereo-kinetic interpretation strives to explain the results within the framework of Euclidean geometry. It suggests that the simplest Euclidean invariant, motion at constant Euclidean speed, although on a slanted plane, lies at the heart of the visual-kinetic illusion.

However, this Euclidean interpretation does not hold, and the earlier results of Viviani and Stucchi (1992) and Levit-Binnun et al. (2006) and recent findings on 3D movement production (Pollick et al. 2009; Maoz et al. 2009) together with our results here suggest that the most parsimonious interpretation of planar and spatial results may be found in the non-Euclidean equi-affine framework, or even more general and abstract geometries (Bennequin et al. 2009). As was demonstrated over the last decade and a half, the many phenomena apparently obeying the two-thirds power law, in both motion perception and production, can all be alternatively explained as stemming from the simplest equi-affine-geometric invariant: movement at constant equi-affine speed (Flash and Handzel 1996, 2007; Pollick and Sapiro 1997; Handzel and Flash 1999; Bennequin et al. 2009). Population-vector analysis of single cell activity in monkey motor cortex during motion execution suggests the direct representation of constant equi-affine speed in the primate brain (Schwartz 1994; Schwartz and Moran 1999, 2000). Polyakov et al. (2009a,b) have used parabolic constant-equi-affine-motion segments as potential building blocks of movements and related them to the underlying neural activity.

In addition, an fMRI study in humans showed that brain activity when viewing visual motions with constant equi-affine speed was stronger and more widespread than for other types of movement (Dayan et al. 2007). A large network of cortical brain-regions involving both visual (e.g., superior temporal sulcus) and motor areas (e.g., dorsal and ventral premotor cortex, caudal cingulate zone, supplementary motor area), frontal regions (Brodmann area 44 of the inferior frontal gyrus), as well as subcortical areas (e.g., cerebellum and basal ganglia) responded more strongly to movement complying with the two-thirds power law compared with other types of motion. Another fMRI study suggested that the observation of human movement complying with the two-thirds power law preferentially activated frontal brain regions, including motor-related ones associated with motion perception and motor expertise (Casile et al. 2010). These results provide more direct neural evidence for the response of multiple brain areas to the two-thirds power law, possibly indicating the importance of equi-affine velocity and equi-affine transformations in neural computations underlying movement perception and production. An earlier fMRI study demonstrated the selective responses of several temporal brain regions to human actions and to point-light displays, highlighting the importance of coding the form and motion of biological movements (Pesuskens et al. 2005). Preliminary attempts have also been reported in Vangeneugden (2010) to test whether neurons that were responsive to visual dynamic-action in temporal brain regions also respond to the two-thirds power law during the observation of human locomotion, using the experimental paradigm detailed in Vangeneugden et al. (2011). These attempts have not yielded clear-cut results so far. We are currently making further efforts in this direction, involving both fMRI and EEG studies aimed at identifying the neural correlates of the psychophysical findings reported here and in earlier studies.

However, there are also well-known systematic deviations from the constant equi-affine speed-invariance. The most striking may be the systematic deviations in the production of the paradigmatic shape on which the two-thirds power law was initially demonstrated, the planar ellipse (Wann et al. 1988), especially when it is traced freely in space (Sternad and Schaal 1999). The same holds for motion perception, where what subjects perceive as the most uniform speed profile was found to depend on the eccentricity the elliptical paths (Viviani and Stucchi 1992; Levit-Binnun et al. 2006). Similarly, there was a significant difference between the power law exponents resulting in most uniform motion over planar elliptical and random-scribbling paths (Viviani and Stucchi 1992).

More recently, careful analysis of spatial hand trajectories during motion production showed that the majority of the variance in the movement speed could be explained using the local dependence of the speed on geometry in accordance with constant equi-affine speed. Yet, a large part of the remaining variance depended on the global geometric form of the traced path, which is incompatible with the simple constant-speed invariance of equi-affine geometry (Maoz et al. 2009). It was therefore suggested that the equi-affine geometry might not be broad enough to completely encompass the richness and diversity of human movement production. Our finding here that subjects chose significantly different power law exponents across some 3D shapes for the coupled curvature-torsion power law as well as for the torsion-only power law in *experiment 1* is in line with this idea. Our results indicate that global geometry may play a role in motion perception, as well as in the production of spatial motion. They also suggest that it is necessary to venture beyond equi-affine geometry to explain the finer details of human movement perception.

Bennequin et al. (2009) have recently done just that. They showed that the duration and compositionality of drawing movements and locomotive trajectories arise from cooperation among constant-speed movements in the Euclidean, equi-affine, and full-affine geometries. Most dominant is the equi-affine geometry, with full-affine geometry the second most influential. This suggested that a more complete account of human movement production may require incorporating full-affine geometry, which is more general than equi-affine geometry. In another recent study (Pham and Bennequin 2012), a new direct test of affine invariance was designed and applied to scribbling movements. Segmenting scribbling movements into randomly selected segment pairs, the path of the first segment in the pair was geometrically transformed to best match the path of the second segment in the pair. The optimal transformation was also used to transform the timing parameterization of the first and second segments and hence the velocity profiles of the two segments. This analysis showed that when two path segments are similar under transformations belonging to the affine group, their time parameterizations are also similar. Hence, this study directly demonstrated the existence of affine invariance in the production of hand movements. It is therefore possible that the dependence of the speed profile that is perceived as most uniform on the global geometry of the path in our results necessitates the integration of a full-affine invariant into the explanation of human perception of 3D motion.

In our results, the speed profiles perceived as most uniform for the FPE in *experiment 1* were generated by a power-law exponent that was very close to −1/3. In contrast, the corresponding power-law exponents in *experiment 2* depended on the viewing distance (rather than the viewing angle) but were always less negative and farther from −1/3 than that of *experiment 1*. Various factors that were examined in 3D motion perception studies, such as viewing conditions, the integration of different monocular and binocular cues, and accumulated past experience (Pierce et al. 2013; Purves et al. 2010), may have contributed to this difference.

However, an arguably more-interesting and parsimonious explanation relies on non-Euclidean geometry and focuses on the difference in distance between the subjects and the screen in our two experiments. In vision, affine transformations are obtained when a planar object is rotated and translated in space and then projected onto the retina via parallel projection. This is a particularly good model of the human visual system when the object is flat enough, and far away from the eye (Pollick and Sapiro 2009). Thus, in our study, the visual judgment of motion could have been better approximated by affine transformations for the greater distances from the eye in *experiment 1* compared with *experiment 2*. If so, our subjects may have favored constant affine speed as most uniform for the FPE and HPE, which is similar to the two-thirds power law for an ellipse. This could explain the nearly constant equi-affine speed profile that they deemed to be most uniform for the FPE and HPE. As for the ellipses in *experiment 2*, these were viewed from closer up, and hence the projection would have been more polar and less parallel, resulting in a speed profile that is not so close to the two-thirds power law.

A full account of our perceptual findings might, in fact, require a mixture of several geometries, similarly to the case of movement production (Bennequin et al. 2009). In this case, the weight of the affine geometry component could increase the farther the observers are from the object, thus influencing the speed profile perceived as most uniform. If the human brain does use a combination of geometries, Euclidean, equi-affine, affine, and possibly even projective, to plan, control, and perceive movements, only a theory of movement production and perception that correctly incorporates all these geometries will be broad enough to completely capture the fine details of these processes.

#### Conclusions.

We have examined the role of constant equi-affine speed as an invariant in planar and spatial motion perception, generalizing the kinetic-visual illusion from 2D to 3D. 3D movement along spatial paths of different shapes appeared most uniform for speed profiles that were generally closer to constant spatial equi-affine speed than to constant Euclidean speed. We also demonstrated the specific impact of path torsion on the speed profiles selected as most uniform. However, the equi-affine framework did not account for all the variability in the speed profiles selected as most uniform among paths of different shapes. In addition, investigating the perception of motion along the more extensively studied planar elliptical paths, we established an effect of viewing distance on the speed profile reported as most uniform. Interestingly, this effect was hardly present when uniformly scaling the paths on the screen, which should have resulted in a similar projection on the retina.

These results unlikely depend on a single more-mundane effect like screen resolution, perimeter, flatness cues, or sensitivity and noise in the visual system, but they might possibly stem from the right combination of these effects. Nevertheless, a much more exciting, and arguably plausible, driving force for our results is the existence of non-Euclidean geometric invariants that underlie both motion production and perception. In fact, our results cannot be explained only within the framework of equi-affine geometry, and their full description may require a combination of Euclidean, equi-affine, full-affine, and even projective geometries. Our findings thus contribute to the growing evidence for the potential role of various non-Euclidean geometries in perception and production of movement by the central nervous system.

## GRANTS

U. Maoz was partially supported by the Dean Fellowship of the Faculty of Mathematics and Computer Science of the Weizmann Institute of Science and by a fellowship from the Van Leer Jerusalem Institute. This research was supported in part by grants from the Ralph Schlaeger Charitable Foundation and Florida State University's Big Questions in Free Will Initiative (both to U. Maoz). This research was also supported in part by Human Frontier of Science Program HFSP-RGP0054/2004-C and FP6–2005-NEST-Path Imp 043403 and by the European Union FP6 COBOL and FP7 Integrated Project VERE (No. 257695 to T. Flash). T. Flash is an incumbent of the Dr. Hymie Moross professorial chair.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: U.M. and T.F. conception and design of research; U.M. performed experiments; U.M. analyzed data; U.M. and T.F. interpreted results of experiments; U.M. prepared figures; U.M. and T.F. drafted manuscript; U.M. and T.F. edited and revised manuscript; U.M. and T.F. approved final version of manuscript.

## ACKNOWLEDGMENTS

We thank Ronit Fuchs, Daniel Bennequin, and Alain Berthoz for meaningful discussions and advice. Uri Maoz is presently at the Division of Biology, California Institute of Technology, Pasadena, CA.

## Appendix

The appendix presents various results on a subject-by-subject basis. First, in Table A1, are the two-thirds power-law exponents that the subjects chose as resulting in most uniform movement for the FPE and HPE in *experiment 1*. The exponents chosen for the BE, SH, and FE with the coupled curvature-torsion power-law and the torsion-only power-law then follow in Tables A2 and A3, respectively. Table A4 then gives the exponents that the subjects chose in *experiment 2* over the three distances from the screen. Last, Table A5 gives the formulas we used to create each of the shapes presented to the subjects in *experiment 1*.

## Footnotes

↵1 Curvature is the inverse of the radius of curvature and is a purely geometrical property of the path, irrespective of speed along the path. Moreover, curvature at every point along a planar path completely determines the path up to Euclidian transformations (which include translations, rotations and reflections; Oprea 1997).

↵2 For an in-depth explanation of equi-affine geometry and equi-affine speed, see (Flash and Handzel 2007) and (Maoz et al. 2009) with its Supplemental Information.

↵3 Torsion, like curvature, is a purely geometrical property of the path, irrespective of the speed profile over the path. Whereas curvature at every point along a planar path is enough to completely determine it up to Euclidian transformations, only curvature and torsion together determine a 3D curve in the same manner (Oprea 1997).

↵4 The viewing distance is the distance from the observer's eyes to the center of the motion path, in the depth direction.

↵5 The SH is not continuous in nature. When the ball reached the last sample in this path, it immediately restarted from the first sample.

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