^{1}

^{2}

^{3}

^{2}

^{*}

Conceived and designed the experiments: DB AB TF. Performed the experiments: AB. Analyzed the data: DB RF TF. Contributed reagents/materials/analysis tools: RF. Wrote the paper: DB RF AB TF. Developed the mathematical models: DB RF TF.

The authors have declared that no competing interests exist.

Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartan's moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limaçons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations.

No existing theory successfully accounts for several amazing properties of biological movements: dependence of movement speed on path curvature, isochrony (movement duration is nearly independent of its size) and the construction of more complex movements from simpler building blocks. Here we present a new theory of movement generation, based on movement invariance with respect to geometrical transformations. Several types of transformations are considered. Euclidian transformations preserve lengths and angles; affine transformations, which are less restricted, preserve parallelisms between lines, while equi-affine transformations preserve both parallelism and area. Each geometry is associated with a different measure of distance along curves. Movement timing is continuously prescribed by the brain by combining different “geometrical times” each assumed to be proportional to the measure of distance of the corresponding geometry. Movements are constructed by using a series of instantaneous (Cartan) coordinate frames. The predictions of the theory compared well with experimental observations of human drawing and walking. Equi-affine geometry was found to play a dominant role in both tasks and is complemented by affine geometry during drawing and by Euclidian geometry during locomotion. The proposed theory has far reaching implications with respect to brain representations of motion for both action production and perception.

As a first approximation, perceived physical space is assumed to be Euclidian. Yet, psychophysical studies of visual perception, drawing movements and locomotion indicate important departures from Euclidian geometry

Most important in affine geometry is the set of affine transformations, which are transformations of space or of a plane transforming straight lines into straight lines and parallel lines into parallel lines. Concretely, affine transformations are obtained by composing together translations and linear mappings which include rotations, stretching and dilatations. A property of a geometrical shape is said to be affine invariant when it is preserved under all possible affine transformations. For instance, being a closed curve is an affine invariant property but enclosing an area equal to

Since we are interested in motion timing, it is important to understand the concept of invariant duration with respect to a given set of transformations. For instance, a timing rule for a given set of curve segments is affine invariant if the duration spent moving along any arc of any one curve is equal to that spent moving along the image of this segment obtained by using any affine transformation. Thus, if timing were a totally affine invariant for all possible planar movements, all elliptical trajectories, for example, would have had the same total duration, generating a complete isochrony. We will show how such an invariance follows from a particular dependence of motor timing on the curvature of trajectories.

The influence of path curvature on movement velocity is well known

Examining the dependence of

The 2/3 power law was extended to human locomotion

As part of our new approach, we treat movement generation as being based on full affine geometry, without making specific choices of units of length or area. This allows us to compare the influences of affine, equi-affine, and Euclidian geometries on the temporal properties of the movements. We suggest that in each of these geometries time is proportional to a specific “measure of distance along the curve” in that particular geometry. In addition, we deduce the variation of the velocity gain factor from the need for full-affine invariance. This results in a local form of isochrony. For instance, for movements along ellipses, a full affine invariance predicts the same 2/3 principle as equi-affine invariance. But, in addition, it predicts that through appropriate adaptation of the “velocity gain factor” movement duration will be the same for all ellipses in the plane.

However, total isochrony may sometimes lead to paradoxical behavior, and we therefore hypothesize that the brain takes advantage of the existence of several possible geometries rather than using a single geometry. Hence, we suggest that movement timing is continuously prescribed and realized according to an equilibrium between affine and Euclidian geometries with Equi-affine transformations, which are the area-preserving affine transformations, playing an essential, if not dominant, role

The idea that geometric invariance is of great importance in prescribing the principles underlying perception and action is quite old

It is natural to propose that, in the same way, the brain uses several levels of representation and processing in planning any particular motion. At each level the computation is organized by respecting certain symmetries. Approaching the level of motor execution fewer and fewer possibilities are allowed, thus reducing the initial larger group of symmetry of all possible movements into smaller groups. This hierarchy of decisions in motion planning and execution is reflected in the representation of space through the performed movement.

In the following mathematical section, which discusses full affine, equi-affine and Euclidian geometries, we use Cartan's moving frame method

From the work of Galois

Curves can be analyzed differently in different geometries (see Monge

The essence of Cartan's moving frame method is that it creates a correspondence between the different orders of description of trajectories and the possible coordinate frames on the plane. This method specifies which geometrical transformations of frames allow identification of the trajectories that are indistinguishable at a given order (see section A in

At each point in time, locations within the plane are represented by coordinates in a moving frame. The motion along any curve is described by the equations representing the new infinitesimal coordinate frame (the new location and the new basis vectors) within the instantaneous current frame. When the moving frame is the canonical moving frame, the only remaining varying coefficient is the

The choice of parametrization of the curve necessary for deriving the canonical form of the moving frame gives a unique parameter, which is also the only parameter invariant under any transformation belonging to the group

To connect this description with known kinematic notions, choosing time to be proportional to the Euclidian arc length

Given a point

The

The mathematical expression for the equi-affine curvature when expressed as a function of Euclidian radius of curvature

Focusing now on the canonical full affine moving frame, the derivative of the full affine arc-length

When time is set proportional to the full affine arc-length

The canonical

Another kind of curvature appears for the full affine frame. This full-affine curvature remains unchanged under all full affine transformations. It determines the relative variation of the velocity gain factor

Two kinds of special points generically arise along a path immersed within the affine plane: ordinary inflection points (at which the Euclidean curvature equals zero) and ordinary parabolic points (at which the equi-affine curvature equals zero). Near ordinary inflection points, when the distance

However, as shown in section A in

Summarizing the main results: at each point along a parameterized curve, locations within the plane are represented by coordinates in a moving frame. The motion along the curve is then expressed by equations representing the new location and the new basis vectors within the present frame. Given a group

We propose here that the brain selects movement timing and duration according to a principle of geometrical invariance. A few principles are necessary to derive timing and kinematics of motion from geometry. As mentioned above, all the geometries considered here define canonical coordinates along curves:

By definition, we will call _{0}Δσ. For movements with equi-affine invariance

Let us remark that monotony necessarily neglects the fact that the motions start and end at rest with zero velocities, accelerations, and higher order derivatives of position (i.e., that we also make discrete movements). Because of the presence of such boundary conditions, the model must be generalized by considering the canonical parameters

In many cases, only one geometrical parameter is insufficient for deriving movement kinematics and it is both necessary and useful to use a combination of several geometrical parameters.

Even for periodic motions the presence of singularities implies that monotony cannot be obeyed. For instance, the full affine parameter expands at an ordinary inflection point. Thus, if

Combining several geometrical timing parameters offers greater flexibility and adaptivity to the motion planning strategy. Consequently, our general hypothesis is that

That is, during some portion of any given movement the velocity will be more affine, while during another portion it will be more equi-affine or more Euclidian. This is formulated by expressing

The consequence of this hypothesis is the existence of a succession of segments belonging more or less to different fixed combinations of geometries. It is natural to expect that the existence of singular points such as inflection and parabolic points implies the presence of extended segments in their vicinity, during which a mixture of geometries are employed. Note that the global shape of a figural form often forces the presence of such singularities, thus we predict that the global shape of a trajectory will influence its local kinematics. When moving from one movement segment to the next, the transition between such segments should be smooth. Hence, all of the above additional guidelines can be summarized as giving rise to

To quantify the combination of geometries in selecting movement timing and total duration, we need to understand the different and modifiable weights attributed by the motor system to the various possible purely geometrical rules. For this purpose we propose an equation having an exponential form.

Let

We examined the recorded tangential Euclidian velocity

The above equation (10) can be rephrased using the following tensorial equation for movement duration:

Observe that, although affine transformations introduce additional complexities to the computations, the actual hypothesis to be tested is contained in equations (0), (10) above, which are easy to understand. Note that all

We have also tested less stringent consequences of using mixtures of invariance. In particular, we tested the possibility that during specific segments one geometry becomes more dominant than the others. A vivid manifestation of the existence of a pure geometric velocity can be obtained by comparing the times

Suppose that the time spent on a segment of a curve is affine invariant, even if affine velocity is not constant we have

Suppose now that the law for duration is purely equi-affine. If

When Euclidian geometry is the dominant kinematic law, if we denote the lengths of the arcs traveled by

Now suppose that a curve

The data used to test our hypotheses were derived from recorded hand movements and locomotion generated along

Three different tests were conducted. The first, using elliptical hand trajectories, tested whether an alternation between Euclidian and affine geometries better explains the experimentally observed relation between Euclidian curvature and velocity than describing the whole elliptical movement as obeying a single power law with constant exponent. This test also examined the limitations of the validity of the isochrony principle by investigating the relation between total duration and perimeter and the relation between the enclosed area and gain factor. The second test used trajectories generated by human subjects while tracking geometrically prescribed complex figural forms - cloverleaves, lemniscates and limaçons - during both drawing and locomotion. This experiment tested whether the proposed tensorial formulae (8), (9) can successfully account for the experimentally observed movement durations. We also examined whether it is possible to distinguish between movement duration during drawing and locomotion based on the different degrees of influence of the different geometries, i.e. whether both tasks are based on similar principles of invariance but arise from different mixtures of geometrical invariance. The third test, applied to the same data as the second test, examined the validity of equation (13) with respect to the ratios between the durations of the movements along the large versus the small loops of the lemniscates and limaçons. It also aimed at confirming the differences between drawing and locomotion identified by the second test with respect to the influence of the different geometries on the durations of movement along large versus small segments.

Elliptical trajectories with different eccentricities, perimeters and performed under various speed conditions were recorded from three subjects (see section

Full isochrony predicts the following linear relation between the velocity gain factor _{3}'s movements also tended to show full isochrony (see below).

Speed | Subject | |||

Slow | S1 | 0.12 | −1.56 | 0.46 |

S2 | 0.17 | −1.30 | 0.91 | |

S3 | 0.28 | 0.29 | 0.93 | |

Natural | S1 | 0.14 | −0.78 | 0.45 |

S2 | 0.23 | −0.35 | 0.87 | |

S3 | 0.33 | 1.01 | 0.96 | |

Fast | S1 | 0.18 | 0.17 | 0.59 |

S2 | 0.21 | 0.35 | 0.80 | |

S3 | 0.28 | 1.32 | 0.95 |

If

Empirical values (blue) of the pairs

Eccentricity | Subject | Piecewise- Linear |
Piecewise- Linear |
Power- Law |
Power- Law |
Probability | |

Small | S1 | 0.11 | 6.04 | −2.23 | −0.23 | 0.65 | 0.87 |

S2 | 0.15 | 5.24 | −2.12 | −0.28 | 0.72 | 0.82 | |

S3 | 0.32 | 6.19 | −1.31 | −0.26 | 0.69 | 0.85 | |

Medium | S1 | 0.19 | 16.09 | −1.70 | −0.14 | 0.95 | 0.58 |

S2 | 0.19 | 13.10 | −1.74 | −0.17 | 0.85 | 0.72 | |

S3 | 0.49 | 9.28 | −0.50 | −0.28 | 0.89 | 0.68 | |

Circle | S1 | 0.15 | 24.61 | −1.98 | −0.07 | 0.95 | 0.56 |

S2 | 0.17 | 27.12 | −1.99 | −0.05 | 0.96 | 0.54 | |

S3 | 0.39 | 17.51 | −0.68 | −0.23 | 0.97 | 0.53 |

Our results thus confirm the existence of heterogeneous geometry and quantify a trajectory segmentation scheme compatible with the presence of separate equi-affine (or affine) and Euclidian segments during the generation of elliptical trajectories.

To further investigate the influence of different geometrical representations on the movements, we used a linear regression analysis of

Log movement time (T) is plotted versus log perimeter (P). The regression lines between log T and Log P are shown for each subject

Subject | speed | |||

S1 | Slow | 0.49 | 2.48 | 0.62 |

Natural | 0.55 | 1.85 | 0.76 | |

Fast | 0.44 | 0.96 | 0.80 | |

S2 | Slow | 0.37 | 2.38 | 0.78 |

Natural | 0.29 | 1.56 | 0.71 | |

Fast | 0.32 | 0.80 | 0.70 | |

S3 | Slow | 0.11 | 1.02 | 0.14 |

Natural | 0.041 | 0.44 | 0.049 | |

Fast | 0.13 | 0.02 | 0.5 |

In sections B.2 and B.3 in

We next examined a series of drawings of lemniscates, limaçons and cloverleaves (from

Several data sets were used in the computation of the different models and in the statistical tests. For drawing, all the data points were used. The velocity profiles of locomotion displayed oscillations due to the stepping movements. To eliminate these and to derive the subject's transport velocity during locomotion, we disregarded the velocity components due to the presence of steps. Thus, for locomotion we consider two data sets. One data set, the “stepwise sampled data set” (SSDS), consisted of the experimental data corresponding to time instants at which the body position (with respect to a point

Then, based on the model for the mixture of geometries, hypothesized tangential velocity profiles were constructed using the following procedure (for further details see section

1) By comparing the known experimental velocity and the three computed monotonic velocities (

Altogether, seven techniques were used to find segments during which one or all three

2) For each scenario, we computed a smooth cubic spline interpolation of the

3) The values for the three remaining constants

4) We chose the best result among those constructed using the above seven scenarios. We call this velocity

Note that we used the above algorithm, since at present, except for parabolic and inflection points, the model does not predict which geometrical combination should be realized for any given curve. However, we have tested the non-triviality of the model predictions using three different statistical measures: 1) Assessing the significance (or the statistical non-triviality) of the existence of special segments using an F-test (details are given below and in section

First we refer to the non-triviality of the special segments found by the data analysis. The SSDS data were considered for the locomotion task. For all trials we computed the squared distance

Considering the respective numbers of statistical degrees of freedom, we applied a Fisher test to the ratio

Exp | Shape | Test 1: % Significance | Test 2: % Significance |

Drawing | 100% | 77.78% | |

100% | 100% | ||

100% | 66.67% | ||

100% | 83.33% | ||

100% | 66.67% | ||

100% | 55.56% | ||

100% | 100% | ||

100% | 100% | ||

100% | 100% | ||

Locomotion | 100% | 75.86% | |

100% | 62.07% | ||

100% | 60% | ||

100% | 63.33% | ||

100% | 83.33% | ||

96.67% | 96.67% | ||

100% | 92.59% |

We also calculated the Akaike Information scores

For locomotion, we examined the success of the four models by computing two sets of AIC scores for the velocity profiles predicted by these models. One set of scores was derived by applying the models to the SSDS data points (“SSDS-scores”), while the second set of scores was derived using the CSDS (“CSDS-scores”). The main difference between the AIC scores calculated for the two data sets is that for CSDS, the experimental velocity profiles contained velocity components due to stepping that are not accounted for by the combination of geometries nor by the minimum-jerk models. The CSDS data set contained more samples than the SSDS data set. For SSDS, the number

For the drawing data, the AIC scores for the four different models are shown in

(A) Akaike's Information Criterion (

The

For the locomotion data (both the SSDS and the CSDS), the AIC scores for the four different models are shown in

(A) Akaike's Information Criterion (

Taken together, the probability that the combined velocity model is better than the minimum-jerk model for drawing was greater than 0.9 for

In conclusion, for most of the drawing as well as locomotion trajectories, the minimum-jerk and the combined velocity models obtained the best

We computed the coefficients of determination,

Every row shows an example of the second repetition of a drawing trial. First row, drawing of a cloverleaf; second row, drawing of an oblate limaçon; third row, drawing of an asymmetric lemniscate. Panels (A), (D) and (G) show the paths drawn by the subject. The colors marked on the paths represent the Euclidian curvature. Blue segments have relatively low curvature (∼0), red segments have a higher curvature (∼0.75). Color scale is shown at the top of the panel. Panels (B), (E) and (H) show the velocity profiles of the drawing. Red, experimental velocity profile; blue, velocity profile predicted by the model of the combination of geometries. Panels (C), (F) and (I) show values of the

Every row shows an example of the second repetition of a locomotion trial. First row, walking around a cloverleaf. Second row, walking along an oblate limaçon. Third row, walking around an asymmetric lemniscate. Panels (A), (D) and (G) show the paths drawn by the subject. The colors on the paths represent the Euclidian curvature; Blue, segments with a relatively low curvature (∼0); red, segments with a higher curvature (∼0.75). Color scale is shown in the panel. Panels (B), (E) and (H), the velocity profiles of the drawing. Red, experimental velocity profile; blue, the velocity profile predicted by the model of the combination of geometries. Panels (C), (F) and (I) show values of the

The

Summary of the

Exp | Shape | ||||

Drawing | 0.90±0.05 | 0.86±0.06 | 0.87±0.03 | −0.11±0.69 | |

0.94±0.03 | 0.92±0.05 | 0.90±0.02 | 0.52±0.17 | ||

0.91±0.06 | 0.93±0.03 | 0.91±0.01 | 0.46±0.20 | ||

0.91±0.02 | 0.90±0.02 | −2.87±0.36 | −12.38±3.01 | ||

0.84±0.03 | 0.88±0.02 | −2.82±0.70 | −11.22±1.45 | ||

0.75±0.05 | 0.83±0.04 | −3.12±0.85 | −10.35±1.92 | ||

0.86±0.27 | 0.82±0.32 | 0.80±0.28 | 0.71±0.37 | ||

0.97±0.01 | 0.95±0.02 | 0.88±0.01 | 0.91±0.02 | ||

0.97±0.01 | 0.95±0.02 | 0.87±0.01 | 0.93±0.02 | ||

Locomotion | 0.86±0.07 | 0.74±0.18 | 0.83±0.10 | −4.27±8.38 | |

0.79±0.09 | 0.41±0.60 | 0.47±0.80 | −8.09±13.26 | ||

0.79±0.16 | 0.15±0.51 | 0.40±0.46 | −12.56±11.16 | ||

0.48±0.27 | 0.20±0.47 | −0.44±1.22 | −7.80±6.95 | ||

0.51±0.18 | 0.08±0.53 | −8.22±5.15 | −20.62±12.90 | ||

0.54±0.23 | 0.13±0.21 | −7.31±3.22 | −21.34±6.55 | ||

0.60±0.17 | 0.09±0.31 | −6.51±2.23 | −21.19±5.01 |

The mean values of the

Exp | Shape | mean of |
mean of |
mean of |

Drawing | cloverleafs | 0.25±0.14 | 0.64±0.22 | 0.11±0.12 |

Limaçons | 0.52±0.14 | 0.46±0.14 | 0.02±0.03 | |

Lemniscates | 0.09±0.01 | 0.79±0.09 | 0.12±0.09 | |

Locomotion | cloverleafs | 0.16±0.13 | 0.66±0.18 | 0.18±0.11 |

Limaçons | 0.08±0.11 | 0.56±0.19 | 0.36±0.18 | |

Lemniscates | 0.09±0.04 | 0.43±0.13 | 0.48±0.12 |

The average values of the

To investigate movement segmentation we examined the influence of the various geometries on the different parts of the three figural forms. To this end, we inspected the values of the _{0} = 0.2, _{1} = 0.5, and _{2} = 0.3, the red area will appear between the abscissa and the line parallel to it passing through the value of 0.2, the green area will appear between that line and another line parallel to the abscissa through the value of 0.7, while the blue area will appear above the latter line).

As can be seen from the drawing example of the limaçon, Euclidian geometry has no influence on the theoretically predicted velocity (there are no blue parts), while in the locomotion example there is no affine influence (no red part). In the lemniscate example, movement trajectories are not segmented at singularity points, as previously suggested in the literature

Examining the different velocity profiles,

We shall now inspect drawing and locomotion of each of these figural forms more closely. Here we mainly use

For cloverleaves the

The limaçons gave high

The

The distribution of values of the

The distribution of the

In conclusion, the equi-affine geometry was almost always the dominant geometry. The mean value of

Using the data from

Both for drawing (3 subjects) and for locomotion (10 subjects), three different ratios

The

There were always regions within the triangle where _{2}−_{0} = some constant) gave high

A second type of statistical analysis was performed over the same three constants

The red dots represent the experimentally measured ratios of movement durations

We present a new theory explaining how the timing of voluntary movements changes according to path geometry. Our model proposes that the velocity along the path is a specific composition of different canonical velocities, a composition that may vary among different segments of the same movement. Both geometrical invariance and movement segmentation are consequences of this principle. The canonical velocities to be combined depend only on path geometry and are defined within three major 2D-spaces: equi-affine, Euclidean and affine spaces. Equi-affine geometry is associated with a measure of area, Euclidian geometry with a measure of distance and affine geometry with the notion of parallelism.

The above notions are illustrated in our analysis of elliptical hand drawings. The trajectories contained two types of curved segments, each displaying different relationships between instantaneous velocity and Euclidian curvature and corresponding either to affine or Euclidian geometries. The Euclidian segments were those portions of the trajectory during which the Euclidian curvature was rather low - below some threshold - while the affine segments corresponded to the more curved portions. Such a description of segments accounts for the observed behavior better than a description based on a single power law (the probability of providing a better model than a single power law model was always higher than 0.64 and up to 0.97 for small ellipses). The observed behavior is a compromise between constant ordinary Euclidian speed and the isochrony principle, which reflects the effect of full affine geometry on motor timing.

For drawing the three figural forms studied here (cloverleaves, lemniscates and limaçons), the comparison of the three canonical velocities with the corresponding experimentally recorded ones strongly supports our new theory. The predicted purely equi-affine and the experimentally recorded velocities were very close for 70% of the time. The disagreement during the remaining 30%, could be systematically explained: here the velocity departed from entirely equi-affine and varied in a direction indicated by full affine or Euclidian velocities, as shown by the velocity profiles (see

To quantify these relations, we analyzed the experimentally recorded trajectories of human drawing or walking along the prescribed figural forms mentioned above. More than 90% of the velocity variance of drawing movements and 60% of the velocity variance for walking (based on the

At first sight, the flexibility offered by three different geometries seems so large that one could imagine that such a model would produce a good fit for any possible movement data set. It was therefore important for us to ask in what way are our results non-trivial? Firstly, the prediction that speed is a weighted product of all three canonical velocities is non-trivial since

The new point of view provided by the geometrical combination of velocities permits us to demonstrate several characteristics of motor timing.

First, we demonstrated that the global shape and size of the trajectory essentially influence motion timing (

Second, we discovered that the main difference between drawing and locomotion was the opposite degree of influence of full affine versus Euclidian geometries. For drawing,

We also applied a more restricted idea of segmentation by studying the effect of alternating between different dominant geometries. As a first approximation we assumed that segments with a constant velocity only within one specified geometry can successfully account for the observed ratios of time spent moving along the large versus the small loops of the complex figural forms as a function of their respective sizes. The observed ratios of movement durations have also corroborated and provided further evidence for our finding that the net balance between Euclidian and affine geometries is totally reversed for drawing versus locomotion.

All these results confirmed our expectation that affine geometry is significant in a theory of movement timing. The canonical velocity of full affine geometry yields the same total time spent on a curve and on the curve obtained through any dilatation. That is, the dominance of affine geometry here corresponds to isochrony, even though it is imperfect. However, we found experimentally that the pure affine geometrical arc-length is generally only a secondary component in determining movement timing, although it is always present during drawing movements. The notable exception is hand drawing of limaçons, where full affine canonical velocity alone explains the entire timing pattern very well. These results point to the important role of equi-affine geometry in motor timing.

The equivalence of the 2/3rd power to moving at a constant equi-affine speed

A first possibility is that the equi-affine invariant parameter may be computationally simpler. This invariant parameter is of third order, i.e., the order of the variation of acceleration, namely jerk. It could be coded by proprioceptive or vestibular information especially during locomotion

A second possibility is the probable importance of area perception during motion, and we know that equi-affine transformations preserve areas. The amplitude of a piece of motion can be judged from the area enclosed by the corresponding segment of the trajectory and by the straight segment joining its initial and final positions. As we have seen in sec∶quantitative laws (see also E in

A third explanation may correspond to the link between equi-affine invariance, the optimization of smoothness and the minimum jerk principle (cf.

The fourth explanation for the dominance of equi-affine geometry is based on dimensional analysis which provides a completely different direction of support for the 2/3 law. Let us suppose that during motion, the total variation of energy

All the above explanations for the dominance of equi-affine geometry in movement timing arise from some sort of invariance. However, in the framework of our theory, it is natural to propose that the main reason for the dominance of the equi-affine geometry (and consequently the 2/3 law) is that it offers an excellent compromise between full affine invariance and the reduction of computational complexity.

We propose that movement duration is determined by invariance and computation. In the present framework, movement duration is related to space. This agrees well with Piaget's

Our suggestions also fit those of Bernstein

Several other recent studies have also reported strong departures from the two-thirds power law

Schaal and Sternad

We suggest that the geometric combinations we show are also affected by the geometrical transformations from joint to hand trajectories. Hence, our model, though considering movement duration only from the point of view of hand trajectories, must be further developed to consider Schaal and Sternad's approach and results. We believe that the motor system has evolved to make simplifications in motion planning compatible with the biomechanical characteristics of the musculoskeletal system.

Examining the generation of different patterns of complex figural forms in various tasks and conditions (tracking, drawing from memory, tracing) Flanders et al.

This points to many possible extensions of our study. In fact, even if a combination of geometries accounts for the link between geometry and movement duration, we suspect that the rules dictating the mixture of geometrical timing parameters chosen by the brain may depend on external or cognitive factors. It will be particularly interesting to examine in what ways cognitive factors and learning

Our present study is limited to 2D motion. Future work should deal with 3D Motion, as well as with movements performed in different orientations, as in

One limitation of the present study is that we only tested periodic arm movements and locomotion trajectories. However, as suggested above, our principles may also be applied to discrete movements that start and end at rest, or to trajectories containing reversals of movement direction. In such cases all the different geometries are expected to be combined and one may need models that no longer assume constant canonical velocities. Thus new kinds of segments are expected to emerge which depend on the particular velocities combined and on the values of the different geometrical curvatures associated with these velocities.

The principal limitation of the present study is that the tests of the theory have not dealt with the question of which geometrical paths the trajectories should follow. We have only dealt with the question of which velocity is chosen along a prescribed path, as a function of the geometrical form being followed. It is probably not difficult to propose which special paths should be selected, depending on the degree of geometric invariance and symmetry they offer. For instance, as suggested in

Some optimization principles predict the complete actual trajectories. Using only via points and end-points, the minimum jerk principle successfully accounted for both the trajectory path and velocity of curved and drawing movements

The minimum variance principle is grounded on Fitts

The use of invariance or mixtures of invariance as proposed here is only a constraint. To realize the actual movements, subjects must apply tools other than, but compatible with, invariance. For example, the geometrical invariance principles can be used together with optimization principles to solve redundancy problems at the task level. Even more importantly, the anticipation of singular points before and during movement generation can help particularly in determining where the motion should be segmented or the precise combination of the canonical geometrical parameterizations to be used. More generally, geometry may indicate in what parametric space or coordinate frames motor commands for movement generation should be planned. Our suggestion does not contradict the need for online optimization of ongoing movements. When needing to anticipate or to respond optimally to trajectory perturbations, optimal feedback, control theory can complement our formulation of invariance principles

We emphasize that our model relates naturally to the neural encoding of movement because it suggests the possibility that different neural populations represent movement kinematics in terms of the different geometries or combined geometrical representations:

Evidence has accumulated for the use of different “reference frames” in movement planning (

The notion of moving frames (as in section Mathematical preliminaries), particularly the affine geometrical representation, may throw new light on the currently available neurophysiological observations and on the roles of different cell populations in representing movement. Schwartz and colleagues

In a recent study Polyakov et al.

Consistent with the theory presented here we speculate that there must be many dynamically interconnected neuronal populations, either within one area or more probably within different areas, which use different geometrical representations. These assemblies would be selective for parameters intrinsic to a particular geometry. Some supporting evidence has been obtained in a recent fMRI study

Analyzing how children draw simple ellipses Viviani and Schneider

Piaget and Inhelder

Our data on voluntary movements suggest a different order of development of the different geometries. Implicit motions, unlike explicit or iconic descriptions, seem to be acquired initially using more Euclidian reference frames. This is suggested by the exponent

Every action is a specific solution to a problem. What is

In full affine geometry, time is a pure number (e.g., going around any ellipse takes

The principle of invariance is also compatible with different optimization principles such as the minimum-jerk

In experimental test no. 1 three young adult men were instructed to draw 10 types of ellipses at 3 different speeds, slow, natural and fast. The ellipses, prescribed in advance, had 3 different eccentricities,

In experimental tests no's 2 and 3, we analyzed a series of drawings and locomotion trajectories of cloverleaves, limaçons and lemniscates, taken from the studies of Viviani and Flash

For drawing, the trajectories were those of the stylus position along the tablet. For locomotion the trajectories measured were those of the orthogonal projection

In both analyses we started with a collection of point coordinates

These calculations were conducted on the entire N samples obtained from the drawing and locomotion data. For locomotion we called this data set the complete sampled data set (CSDS). For locomotion, we also extracted the data corresponding to positions where the point

The velocity

To choose the mixture of these uniform velocities which results in the predicted combined velocity, we looked for segments of the experimental velocity during which we could set at least one of the weight function

In the first algorithm we used a linear regression in logarithms of velocity and found segments between points where we could determine

For the second algorithm we

In the third and fourth algorithms we considered the combination of affine and Euclidian geometries and the equi-affine and Euclidian velocities respectively. The equations used were, respectively,

The velocities constructed in cases two, three and four were marked as

All these algorithms were based on the following arguments. First, we expected to find segments during which a constant combination of geometries appears the primary source for movement segmentation. Second, we had no reason to believe that constant combinations of all three geometries would appear at the same time, so we looked for two-by-two constant combinations. However, to reduce the number of parameters, based on the data we limited the algorithm to the same pair of geometries all along the trajectory.

This procedure required verification that these segments (primary and secondary) were statistically non-trivial. We therefore used a Fisher's test (see below), as explained in section Experimental tests. Our modeling approach also required verification that the success of the model was not only a consequence of our using a large number of fitted parameters. For this purpose we used the Akaike criterion (AIC) as explained in section Experimental tests.

The F-test: The data used were those of the logarithms of the velocities. For each curve and for each of the seven computational scenarios, let

For drawing, 61 of 78 trials (78%) showed a

The Akaike test

Log γ vs. Log A for each subject. All the repetitions for each ellipse size and speed condition are grouped into a single dot, the y-axis the log γ values. The diamond shape plotted around the mean value ±1 displays the standard deviation for both axes. The results for each subject are shown in different figures. In all figures, blue represents slow drawing speed; green, the natural speed; red, fast drawing speed. The dashed lines gives the regression lines separately for each speed. The parameters of the lines are given in

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The geometrical shapes used in the second and third tests. The analytical shapes and the parametric equations of the asymmetrical lemniscate, the oblate limaçon and the cloverleaf used in the drawing and locomotion experiments analyzed by the second and third tests.

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The reference points used in the locomotion experiments. The M and R reference points marked on the subject's body for the locomotion experiments.

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Examples of the experimental data and the geometrically based predicted velocities for the drawing movements. Every row is an example of the second repetition of a trial. First row, typical example of drawing a cloverleaf; second row, drawing an oblate limaçon; third row, drawing an asymmetric lemniscate. Panels (A),(D) and (G) show the paths drawn by the subject, colors represent the Euclidian curvatures along the curves: blue, low curvature; red, high curvature. Panels (B), (E) and (H) show the velocity profiles of the movements: red, experimental velocity used by the subject; green, the velocity profile under Euclidian parameterization; black, the velocity profile under equi-affine parameterization, blue, the velocity profile under affine parameterization. Panels (C), (F) and (I) show the curvatures of the curve; green, Euclidian curvature; black the equi-affine curvature; blue, the affine curvature.

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Examples of the experimental data and the geometrically based predicted velocities for the locomotion experiment using the M-point. Every row gives an example of the second repetition of a trial; first row, a cloverleaf; second row, an oblate limaçon; third row, an asymmetric lemniscate. Panels (A),(D) and (G) show the gait paths generated by the subject. The colors used in plotting the paths represent the Euclidian curvature along the path; blue, low curvature values; red; high curvature values. Panels (B), (E) and (H) show the velocity profiles of the movements. Red, the velocity used by the subject; green, the velocity profile under Euclidian parameterization; black, the velocity profile under equi-affine parameterization; blue, the velocity profile under affine parameterization. Panels (C), (F) and (I) show the curvatures of the curve; green, Euclidian curvature; black, the equi-affine curvature; blue, the affine curvature.

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Examples of the β functions and the combined velocities for the drawing experiment. Every row gives an example for the second repetition of a trial. The first row, cloverleaf; second row, an oblate limaçon; third row, an asymmetric lemniscate. Panels (A),(D) and (G) are the paths drawn by the subject. The colors marked on the paths represent the β functions: blue, a part that is more Euclidian; green, a part that is more equi-affine; red, a part that is more affine. The full range of colors and their relation to the values of β_{0}, β_{1} and β_{2} can be seen in _{0} function; green area, the values of the β_{1} function; blue area, the values of the β_{2} function.

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Examples of the β functions and the combination velocity of the locomotion. Every row gives an example of the second repetition of a trial. The first row, a cloverleaf; second row, an oblate limaçon; third row, an asymmetric lemniscate. Panels (A),(D) and (G) show the path generated by the subject. The colors on the paths represent the β functions: blue, a part that is more Euclidian; green, a part that is more equi-affine; red, a part that is more affine. The full range of colors and their relation to the values of β_{0}, β_{1} and β_{2} can be seen in _{0} function; green area, the values of the β_{1} function; blue area, the values of the β_{2} function.

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Color map for the values of the βs. Every point in the triangle represents a specific relation between β_{0}, β_{1} and β_{2} values shown in the color corresponding to this combination.

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Results in the locomotion experiments using the R-point. Panels (A) and (B) represent the R^{2} and AIC scores for the CSDS (all data) of the locomotion R-point, respectively, for the combined velocity (red bars), minimum-jerk velocity (green bars), constant equi-affine velocity (yellow bars) and constant affine velocity (cyan bars). The probability of the combined velocity being a better model than the minimum-jerk model for the different figural forms is shown in Panel (C). Panels (D) and (E) represent the values of the functions β_{0}, β_{1} and β_{2} averaged over all trials. The results are presented for the R-point at the level of the figural form. The cloverleaf form is marked by _{1},_{2},_{3} and _{1},_{2},_{3} represent the limaçon and the lemniscate templates, respectively, according to the ascending ratio of the large to the small loops.

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The mean values of the β functions for the different figural forms and different subjects. The mean values of the functions β_{0}, β_{1} and β_{2} averaged over each trial, summarized over the subjects and the templates of the different figural forms. Every β is displayed in a separate figure. Every color represents a different subject. Every group of bars represents a different figural form. The cloverleaves are marked by _{1},_{2},_{3} in the order of ascending speed for drawing and by _{1},_{2},_{3} and _{1},_{2},_{3} represent the limaçon and the lemniscate templates, respectively, according to the ascending ratio of the large to the small loops.

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The mean values of the β functions averaged over loops and figural forms. The mean values of the functions β_{0}, β_{1} and β_{2} over loops within a trial, summarized over the templates of the figural forms. Every β is displayed in a separate figure. Blue bars, the small loops; red bars, the large loops. Every group of bars represents a different figural form, for notations see

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Supporting Information. Mathematical background, data processing and additional results.

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We thank Halim Hicheur, Uri Maoz, Roni More and Paolo Viviani for their help in collecting and sharing with us the experimental data and movement analysis. We thank Amir Handzel, Felix Polyakov, Quang-Cuong Pham, Jean Jacques Slotine and Moshe Abeles for many fruitful discussions and Jenny Kien for her editorial help. We also warmly thank two anonymous referees for their attentive and inspired corrections and suggestions.