Drawing Movements as an Outcome of the Principle of Least Action


Journal of Mathematical Psychology 45, 43�52 (2001)

Drawing Movements as an Outcome of the
Principle of Least Action

Sergey Lebedev, Wai Hon Tsui, and Peter Van Gelder

Nathan S. Kline Institute for Psychiatric Research and New York University Medical Center

According to the two-thirds power law the cube of the speed of a drawing
movement is proportional to the radius of curvature of the trajectory, and the
coefficient of proportionality has the meaning of mechanical power. We derive
this empirical law from the variational principle known in physics as the
principle of least action. It states that if a movement between two points of
a given path obeys the two-thirds law, then the amount of work required to
execute a trajectory in a fixed time is minimal. In this strict sense one may
say that among infinitely many ways to execute a given path, the central
nervous system chooses the most economical. We show that the kinematic
equations for all drawing movements are solutions of a certain differential
equation with a single (time-variable) coefficient. We consider several special
cases of drawing movements corresponding to simplest forms of this coefficient.
� 2001 Academic Press

Drawing and handwriting involve complex movements executed as a chain of
individually planned simple movements. Figure 1 provides a good example of such
planning: a great artist in a signature-like movement drew a silhouette of a bird
(Clark, 1993). It seems reasonable to think that an internal representation of this
drawing in a form of an image or silhouette was available to the implementation
stage prior to the inception of the movement.1 The segmentation into units seems
to occur at the cusps, points of inflection, or contour discontinuities, and within
each segment the curvature preserves its sign.

It has been empirically established (Lacquaniti, Terzuolo, 6 Viviani, 1983;
Viviani 6 Cenzato, 1985) that within each segment of execution a power-law

doi:10.1006�jmps.1999.1287, available online at http:��www.idealibrary.com on

43 0022-2496�01 �35.00
Copyright � 2001 by Academic Press

All rights of reproduction in any form reserved.

All authors are at the Nathan Kline Institute and the Department of Psychiatry, New York University
Medical Center. We are grateful to Professor Edgar E. Coons of the New York University for insight
on how the central nervous system uses the principle of least action. We also thank the young artist
Jonathan Lester for participation in a pilot experiment, which led us to fruitful ideas. Finally, we are
indebted to Professor Ethibar N. Dzhafarov of Purdue University for his constructive suggestions.

Address correspondence and reprint requests to Peter Van Gelder, Department of Psychiatry, Room
TH-N 416, NYU Medical Center, 550 First Avenue, New York, NY 10016. E-mail: pvg�ipl.med.nyu.edu.

1 Here is how Picasso understands the process: ``Drawing is no joke. There is something very serious and
mysterious about the fact that one can represent a living human being with line alone and create not only
his likeness but, in addition, an image of how he really is. That is the marvel!'' (Clark, 1993).



FIG. 1. Pablo Picasso, Dove, 1961, � 1999 Estate of Pablo Picasso�Artists Rights Society (ARS),
New York. This drawing suggests that the entire silhouette can be decomposed into about 20 segments,
and the segmentation seems to occur at the cuspids and the points of inflection.

relation exists between the angular velocity of a movement and the curvature of the
trajectory

0(t)=K } C(t)2�3, (1)

where t denotes an instant in time, 0(t) is the angular velocity, and C(t) is the
curvature. The coefficient of proportionality K is called the gain factor and remains
approximately constant within each segment of execution (Viviani 6 Cenzato, 1985).
This dependence has been known as the ``two-thirds power law.'' This law is also
known in a different but mathematically equivalent form (Viviani 6 Cenzato, 1985),

V(t)=K } R(t)1�3, (2)

where V(t) and R(t) denote instantaneous values of the tangential velocity of the
movement and the radius of curvature of the trajectory, correspondingly.

Although a given drawing trajectory can be traced in infinitely many different
ways, the nervous system effectively reduces this excess of degrees of freedom
(Viviani 6 Flash, 1995; Viviani 6 Schneider, 1991). In this article we derive the
two-thirds power law from the principle of least action and show the constraints
imposed on the parametric equations of drawing movements. These constraints are

44 LEBEDEV, TSUI, AND VAN GELDER



sufficient to characterize the control system that governs the execution of drawing
movements and the reduction in the degrees of freedom that are available before the
inception of such a movement.

1. TWO-THIRDS POWER LAW AND THE STRATEGY OF LEAST ACTION

Equation (2) can be rewritten in the form

V 3(t)
R(t)

=P, (3)

where P=K 3=const. The left-hand side of (3) can be decomposed into

V 2(t)
R(t)

} V(t) =An(t) } V(t), (4)

where An(t) is the magnitude of the normal or centripetal acceleration, that is, an
acceleration directed toward the center of the curvature of the path. The centripetal
acceleration characterizes the rate of change in the direction of the velocity vector
(Yavorsky 6 Detlaf, 1982). Then (3) takes the form

An(t) } V(t) =P, (5)

which states that in drawing movements the product of the instantaneous values of
centripetal acceleration and the tangential velocity remains constant during segment
execution. This product is known in physics as mechanical power, and it characterizes
the amount of mechanical work per unit of time necessary to keep movement along
the trajectory.

The substitution into (3)

R(t) =
(x* 2+y* 2)3�2

x* y� &y* x�
,

(6)
V=- x* 2+y* 2,

where x=x(t), y=y(t) are the kinematic equations of movement, and x* , y* , x� , y�
denote their first and second derivatives with respect to time, gives

P=x* y� &y* x� , (7)

which is the two-thirds power law in a differential form (see also Viviani 6 Cenzato
(1985)). It is known that along a given path y=y(x), x* y� &y* x� =x* 3y"x (Bronshtein
6 Semendyayev, 1985). Thus (7) can be rewritten in the form

P=x* 3y"x , (8)

where y"x denotes the second derivative of y with respect to x.

45DRAWING MOVEMENTS AND THE PRINCIPLE OF LEAST ACTION



The very fact that the mechanical power remains constant throughout the
movement suggests that there exists a certain criterion from which the two-thirds
power law can be derived. To find this criterion, consider the problem: along
a given path y=f (x), find a movement whose kinematic equations require the
smallest amount of mechanical work necessary to accomplish the movement
between two given points M(x0 , y0) and N(x1 , y1) of a curve in a fixed time
T=T1&T0 . Mathematically, this means that the kinematic equations x=x(t) and
y=y(t) must minimize the integral

|
T1

T0

x* 3y"x dt. (9)

In Appendix A we show that the necessary conditions for an extremum of (9) are
relations (8), that is, x* 3y"x =const.

In the field of theoretical physics the time integral of the form (9) is known as
the action of the system, and the assertion that the action on an actual path is
minimal is called the principle of least action (Cohen, 1981; Kompaneyets, 1961;
Landau 6 Lifshitz, 1969). In this strict sense we may say that in drawing movements
the central nervous system follows the strategy of least action: among infinitely many
ways to execute a given path, the system chooses the most economical. In Section 4 we
illustrate this statement by considering three different ways of drawing a segment of a
Lissajous figure, the movement with a constant velocity, the case when the x and y
coordinates oscillate independently with different angular frequencies (Lissajous
motion), and the movement governed by the strategy of least action.

2. DIFFERENTIAL FORM OF THE TWO-THIRDS POWER LAW AND
ITS PROPERTIES

By the two-thirds power law, the mechanical power of a drawing movement remains
constant during trajectory execution:

x* y� &y* x� =P=const. (10)

To see what kind of constraints it imposes on the parametric equations of movement
we differentiate it with respect to time,

x* y! &y* x! =0 (11)

or

x!
x*

=
y!
y*

, (12)

46 LEBEDEV, TSUI, AND VAN GELDER



and this is another useful form of the two-thirds power law. It states that in draw-
ing movements the ratios of the third to the first derivatives of the two kinematic
equations must be equal. It follows immediately from (12) that the kinematic equa-
tions of movement, x(t), y(t), must simultaneously satisfy the equation

u! +q(t) u* =0, (13)

that is, x! +q(t) x* =0 and y! +q(t) y* =0.
Equation (13) is a third-order linear ordinary differential equation. It is homo-

geneous (right-hand term equals zero), and its solutions depend solely on the
coefficient q(t). Thus, all drawing movements can be classified with respect to the
form of this coefficient. In the next section we consider three simple special cases
of q(t).

It can be seen from (13) that u=const is one of its solutions. Then its general
solution is of the form

u=C0+C1 w1(t) +C2 w2(t), (14)

where w1(t), w2(t) are any linearly independent particular solutions of (13), and C0 ,
C1 , C2 are arbitrary constants. Without loss of generality we assume C0=0. Then

x=Cx 1 w1(t) +Cx 2 w2(t)

y=Cy 1 w1(t)+Cy 2 w2(t).

3. CONSTANT-COEFFICIENT CASE: ELLIPSES, HYPERBOLAS, AND PARABOLAS

The simplest case of (13) is obtained by putting q(t) =const. If this constant is
nonzero, then we have (see, e.g., Korn 6 Korn, 1968)

Class 1: q=|2>0

x=A1 sin(|t+.1) (15)

y=A2 sin(|t+.2),

Class 2: q=&|2<0

x=A1 e
|t+A2 e

&|t (16)

y=B1 e
|t +B2 e

&|t.

In the case when q=0, direct integration leads to

Class 3: x=A0 tA1 t+A2 t
2

y=B0 tB1 t+B2 t
2.

47DRAWING MOVEMENTS AND THE PRINCIPLE OF LEAST ACTION



or, by assigning the initial positions to zero,

x=A1 t+A2 t
2

(17)
y=B1 t+B2 t

2.

The motion described by (15) is called an elliptically polarized oscillation; the
term is due to the fact that the path-equation in the Cartesian coordinate system
is an ellipse (Yavorsky 6 Detlaf, 1982). The path-equation of (16) is a hyperbola,
and the path-equation of (17) is a parabolic path. Thus, whenever the coefficient q
in (13) is a constant, the two-thirds law allows the motor planning only in terms
of the second-order curves: ellipses, hyperbolas, parabolas, or their degenerated
cases, e.g., straight lines.

The vast majority of experimental studies exploit the elliptic trajectory. In
the study of drawing movements in children, Viviani 6 Schneider (1991) con-
sidered movements along an elliptic path with tangential velocity of the form
V(t)=K } R(t) ;, K, ;>0. They define this class of movements as generalized
Lissajous elliptic motion (GLEM model). When ;=1�3 we have the two-thirds
power law movement along an elliptic trajectory, that is, mutually perpendicular
harmonic oscillations of equal frequency, or our Class 1. The two other classes have
never been used.

4. LISSAJOUS MOTIONS DO NOT OBEY THE TWO-THIRDS POWER LAW

The class of Lissajous motions is defined as harmonic oscillations of the form

x=A1 sin(| 1 t+.1)
(18)

y=A2 sin(|2 t+.2),

with commensurable angular frequencies

|1
|2

=
n1
n2

,

where n1 , n2 are some integers (e.g., Symon, 1953; Yavorsky 6 Detlaf, 1982). The
closed paths of such motions are called Lissajous figures. Figure 2 presents one such
path with |2=2|1 .

By taking first and third derivatives of Eq. (18) one can see that whenever |1{|2 ,
Eq. (12) does not hold. Thus, with the exception of simple harmonic oscillations of
equal angular frequencies, Lissajous motions do not obey the two-thirds power law.
In Appendix B we derive equations necessary to draw Lissajous figures obeying the
two-thirds power law, and Fig. 2 provides an example.

48 LEBEDEV, TSUI, AND VAN GELDER



FIG. 2. (A) Lissajous figure: x=25 sin(0.2?t+0), y=19 sin(0.4?t+180). The figure consists of four
identical, mirror-transformed segments. If a movement follows the above equations, it will have a period

of 10 s. The data present counterclockwise movements along the lower right segment beginning from the

center. All movements must be accomplished in 2.5 s. (B) The mechanical power of movements as a

function of time for constant velocity movement (short-dashed line), Lissajous movement (long-dashed

line), and the optimal, two-thirds power law movement (solid line). (C) Mechanical work as a function

of time, for each of the movements. It can be seen that the integral assumes a minimum value for the

two-thirds power law, and that the constant velocity strategy requires the greatest amount of mechanical

work. (D) Velocities of the respective movements.

5. DISCUSSION

It is generally accepted in the fields of drawing motions, handwriting, and similar
movements, that motor planning takes place prior to the inception of movement:
a complex trajectory is planned as a chain of concatenated segments, and each
segment is executed by the two-thirds power law (Massey, Lurito, Pellitzer, 6
Georgopoulos, 1992; Schwartz, 1994; Viviani 6 Flash, 1995; Wolpert, Ghahramany,
6 Jordan, 1995). One of the articles (Massey et al., 1992) ends with this statement:
``However, the neural basis which would explain why this relation follows a power

49DRAWING MOVEMENTS AND THE PRINCIPLE OF LEAST ACTION



law of relatively constant exponent remains to be explored.''2 The principle of least
action is the most general principle to describe the behavior of any mechanical
system (Landau 6 Lifshitz, 1969). In drawing movements we deal with a mechanical
system of a special nature: it is planned and governed by the central nervous system
(CNS). From the principle of least action it follows that the CNS does not impose
the power law directly, but follows the strategy of accomplishing the desired goal
in a preset time with the minimum mechanical work required.

In drawing elliptic templates, both accuracy and consistency of drawing increase
with age (Viviani 6 Schneider, 1991). Rhesus monkeys must be trained properly,
with rewards, before they are able to draw a spiral, but after successful training they
perform the task as humans do, in compliance with the two-thirds power law
(Schwartz, 1994). These facts suggest that the two-thirds power law is the result of
the principle of least action acquired through a training process: among infinitely
many possible ways to move along a given path, the CNS chooses the one which
minimizes the amount of mechanical work needed to perform the desired task.

APPENDIX A: DERIVATION OF THE TWO-THIRDS LAW FROM THE
PRINCIPLE OF LEAST ACTION

We show here that if the functional

I=|
T1

T0

x* 3y"x dt (A1.1)

is minimal, then the Lagrangian function must be a constant, or

x* 3y"x=const. (A1.2)

The Euler�Lagrange equation is

Fx &
d
dt

Fx* =0, (A1.3)

where Fx and Fx* are partial derivatives of the Lagrangian function in (A1.1) with
respect to x and x* .

For the function x* 3y"x we have

Fx =x*
3yx$$$ , Fx* =3x*

2y"x , and
d
dt

Fx* =6x* x� y"x+3x*
3yx$$$;

50 LEBEDEV, TSUI, AND VAN GELDER

2 A similar question could have been applied to Kepler's famous two-thirds power law (known as
Kepler's third law of planetary motion), which states that the major semiaxes of elliptic orbits of the
planets are proportional to the periods of revolution of the planets around the sun to a power of two-
thirds. Nobody understood why the planets follow this law until the Universal Law of Gravitation was
discovered, from which all Kepler's Laws follow with necessity (Motz 6 Duveen, 1977).



hence, (A1.3) is

3x* x� y"x +x*
3yx$$$=0, (A1.4)

By multiplying both sides of (A1.4) by x* one gets

3x* 2x� y"x +x*
4yx$$$=0, (A1.5)

which is equivalent to

d
dt

(x* 3y"x) =0 (A1.6)

or

x* 3y"x=const, (A1.7)

which is the two-thirds power law.

APPENDIX B: DRAWING LISSAJOUS FIGURES WITH RESPECT TO
THE TWO-THIRDS POWER LAW

Here we derive the differential equation for drawing the Lissajous contour in
Fig. 2 in accordance with the two-thirds power law (solid lines), and we explain
how these lines have been obtained. By the two-thirds power law we have

x* 3y"x=P=const. (A2.1)

To obtain the expression for y"x , we first eliminate time from the kinematic equations
of Lissajous motion (Eq. (18)),

x=A1 sin(|1 t+.1)

y=A2 sin(|2 t+.2).

We get

y=A2 sin \$ arcsin
x

A1
+,+ , (A2.2)

where

$=
n2
n1

, ,=.2&$.1 . (A2.3)

51DRAWING MOVEMENTS AND THE PRINCIPLE OF LEAST ACTION



Then, after differentiating (A2.2) twice with respect to x we get

y"x =
A2 $x

(A 21&x
2)3�2

cos \$ arcsin
x

A1
+,+

&
A2 $

2

A 21&x
2 sin \$ arcsin

x
A1

+,+ . (A2.4)
Equation (A2.1) can be rewritten in the form:

x* = 3�
P
y"x

. (A2.5)

To produce Fig. 2 (solid lines) we integrated (A2.5) by the Runge�Kutta fourth-
order method (Jeffrey, 1995).

REFERENCES

Bronshtein, I. N., 6 Semendyayev, K. A. (1985). Handbook of mathematics. New York: Van Nostrand.

Clark, H. (Ed.) (1993). Picasso: In his words. San Francisco: Collins.

Cohen, B. I. (Ed.). (1981). The conservation of energy and the principle of least action. New York: Arno.

Jeffrey, A. (1995). Handbook of mathematical formulas and integrals. San Diego: Academic.

Kompaneyets, A. S. (1961). Theoretical physics. New York: Dover.

Korn, G. A., 6 Korn, T. M. (1968). Mathematical handbook for scientists and engineers (2nd ed.). New
York: McGraw�Hill.

Lacquaniti, F., Terzuolo, C., 6 Viviani, P. (1983). The law relating the kinematic and figural aspects of
drawing movements. Acta Psychologica 54(1�3), 115�130.

Landau, L. D., 6 Lifshitz, E. M. (1969). Mechanics. Oxford: Pergamon.

Massey, J. T., Lurito, J. T., Pellizzer, G., 6 Georgopoulos, A. P. (1992). Three-dimensional drawings in
isometric conditions: Relation between geometry and kinematics. Experimental Brain Research, 88(3),
685�690.

Motz, L., 6 Duveen, A. (1977). Essentials of astronomy. New York: Columbia.

Murdock, J. A. (1991). Perturbations: Theory and methods. New York: Wiley.

Schwartz, A. B. (1994). Direct cortical representation of drawing. Science, 265, 540�542.

Symon, K. R. (1953). Mechanics. Reading, MA: Addison�Wesley.

Viviani, P., 6 Cenzato, M. (1985). Segmentation and coupling in complex movements. Journal of
Experimental Psychology: Human Perception and Performance, 11(6), 828�845.

Viviani, P., 6 Flash, T. (1995). Minimum-jerk, two-thirds power law, and isochrony: Converging approaches
to movement planning. Journal of Experimental Psychology: Human Perception H Performance, 21(1),
32�53.

Viviani, P., 6 Schneider, R. (1991). A developmental study of the relationship between geometry and
kinematics in drawing movements. Journal of Experimental Psychology: Human Perception and
Performance, 17(1), 198�218.

Wolpert, D. M., Ghahramani, Z., 6 Jordan, M. I. (1995). An internal model for sensorimotor integration.
Science, 269, 1880�1882.

Yavorsky, B. M., 6 Detlaf, A. A. (1982). A modern handbook of physics (English translation). Moscow:
Mir.

Received: May 12, 1998

52 LEBEDEV, TSUI, AND VAN GELDER


	FIG. 1 
	1. TWO-THIRDS POWER AND THE STRATEGY OF LEAST ACTION 
	2. DIFFERENTIAL FORM OF THE TWO-THIRDS POWER LAW AND ITS PROPERTIES 
	3. CONSTANT-COEFFICIENT CASE: ELLIPSES, HYPERBOLAS, AND PARABOLAS 
	4. LISSAJOUS MOTIONS DO NOT OBEY THE TWO-THIRDS POWER LAW 
	FIG. 2 

	5. DISCUSSION 
	APPENDIX A: DERIVATION OF THE TWO-THIRDS LAW FROM THE PRINCIPLE OF LEAST ACTION 
	APPENDIX B: DRAWING LISSAJOUS FIGURES WITH RESPECT TO THE TWO-THIRDS POWER LAW 
	REFERENCES