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Flocks, Herds, and Schools: A Distributed Behavioral Model


Abstract and Figures

The aggregate motion of a flock of birds, a herd of land animals, or a school of fish is a beautiful and familiar part of the natural world. But this type of complex motion is rarely seen in computer animation. This paper explores an approach based on simulation as an alternative to scripting the paths of each bird individually. The simulated flock is an elaboration of a particle system, with the simulated birds being the particles. The aggregate motion of the simulated flock is created by a distributed behavioral model much like that at work in a natural flock; the birds choose their own course. Each simulated bird is implemented as an independent actor that navigates according to its local perception of the dynamic environment, the laws of simulated physics that rule its motion, and a set of behaviors programmed into it by the "animator." The aggregate motion of the simulated flock is the result of the dense interaction of the relatively simple behaviors of the individual simulated b...
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(~) ~ Computer Graphics, Volume 21, Number 4, July 1987
Flocks, Herds, and Schools: A Distributed Behavioral Model
Craig W. Reynolds
Symbolics Graphics Division
1401 Westwood Boulevard
Los Angeles, California 90024
(Electronic mail: cwr@Symbolics.COM)
The aggregate motion of a flock of birds, a herd of land ani-
mals, or a school of fish is a beautiful and familiar part of the
world. But this type of complex motion is rarely seen in
computer animation. This paper explores an approach based
on simulation as an alternative to scripting the paths of each
bird individually. The simulated flock is an elaboration of a
particle system, with the simulated birds being the particles.
The aggregate motion of the simulated flock is created by a
distributed behavioral model much like that at work in a natu-
ral flock; the birds choose their own course. Each simulated
bird is implemented as an independent actor that navigates ac-
cording to its local perception of the dynamic environment, the
laws of simulated physics that rule its motion, and a set of
behaviors programmed into it by the "animator." The aggre-
gate motion of the simulated flock is the result of the dense
interaction of the relatively simple behaviors of the individual
simulated birds.
Categories and Subject Descriptors: 1.2.10 [Artificial Intelli-
gence]: Vision and Scene Understanding; 1.3.5 [Computer
Graphics]: Computational Geometry and Object Modeling;
1.3.7 [Computer Graphics]: Three-Dimensional Graphics and
Realism--Animation; 1.6.3 [Simulation and Modeling[: Appli-
General Terms: Algorithms, design.
Additional Key Words, and Phrases: flock, herd, school, bird,
fish, aggregate motion, particle system, actor, flight, behav-
ioral animation, constraints, path planning.
The motion of a flock of birds is one of nature's delights.
Flocks and related synchronized group behaviors such as
schools of fish or herds of land animals are both beautiful to
watch and intriguing to contemplate. A flock* exhibits many
contrasts. It is made up of discrete birds yet overall motion
seems fluid; it is simple in concept yet is so visually complex,
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it seems randomly arrayed and yet is magnificently synchro-
nized. Perhaps most puzzling is the strong impression of inten-
tional, centralized control. Yet all evidence indicates that flock
motion must be merely the aggregate result of the actions of
individual animals, each acting solely on the basis of its own
local perception of the world.
One area of interest within computer animation is the de-
scription and control of all types of motion. Computer anima-
tors seek both to invent wholly new types of abstract motion
and to duplicate (or make variations on) the motions found in
the real world. At first glance, producing an animated, com-
puter graphic portrayal of a flock of birds presents significant
difficulties. Scripting the path of a large number of individual
objects using traditional computer animation techniques would
be tedious. Given the complex paths that birds follow, it
doubtful this specification could be made without error. Even
if a reasonable number of suitable paths could be described, it
is unlikely that the constraints of flock motion could be main-
tained (for example, preventing collisions between all birds at
each frame). Finally, a flock scripted in this manner would be
hard to edit (for example, to alter the course of all birds for a
portion of the animation). It is not impossible to script flock
motion, but a better approach is needed for efficient, robust,
and believable animation of flocks and related group motions.
This paper describes one such approach. This approach
assumes a flock is simply the result of the interaction between
the behaviors of individual birds. To simulate a flock we simu-
late the behavior of an individual bird (or at least that portion
of the bird's behavior that allows it to participate in a flock). To
support this behavioral "control structure" we must also simu-
late portions of the bird's perceptual mechanisms and aspects
of the physics of aerodynamic flight. If this simulated bird
model has the correct flock-member behavior, all that should
be required to create a simulated flock is to create some in-
stances of the simulated bird model and allow them to inter-
act. **
Some experiments with this sort of simulated flock are de-
scribed in more detail in the remainder of this paper. The suc-
*In this paperflock refers generically to a group of objects that
exhibit this general class of polarized, noncolliding, aggregate
motion. The term polarization is from zoology, meaning align-
ment of animal groups. English is rich with terms for groups of
animals; for a charming and literate discussion of such words see
An Exultation of Larks. [16]
**This paper refers to these simulated bird-like, "bird-old" objects
generically as "boids" even when they represent other sorts of
creatures such as schooling fish.
~ SIGGRAPH '87, Anaheim, July 27-31, 1987
cess and validity of these simulations is difficult to measure
objectively. They do seem to agree well with certain criteria
[25] and some statistical properties [23] of natural flocks and
schools which have been reported by the zoological and behav-
ioral sciences. Perhaps more significantly, many people who
view these animated flocks immediately recognize them as a
representation of a natural flock, and find them similarly de-
lightful to watch.
The computer graphics community has seen simulated bird
flocks before. The Electronic Theater at SIGGRAPH '85 pre-
sented a piece labeled "motion studies for a work in progress
entitled 'Eurythmy'" [4] by Susan Amkraut, Michael Girard,
and George Karl from the Computer Graphics Research Group
of Ohio State University. In the film, a flock of birds flies up
out of a minaret and, passing between a series of columns, flies
down into a lazy spiral around a courtyard. All the while the
birds slowly flap their wings and avoid collision with their
That animation was produced using a technique completely
unlike the one described in this paper and apparently not spe-
cifically intended for flock modeling. But the underlying con-
cept is useful and interesting in its own right. The following
overview is based on unpublished communications [3]. The
software is informally called "the force field animation sys-
tem." Force fields are defined by a 3 x 3 matrix operator that
transform from a point in space (where an object is located) to
an acceleration vector; the birds trace paths along the "phase
portrait" of the force field. There are "rejection forces"
around each bird and around static objects. The force field
associated with each object has a bounding box, so object in-
teractions can be culled according to bounding box tests. An
incremental, linear time algorithm finds bounding box inter-
sections. The "animator" defines the space field(s) and sets
the initial positions, orientations, and velocities of objects. The
rest of the simulation is automatic.
Karl Sims of MIT's Media Lab has constructed some be-
haviorally controlled animation of groups of moving objects
(spaceships, inchworms, and quadrupeds), but they are not or-
ganized as flocks [35]. Another author kept suggesting [28,
29, 30] implementing a flock simulation based on a distributed
behavioral model.
Particle Systems
The simulated flock described here is closely related to
cle systems
[27], which are used to represent dynamic "fuzzy
objects" having irregular and complex shapes. Particle systems
have been used to model fire, smoke, clouds, and more re-
cently, the spray and foam of ocean waves [27]. Particle sys-
tems are collections of large numbers of individual particles,
each having its own behavior. Particles are created, age, and
die off. During their life they have certain behaviors that can
alter the particle's own state, which consists of
color, opacity,
Underlying the bold flock model is a slight generalization
of particle systems. In what might be called a "subobject sys-
tem," Reeves's dot-like particles are replaced by an entire geo-
metrical object consisting of a full local coordinate system and
a reference tO a geometrical shape model. The use of shapes
instead of dots is visually significant, but the more fundamen-
tal difference is that individual subobjects have a more com-
plex geometrical state: they now have orientation.
Another difference between bold flocks and particle sys-
tems is not as well defined. The behavior of boids is generally
more complex than the behaviors for particles as described in
the literature. The present bold behavior model might be about
one or two orders of magnitude more complex than typical
particle behavior. However this is a difference of degree, not of
kind. And neither simulated behavior is nearly as complex as
that of a real bird.
Also, as presented, particles in particle systems do not in-
teract with one another, although this is not ruled out by defini-
tion. But birds and hence boids must interact strongly in
to flock correctly. Bold behavior is dependent not only on
ternal state
but also on
external state.
Actors and Distributed Systems
The behavioral model that controls the boid's flight and flock-
ing is complicated enough that rather than use an
ad hoc
proach, it is worthwhile to pursue the most appropriate formal
computational model. The behaviors will be represented as
rules or programs in some sense, and the internal state of each
bold must be held in some sort of data structure. It is conve-
nient to encapsulate these behaviors and state as an
the sense of object-oriented programming systems [10, 11,
21]. Each
of these objects needs a computational
to apply the behavioral programs to the internal data.
The computational abstraction that combines process, proce-
dure, and state is called an
[12, 26, 2]. An actor is
essentially a virtual computer that communicates with other
virtual computers by
passing messages.
The actor model has
been proposed as a natural structure for animation control by
several authors [28, 13, 29, 18]. It seems particularly apt for
situations involving interacting characters and behavior simula-
tion. In the literature of parallel and distributed computer sys-
tems, flocks and schools are given as examples of robust
self-organizing distributed systems [15].
Behavioral Animation
Traditional hand-drawn eel animation was produced with a me-
dium that was completely inert. Traditional computer anima-
tion uses an active medium (computers running graphics
software), but most animation systems do not make much use
of the computer's ability to automate motion design. Using
different tools, contemporary computer animators work at al-
most the same low level of abstraction as do eel animators.
They tell their story by directly describing the motion of their
characters. Shortcuts exist in both media; it is common for
computer animators and eel animators to use helpers to inter-
polate between specified keyframes. But little progress has
been made in automating motion description; it is up to the
animator to translate the nuances of emotion and characteriza-
tion into the motions that the character performs. The animator
cannot simply tell the character to "act happy" but must tedi-
ously specify the motion that conveys happiness.
Typical computer animation models only the shape and
physical properties of the characters, whereas
animation seeks to model the behavior of the
character. The goal is for such simulated characters to handle
many of the details of their actions, and hence their motions.
include a whole range of activities from sim-
ple path planning to complex "emotional" interactions be-
tween characters. The construction of behavioral animation
characters has attracted many researchers [19, 21, 13, 14, 29,
(~ ~ Computer Graphics, Volume 21, Number 4, July 1987
ii llll ii
30, 41, 40], but it is still a young field in which more work is
Because of the detached nature of the control, the person
who creates animation with character simulation might not
strictly be an
Traditionally, the animator is directly
responsible for all motion in animation production [40]. It
might be more proper to call the person who directs animation
via simulated characters a
since the animator is
less a designer of motion and more a designer of behavior.
These behaviors, when acted out by the simulated characters,
lead indirectly to the final action. Thus the animator's job be-
comes somewhat like that of a theatrical director: the charac-
ter's performance is the indirect result of the director's
instructions to the actor. One of the charming aspects of
work reported here is not knowing how a simulation is going to
proceed from the specified behaviors and initial conditions;
there are many unexpected, pleasant surprises. On the other
hand, this charm starts to wear thin as deadlines approach and
the unexpected annoyances pop up. This author has spent a lot
of time recently trying to get uncooperative flocks to move as
intended ("these darn boids seem to have a mind of their
own! ").
speed. A minimum speed
can also be specified but defaults to
zero. A
maximum acceleration,
expressed as a fraction of the
maximum speed, is used to truncate over-anxious requests for
acceleration, hence providing for smooth changes of speed and
heading. This is a simple model of a creature with a finite
amount of available energy.
Many physical forces are not supported in the current bold
is modeled but used only to define banking
behavior. It is defined procedurally to allow the construction of
arbitrarily shaped fields. If each bold was accelerated by grav-
ity each frame, it would tend to fall unless gravity was coun-
tered by
Buoyancy is aligned against gravity,
but aerodynamic lift is aligned with the boid's local "up" di-
rection and related to velocity. This level of modeling leads to
effects like normally level flight, going faster when flying
down (or slower up), and the "stall" maneuver. The speed
limit parameter could be more realistically modeled as a fric-
a backward pointing force related to velocity. In
the current model steering is done by directing the available
in the appropriate direction. It would be more realistic to
separately model the
thrusting forces and the
steering forces, since they normally have different magnitudes.
Geometric Flight
A fundamental part of the bold model is the geometric ability
The motion of the members of a simulated school or
herd can be considered a type of "flying" by glossing over the
considerable intricacies of wing, fin, and leg motion (and in
the case of herds, by restricting freedom of motion in the third
dimension). In this paper the term
geometric flight
refers to a
certain type of motion along a path: a dynamic, incremental,
rigid geometrical transformation of an object, moving along
and tangent to a 3D curve. While the motion is rigid, the ob-
ject's underlying geometric model is free to articulate or
change shape within this "flying coordinate system." Unlike
more typical animated motion along predefined spline curves,
the shape of a flight path is not specified in advance.
Geometric flight is based on incremental translations along
the objecrs "forward direction" its local positive Z axis.
These translations are intermixed with
about the local X and Y axes
which realign
the global orientation of the local Z axis. In real flight, turning
and moving happen continuously and simultaneously. Incre-
mental geometric flight is a discrete approximation of this;
small linear motions model a continuous curved path. In ani-
mation the motion must increment at least once per frame.
Running the simulation at a higher rate can reduce the discrete
sampling error of the flight model and refine the shape of mo-
tion blur patterns.
Flight modeling makes extensive use of the object's own
coordinate system. Local space represents the "boid's eye
view;" it implies measuring things relative to the boid's own
position and orientation. In Cartesian terms, the lefl/right axis
is X, up/down is Y, and forward/back is Z. The conversion of
geometric data between the local and global reference frames is
handled by the geometric operators
It is
convenient to use a local scale so that the unit of length of the
coordinate system is one
body length.
Biologists routinely
specify flock and school statistics in terms of body lengths.
Geometric flight models conservation of momentum. An
object in flight tends to stay in flight. There is a simple model
of viscous speed damping, so even if the bold continually ac-
celerates in one direction, it will not exceed a certain
Geometric flight relates translation, pitch, and yaw, but does
not constrain
the rotation about the local Z axis. This
degree of freedom is used for
the object to
align the local Y axis with the (local XY component of the
total) acceleration acting upon it. Normally banking is based
on the lateral component of the acceleration, but the tangential
component can be used for certain applications. The lateral
components are from steering and gravity. In straight flight
there is no radial force, so the gravitational term dominates and
banking aligns the object's -Y axis with "gravitational down"
direction. When turning, the radial component grows larger
and the "accelerational down" direction swings outward, like a
pendulum hanging from the flying object. The magnitude of
the turning acceleration varies directly with the object's veloc-
ity and with the curvature of its path (so inversely with the
radius of its turn). The limiting case of infinite velocity resem-
bles banking behavior in the absence of gravity. In these cases
the local + Y (up) direction points directly at the center of
curvature defined by the current turn.
Figure 1.
With correct banking (what pilots call a
coordinated turn)
the object's local space remains aligned with the "perceptual"
or "accelerational" coordinate system. This has several advan-
tages: it simplifies the bird's (or pilot's) orientation task, it
~/~ SIGGRAPH '87, Anaheim, July 27-31, 1987
keeps the lift from the airfoils of the wings pointed in the most
efficient direction ("accelerational up"), it keeps the passen-
gers' coffee in their cups, and most importantly for animation,
it makes the flying boid fit the viewer's expectation of how
flying objects should move and orient themselves. On the other
hand, realism is not always the goal in animation. By simply
reversing the angle of bank we obtain a cartoony motion that
looks like the object is being flung outward by the centrifugal
force of the turn.
Boids and Turtles
The incremental mixing of forward translations and local rota-
tions that underlies geometric flight is the basis of "turtle
graphics" in the programming language Logo [5]. Logo was
first used as an educational tool to allow children to learn ex-
perimentally about geometry, arithmetic, and programming
[22]. The Logo turtle was originally a little mechanical robot
that crawled around on large sheets of paper laid on the class-
room floor, drawing graphic figures by dragging a felt tip
marker along the paper as it moved. Abstract turtle geometry is
a system based on the frame of reference of the turtle, an ob-
ject that unites position and heading. Under program control
the Logo turtle could move forward or back from its current
position, turn left or right from its current heading, or put the
pen up or down on the paper. The turtle geometry has been
extended from the plane onto arbitrary manifolds and into 3D
space [1]. These "3d turtles" and their paths are exactly equiv-
alent to the boid objects and their flight paths.
Natural Flocks, Herds, and Schools
"... and the thousands of fishes moved as a huge beast, piercing
the water. They appeared united, inexorably bound to a common
fate. How comes this unity?"
--Anonymous, 17th century (from Shaw)
For a bird to participate in a flock, it must have behaviors
that allow it to coordinate its movements with those of its
flockmates. These behaviors are not particularly unique; all
creatures have them to some degree. Natural flocks seem to
consist of two balanced, opposing behaviors: a desire to stay
close to the flock and a desire to avoid collisions within the
flock [34]. It is clear why an individual bird wants to avoid
collisions with its flockrnates. But why do birds seem to seek
out the airborne equivalent of a nasty traffic jam? The basic
urge to join a flock seems to be the result of evolutionary
pressure from several factors: protection from predators, statis-
tically improving survival of the (shared) gene pool from at-
tacks from predators, profiting from a larger effective search
pattern in the quest for food, and advantages for social and
mating activities [33].
There is no evidence that the complexity of natural flocks
is bounded in any way. Flocks do not become "full" or "over-
loaded" as new birds join. When herring migrate toward their
spawning grounds, they run in schools extending as long as 17
miles and containing millions of fish [32]. Natural flocks seem
to operate in exactly the same fashion over a huge range of
flock populations. It does not seem that an individual bird can
be paying much attention to each and every one of its flock-
mates. But in a huge flock spread over vast distances, an indi-
vidual bird must have a localized and filtered perception of the
rest of the flock. A bird might be aware of three categories:
itself, its two or three nearest neighbors, and the rest of the
flock [23].
These speculations about the "computational complexity"
of flocking are meant to suggest that birds can flock with any
number of flockmates because they are using what would be
called in formal computer science a constant time algorithm.
That is, the amount of "thinking" that a bird has to do in order
to flock must be largely independent of the number of birds in
the flock. Otherwise we would expect to see a sha W upper
bound on the size of natural flocks when the individual birds
became overloaded by the complexity of their navigation task.
This has not been observed in nature.
Contrast the insensitivity to complexity of real flocks with
the situation for the simulated flocks described below. The
complexity of the flocking algorithm described is basically
O(N2). That is, the work required to run the algorithm grows
as the square of the flock's population. We definitely
do see
upper bound on the size of simulated flocks implemented as
described here. Some techniques to address this performance
issue are discussed in the section Algorithmic Considerations.
Simulated Flocks
To build a simulated flock, we start with a boid model that
supports geometric flight. We add behaviors that correspond to
the opposing forces of collision avoidance and the urge to join
the flock. Stated briefly as rules, and in order of decreasing
precedence, the behaviors that lead to simulated flocking are:
1. Collision Avoidance: avoid collisions with nearby
2. Velocity Matching: attempt to match velocity with nearby
3. Flock Centering: attempt to stay close to nearby flockmates
Velocity is a vector quantity, referring to the combination of
heading and speed. The manner in which the results from each
of these behaviors is reconciled and combined is significant
and is discussed in more detail later. Similarly, the meaning
nearby in these rules is key to the flocking process. This is also
discussed in more detail later, but generally one boid's aware-
ness of another is based on the distance and direction of the
offset vector between them.
Static collision avoidance and dynamic velocity matching
are complementary. Together they ensure that the members of a
simulated flock are free to fly within the crowded skies of the
flock's interior without running into one another. Collision
avoidance is the urge to steer away from an imminent impact.
Static collision avoidance is based on the relative position of
the flockmates and ignores their velocity. Conversely, velocity
matching is based only on velocity and ignores position. It is a
predictive version of collision avoidance: if the boid does a
good job of matching velocity with its neighbors, it is unlikely
that it will collide with any of them any time soon. With veloc-
ity matching, separations between boids remains approximately
invariant with respect to ongoing geometric flight. Static colli-
sion avoidance serves to establish the minimum required sepa-
ration distance; velocity matching tends to maintain it.
Flock centering makes a boid want to be near the center of
the flock. Because each boid has a localized perception of the
world, "center of the flock" actually means the center of the
nearby flockmates. Flock centering causes the boid to fly in a
direction that moves it closer to the centroid of the nearby
boids. If a boid is deep inside a flock, the population density in
its neighborhood is roughly homogeneous; the boid density is
~' Computer Graphics, Volume 21, Number 4, July 1987
approximately the same in all directions. In this case, the cem
troid of the neighborhood boids is approximately at the center
of the neighborhood, so the flock centering urge is small. But
if a boid is on the boundary of the flock, its neighboring boids
are on one side. The centroid of the neighborhood boids is
displaced from the center of the neighborhood toward the body
of the flock. Here the flock centering urge is stronger and the
flight path will be deflected somewhat toward the local flock
Real flocks sometimes split apart to go around an obstacle.
To be realistic, the simulated flock model must also have this
ability. Flock centering correctly allows simulated flocks to
bifurcate. As long as an individual boid can stay close to its
nearby neighbors, it does not care if the rest of the flock turns
away. More simplistic models proposed for flock organization
(such as a
central force
model or a
follow the designated leader
model) do not allow splits.
The flock model presented here is actually a better model
of a school or a herd than a flock. Fish in murky water (and
land animals with their inability to see past their herdmates)
have a limited, short-range perception of their environment.
Birds, especially those on the outside of a flock, have excellent
long-range "visual perception." Presumably this allows widely
separated flocks to join together. If the flock centering urge
was completely localized, when two flocks got a certain dis-
tance apart they would ignore each other. Long-range vision
seems to play a part in the incredibly rapid propagation of a
"maneuver wave" through a flock of birds. It has been shown
that the speed of propagation of this wavefront reaches three
times the speed implied by the measured startle reaction time
of the individual birds. The explanation advanced by Wayne
Ports is that the birds perceive the motion of the oncoming
"maneuver wave" and time their own turn to match it [25].
Potts refers to this as the "chorus line" hypothesis.
Arbitrating Independent Behaviors
The three behavioral urges associated with flocking (and others
to be discussed below) each produce an isolated suggestion
about which way to steer the boid. These are expressed as
acceleration requests.
Each behavior says: "if I were in
charge, I would accelerate in
direction." The acceleration
request is in terms of a 3D vector that, by system convention,
is truncated to unit magnitude or less. Each behavior has sev-
eral parameters that control its function; one is a "strength," a
fractional value between zero and one that can further attenuate
the acceleration request. It is up to the
navigation module
the boid brain to collect all relevant acceleration requests and
then determine a single behaviorally desired acceleration. It
must combine, prioritize, and arbitrate between potentially
conflicting urges. The
pilot module
takes the acceleration de-
sired by the navigation module and passes it to the
flight mod-
which attempts to fly in that direction.
The easiest way to combine acceleration requests is to aver-
age them. Because of the included "strength" factors, this is
actually a weighted average. The relative strength of one be-
havior to another can be defined this way, but it is a precarious
interrelationship that is difficult to adjust. An early version of
the boid model showed that navigation by simple weighted av-
eraging of acceleration requests works "pretty well." A bold
that chooses its course this way will fly a reasonable course
under typical conditions. But in critical situations, such as po-
tential collision with obstacles, conflicts must be resolved in a
timely manner. During high-speed flight, hesitation or indeci-
sion is the wrong response to a brick wall dead ahead.
The main cause of indecision is that each behavior might
be shouting advice about which way to turn to avoid disaster,
but if those acceleration requests happen to lie in approxi-
mately opposite directions, they will largely cancel out under a
simple weighted averaging scheme. The boid would make a
very small turn and so continue in the same direction, perhaps
to crash into the obstacle. Even when the urges do not cancel
out, averaging leads to other problems. Consider flying over a
gridwork of city streets between the skyscrapers; while "'fly
north" or "fly east" might be good ideas, it would be a bad
idea to combine them as "fly northeast."
Techniques from artificial intelligence, such as expert sys-
tems, can be used to arbitrate conflicting opinions. However, a
less complex approach is taken in the current implementation.
Prioritized acceleration allocation
is based on a strict priority
ordering of all component behaviors, hence of the consider-
ation of their acceleration requests. (This ordering can change
to suit dynamic conditions.) The acceleration requests are con-
sidered in priority order and added into an accumulator. The
of each request is measured and added into another
accumulator. This process continues until the sum of the accu-
mulated magnitudes gets larger than the
maximum acceleration
value, which is a parameter of each boid. The last acceleration
request is trimmed back to compensate for the excess of accu-
mulated magnitude. The point is that a fixed amount of accel-
eration is under the control of the navigation module; this
acceleration is parceled out to satisfy the acceleration request
of the various behaviors in order of priority. In an emergency
the acceleration would be allocated to satisfy the most pressing
needs first; if all available acceleration is "'used up," the less
pressing behaviors might be temporarily unsatisfied. For exam-
ple, the flock centering urge could be correctly ignored tempo-
rarily in favor of a maneuver to avoid a static obstacle.
Simulated Perception
The bold model does not directly simulate the senses used by
real animals during flocking (vision and hearing) or schooling
(vision and fishes' unique "lateral line" structure that provides
a certain amount of pressure imaging ability [23, 24]). Rather
the perception model tries to make available to the behavior
model approximately the same information that is available to a
real animal as the end result of its perceptual and cognitive
This is primarily a matter of filtering out the surplus infor-
mation that is available to the software that implements the
boid's behavior. Simulated boids have direct access to the geo-
metric database that describes the exact position, orientation,
and velocity of all objects in the environment. The real bird's
information about the world is severely limited because it per-
ceives through imperfect senses and because its nearby flock-
mates hide those farther away. This is even more pronounced in
herding animals because they are all constrained to be in the
same plane. In fish schools, visual perception of neighboring
fish is further limited by the scattering and absorption of light
by the sometimes murky water between them. These factors
combine to strongly localize the information available to each
Not only is it unrealistic to give each simulated boid perfect
and complete information about the world, it is just plain
wrong and leads to obvious failures of the behavior model.
Before the current implementation of localized
flock centering
behavior was implemented, the flocks used a central force
model. This leads to unusual effects such as causing all mem-
bers of a widely scattered flock to simultaneously converge
~ SIGGRAPH '87, Anaheim, July 27-31, 1987
toward the flock's centroid. An interesting result of the experi-
ments reported in this paper is that the aggregate motion that
we intuitively recognize as "flocking" (or schooling or herd-
ing) depends upon a limited, localized view of the world.
The behaviors that make up the flocking model are stated
in terms of "nearby flockmates." In the current implementa-
tion, the neighborhood is defined as a spherical zone of sensi-
tivity centered at the boid's local origin. The magnitude of the
sensitivity is defined as an inverse exponential of distance.
Hence the neighborhood is defined by two parameters: a radius
and exponent. There is reason to believe that this field of sensi-
tivity should realistically be exaggerated in the forward direc-
tion and probably by an amount proportional to the boid's
speed. Being in motion requires an increased awareness of
what lies ahead, and this requirement increases with speed. A
forward-weighted sensitivity zone would probably also im-
prove the behavior in the current implementation of boids at
the leading edge of a flock, who tend to get distracted by the
flock behind them. Because of the way their heads and eyes are
arranged, real birds have a wide field of view (about 300 de-
grees), but the zone of overlap from both eyes is small (10 to
15 degrees). Hence the bird has stereo depth perception only
in a very small, forward-oriented cone. Research is currently
under way on models of forward-weighted perception for
In an early version of the flock model, the metrics of at-
traction and repulsion were weighted linearly by distance. This
spring-like model produced a bouncy flock action, fine per-
haps for a cartoony characterization, but not very realistic. The
model was changed to use an inverse square of the distance.
This more gravity-like model produced what appeared to be a
more natural, better damped flock model. This correlated well
with the carefully controlled quantitative studies that Brian
Partridge made of the spatial relationships of schooling fish
[23]; he found that "a fish is much more strongly influenced
by its near neighbors than it is by the distant members of the
school. The contribution of each fish to the [influence] is in-
versely proportional to the square or the cube of the distance."
In previous work he and colleagues [23, 24] demonstrated that
fishes school based on information from both their visual sys-
tem and from their "lateral line" organ which senses pressure
waves. The area of a perspective image of the silhouette of an
object (its "visual angle") varies inversely with the square of
its distance, and that pressure waves traveling through a 3D
medium like water fall off inversely with the cube of the dis-
The boid perception model is quite
ad hoc
and avoids actu-
ally simulating vision. Artificial vision is an extremely com-
plex problem [38] and is far beyond the scope of this work. But
if boids could "see" their environment, they would be better at
path planning than the current model. It is possible to construct
simple maze-like shapes that would confuse the current boid
model but would be easily solved by a boid with vision.
Impromptu Flocking
The flocking model described above gives boids an eagerness
to participate in an acceptable approximation of flock-like mo-
tion. Boids released near one another begin to flock together,
cavorting and jostling for position. The boids stay near one
(flock centering)
but always maintain prudent separa-
tion from their neighbors
(collision avoidance),
and the flock
quickly becomes "polarized"--its members heading in ap-
proximately the same direction at approximately the same
(velocity matching);
when they change direction they do
it in synchronization. Solitary boids and smaller flocks join to
become larger flocks, and in the presence of external obstacles
(discussed below), larger flocks can split into smaller flocks.
For each simulation run, the initial position (within a speci-
fied ellipsoid), heading, velocity, and various other parameters
of the boid model are initialized to values randomized within
specified distributions. A restartable random-number generator
is used to allow repeatability. This randomization is not re-
quired; the boids could just as well start out arranged in a
regular pattern, all other aspects of the flock model are com-
pletely deterministic and repeatable.
When the simulation is run, the flock's first action is a
reaction to the initial conditions. If the boids started out too
closely crowded together, there is an initial "flash expansion"
where the mutual desire to avoid collision drives the boids
radially away from the site of the initial over-pressure. If re-
leased in a spherical shell with a radius smaller than the
"neighborhood" radius, the boids contract toward the sphere's
center; otherwise they begin to coalesce into small flockettes
that might themselves begin to join together. If the boids are
confined within a certain region, the smaller flocks eventually
conglomerate into a single flock if left to wander long enough.
Scripted Flocking
The behaviors discussed so far provide for the ability of indi-
vidual birds to fly and participate in happy aimless flocking.
But to combine flock simulations with other animated action,
we need more direct control over the flock. We would like to
direct specific action at specific times (for example, "the flock
enters from the left at :02.3 seconds into the sequence, turns to
fly directly upward at :03.5, and is out of the frame at :04.0").
The current implementation of the bold model has several
facilities to direct the motion and timing of the flock action.
First, the simulations are run under the control of a general-
purpose animation scripting system [36]. The details of that
scripting system are not relevant here except that, in addition to
the typical interactive motion control facilities, it provides the
ability to schedule the invocation of user-supplied software
(such as the flock model) on a frame-by-frame basis. This
scripting facility is the basic tool used to describe the timing of
various flock actions. It also allows flexible control over the
time-varying values of parameters, which can be passed down
to the simulation software. Finally the script is used to set up
and animate all nonbehavioral aspects of the scene, such as
backgrounds, lighting, camera motion, and other visible ob-
The primary tool for scripting the flock's path is the
tory urge
built into the boid model. In the current model this
urge is specified in terms of a global target, either as a global
direction (as in "going Z for the winter") or as a global
position--a target point toward which all birds fly. The model
computes a bounded acceleration that incrementally turns the
boid toward its migratory target.
With the scripting system, we can
a dynamic pa-
rameter whose value is a global position vector or a global
direction vector. This parameter can be passed to the flock,
which can in turn pass it along to all boids, each of which sets
its own "migratory goal register." Hence the global migratory
behavior of all birds can be directly controlled from the script.
(Of course, it is not necessary to alter all boids at the same
time, for example, the delay could be a function of their
present position in space. Real flocks do not change direction
simultaneously [25], but rather the turn starts with a single
bird and spreads quickly across the flock like a shock wave.)
(~ ~ Computer Graphics, Volume 21, Number 4, July 1987
We can lead the flock around by animating the goal point
along the desired path, somewhat ahead of the flock. Even if
the migratory goal point is changed abruptly the path of each
boid still is relatively smooth because of the flight model's
simulated conservation of momentum. This means that the
boid's own flight dynamics implement a form of smoothing
interpolation between "control points."
Avoiding Environmental Obstacles
The most interesting motion of a simulated flock comes from
interaction with other objects in the environment. The isolated
behavior of a flock tends to reach a steady state and becomes
rather sterile. The flock can be seen as a
solution to
the constraints implied by its behaviors. For example, the con-
flicting urges
of flock centering
collision avoidance
do not
lead to constant back and forth motion, but rather the boids
eventually strike a balance between the two urges (the degree
of damping controls how soon this balance is reached). Envi-
ronmental obstacles and the boid's attempts to navigate around
them increase the apparent complexity of the behavior of the
flock. (In fact the complexity of real flocks might be due
largely to the complexity of the natural environment.)
Environmental obstacles are also important from the stand-
point of modeling the scene in which we wish to place the
flock. If the flock is scripted to fly under a bridge and around a
tree, we must be able to represent the geometric shape and
dimension of these obstacles. The approach taken here is to
independently model the "shape for rendering" and the "shape
for collision avoidance." The types of shapes currently used for
environmental obstacles are much less complicated than the
models used for rendering of computer graphic models. The
current work implements two types of shapes of environmental
collision avoidance. One is based on the
force field
which works in undemanding situations but has some short-
comings. The other model called
is more robust
and seems closer in spirit to the natural mechanism.
The force field model postulates a field of repulsion force
emanating from the obstacle out into space; the boids are in-
creasingly repulsed as they get closer to the obstacle. This
scheme is easy to model; the geometry of the field is usually
fairly simple and so an avoidance acceleration can be directly
calculated from the field equation. These models can produce
good results, such as in "Eurythmy" [4], but they also have
drawbacks that are apparent on close examination. If a boid
approaches an obstacle surrounded by a force field at an angle
such that it is exactly opposite to the direction of the force
field, the boid will not turn away. In this case the force field
serves only to slow the boid by accelerating it backwards and
provides no side thrust at all. The worst reaction to an impend-
ing collision is to fail to turn. Force fields also cause problems
with "peripheral vision" The boid should notice and turn away
from a wall as it flies toward it, but the wall should be ignored
if the boid is flying alongside it. Finally, force fields tend to be
too strong close up and too weak far away; avoiding an obstacle
should involve long-range planning rather than panicky correc-
tions at the last minute.
is a better simulation of a natural bird guided
by vision. The boid considers only obstacles directly in front
of it. (It finds the intersection, if any, of its local Z axis with
the obstacle.) Working in local perspective space, it finds the
silhouette edge of the obstacle closest to the point of eventual
impact. A radial vector is computed which will aim the boid at
a point one body length beyond that silhouette edge (see figure
2). Currently steer-to-avoid has been implemented for several
obstacle shapes: spheres, cylinders, planes, and boxes.
~ SIGGRAPH '87, Anaheim, July 27-31, 1987
sion avoidance for arbitrary convex polyhedral obstacles is be-
ing developed.
Figure 2.
Obstacles are not necessarily fixed in space; they can be
animated around by the script during the animation. Or more
interestingly, the obstacles can be behavioral characters. Spar-
rows might flock around a group of obstacles that is in fact a
herd of elephants. Similarly, behavioral obstacles might not
merely be in the way; they might be objects of fear such as
predators. It has been noted [25] that natural flocking instincts
seem to be sharpened by predators.
Other Applications of the Flock Model
The model of polarized noncolliding aggregate motion has
many applications, visual simulation of bird flocks in computer
animation being one. Certain modifications yield a fish school
model. Further modifications, such as limitation to a 2D sur-
face and the ability to follow the terrain, lead to a herd model.
Imagine a herd of PODA-style legged creatures 19], using Karl
Sims' techniques for locomotion over uneven, complex terrain
|35]. Other applications are less obvious. Traffic patterns,
such as the flow of cars on a freeway, is a flock-like motion.
There are specialized behaviors, such as being constrained to
drive within the lanes, but the basic principles that keep boids
from colliding are just as applicable on the freeway. We could
imagine creating crowds of "extras" (human or otherwise) for
feature films. However the most fun are the offbeat combina-
tions possible in computer graphics by mixing and matching: a
herd of pogo sticks, a flock of Pegasus-like winged horses, or a
traffic jam of spaceships on a 3D interplanetary highway.
One serious application would be to aid in the scientific
investigation of flocks, herds, and schools. These scientists
must work almost exclusively in the observational mode; ex-
periments with natural flocks and schools are difficult to per-
form and are likely to disturb the behaviors under study. It
might be possible, using a more carefully crafted model of the
realistic behavior of a certain species of bird, to perform con-
trolled and repeatable experiments with "simulated natural
flocks." A theory of flock organization can be unambiguously
tested by implementing a distributed behavioral model and sim-
ply comparing the aggregate motion of the simulated flock
with the natural one.
Algorithmic Considerations
A naive implementation of the basic flocking algorithm would
grow in complexity as the order of the square of the flock's
population ("O(N~)"). Basically this is because each bold must
reason about each of the other boids, even if only to decide to
ignore it. This does not say the algorithm is slow or fast,
merely that as the size of the problem (total population of the
flock) increases, the complexity increases even faster. Dou-
bling the number of boids quadruples the amount of time
However, as stated before, real birds are probably not as
sensitive to the total flock population. This gives hope that the
simulated boid could be taught to navigate independently of the
total population. Certainly part of the problem is that we are
trying to run the simulation of the whole flock on a single
computer. The natural solution is to use distributed processing,
as the real flock does. If we used a separate processor for each
boid, then even the naive implementation of the flocking al-
gorithm would be O(N), or linear with respect to the popula-
tion. But even that is not good enough. It still means that as
more boids are added to the flock, the complexity of the prob-
lem increases.
What we desire is a constant time algorithm, one that is
insensitive to the total population. Another way to say this is
that an N 2 algorithm would be OK if there was an efficient way
to keep N very small. Two approaches to this goal are cur-
rently under investigation. One is dynamic spatial partitioning
of the flock; the boids are sorted into a lattice of "bins" based
on their position in space. A boid trying to navigate inside the
flock could get quick access to the flockmates that are physi-
cally nearby by examining the "bins" near its current position.
Another approach is to do incremental collision detection
("nearness testing"). General collision detection is another N ~
algorithm, but if one does collision detection incrementally,
based on a partial solution that described the situation just a
moment before, then the algorithm need worry only about the
changes and so can run much faster, assuming that the incre-
mental changes are small. The incremental collision detection
algorithm used in Girard's PODA system [9] apparently
achieves constant time performance in the typical case.
Computing Environment
The boids software was written in Symbolics Common Lisp.
The code and animation were produced on a Symbolics 3600
Lisp Machine, a high-performance personal computer. The
flock software is implemented in Flavors, the object-oriented
programming extensions to Symbolics Common Lisp. The ge-
ometric aspects of the system are layered upon S-Geometry, an
interactive geometric modeler [37]. Boids are based on the
flavor 3D:OBJECT, which provides their geometric abilities.
The flock simulations are invoked from scripts created and
animated with the S-Dynamics [36] animation system, which
also provided the real-time playback facility used to view the
motion tests. The availability of this graphical toolkit allowed
the author to focus immediately on the issues unique to this
project. One example of the value of this substrate is that the
initial version of the flock model, including implementation,
testing, debugging, and the production of seven short motion
tests was accomplished in the ten days before the SIGGRAPH
'86 conference.
The boid software has not been optimized for speed. But
this report would be incomplete without a rough estimate of the
actual performance of the system. With a flock of 80 boids,
using the naive O(N ~) algorithm (and so 6400 individual boid-
to-boid comparisons), on a single Lisp Machine without any
special hardware accelerators, the simulation ran for about 95
seconds per frame. A ten-second (300 frame) motion test took
about eight hours of real time to produce.
~ Computer Graphics, Volume 21, Number 4, July 1987
Future Work
This paper has largely ignored the internal animation of the
geometrical model that provides the visual representation of
the bold. The original motion tests produced with these models
all show flocks of little abstract rigid shapes that might be
paper airplanes. There was no flapping of wings nor turning of
heads, and there was certainly no character animation. These
topics are all important and pertinent to believable animation of
simulated flocks. But the underlying abstract nature of flocking
as polarized, noncolliding aggregate motion is largely indepen-
dent of these issues of internal shape change and articulation.
This notion is supported by the fact that most viewers of these
simulations identify the motion of these abstract objects as
"flocking'" even in the absence of any internal animation.
But doing a believable job of melding these two aspects of
the motion is more than a matter of concatenating the action of
an internal animation cycle for the character with the motion
defined by geometrical flight. There are important issues of
synchronization between the current state of the flight dy-
namics model, and the amplitude and frequency of the wing
motion cycle. Topics of current development include internal
animation, synchronization, and interfaces between the
simulation-based flock model and other more traditional, inter-
active animation scripting systems. We would like to allow a
skilled computer animator to design a bird character and define
its "wing flap cycle" using standard interactive modeling and
scripting techniques, and then be able to take this cyclic mo-
tion and "plug it in" to the flock simulation model causing the
boids in the flock to fly according to the scripted cycle.
The behaviors that have been discussed in this paper are all
simplistic, isolated behaviors of low complexity. Tile boids
have a geometric and kinematic state, but they have no signifi-
mental state. Real animals have more elaborate, abstract
behaviors than a simple desire to avoid a painful collision; they
have more complex motivations than a simple desire to fly to a
certain point in space. More interesting behavior models would
take into account hunger, finding food, fear of predators, a
periodic need to sleep, and so on. Behavior models of this type
have been created by other investigators [6, 19, 21], but they
have not yet been implemented for the bold model described
This paper has presented a model of polarized, noncolliding
aggregate motion, such as that of flocks, herds, and schools.
The model is based on simulating the behavior of each bird
independently. Working independently, the birds try both to
stick together and avoid collisions with one another and with
other objects in their environment. The animations showing
simulated flocks built from this model seem to correspond to
the observer's intuitive notion of what constitutes "flock-like
motion." However it is difficult to objectively measure how
valid these simulations are. By comparing behavioral aspects
of the simulated flock with those of natural flocks, we are able
improve and refine the model. But having approached a certain
level of realism in the model, the parameters of the simulated
flock can be altered at will by the animator to achieve many
variations on flock-like behavior.
I would like to thank flocks, herds, and schools for existing;
nature is the ultimate source of inspiration for computer graph-
ics and animation. I would also like to acknowledge the contri-
butions to this research provided by workers in a wonderfully
diverse collection of pursuits:
To the natural sciences of behavior, evolution, and zoology:
for doing the hard work, the Real Science, on which this com-
puter graphics approximation is based. To the Logo group who
invented the appropriate geometry, and so put us in the driver's
seat. To the Actor semantics people who invented the appropri-
ate control structure, and so gave the bold a brain. To the many
developers of modern Lisp who invented the appropriate pro-
gramming language. To my past and present colleagues at
MIT, III, and Symbolics who have patiently listened to my
speculations about flocks for years and years before I made my
first bold fly. To the Graphics Division of Symbolics, Inc.,
who employ me, put up with my nasty disposition, provide me
with fantastic computing and graphics facilities, and have gen-
erously supported the development of the work described here.
And to the field of computer graphics, for giving professional
respectability to advanced forms of play such as reported in
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... Models of collective motion based in self-propelled particles and interacting with neighbouring particles via these three kinds of social interactions are able to reproduce typical features observed in moving animals such as polarized movement, where individuals move globally * in the same direction; swarming, where individuals aggregate in a group but move in random directions; and milling, where individuals rotate around a core [5,6]. Although alignment forces are a straight forward way to induce global orientational state, they are not a necessary condition [3,[7][8][9][10][11]. ...
Full-text available
Animals moving together in groups are believed to interact among each other with effective social forces, such as attraction, repulsion and alignment. Such forces can be inferred using 'force maps', i.e. by analysing the dependency of the acceleration of a focal individual on relevant variables. Here we introduce a force map technique for alignment depending on relative velocities between an individual and its neighbours. After the force map approach is validated with an agent-based model, we apply it to experimental data of schooling fish, where we observe signatures of an effective alignment force with faster neighbours, and an unexpected anti-alignment with slower neighbours. Instead of an explicit anti-alignment behaviour, we suggest that the observed pattern is a result of a selective attention mechanism, where fish pay less attention to slower neighbours. We present support for this hypothesis both from agent-based modeling, as well as from exploring leader-follower relationships between faster and slower neighbouring fish in the experimental data.
... Numerous situations require individuals to organize themselves into such groups, to execute an action or acquire a resource. In the animal kingdom, many species gather into flocks, herds, or schools for traveling purposes (Okubo 1986;Reynolds 1987) and predators assemble packs for hunting (Creel and Creel 1995;Gittleman 1989). In the human world, children groups gather in the schoolyard to engage in common activities (Moody 2001), sports teams emerge to provide opportunities for shared free-time activities (Lazarsfeld and Merton 1954;Putnam 2000), and project teams assemble spontaneously to tackle organizational tasks (Guimera et al. 2005;Zhu, Huang, and Contractor 2013). ...
Despite the central role of self-assembled groups in animal and human societies, statistical tools to explain their composition are limited. The authors introduce a statistical framework for cross-sectional observations of groups with exclusive membership to illuminate the social and organizational mechanisms that bring people together. Drawing from stochastic models for networks and partitions, the proposed framework introduces an exponential family of distributions for partitions. The authors derive its main mathematical properties and suggest strategies to specify and estimate such models. A case study on hackathon events applies the developed framework to the study of mechanisms underlying the formation of self-assembled project teams.
Full-text available
Flocking guidance addresses a challenging problem considering the navigation and control of a group of passive agents. To solve this problem, shepherding offers a bio-inspired technique for navigating such a group of agents using external steering agents with appropriately designed movement law. Although most shepherding research is mainly based on the availability of centralized instructions, these assumptions are not realistic enough to solve some emerging application problems. Therefore, this paper presents a decentralized shepherding method where each steering agent makes movements based on its own observation without any inter-agent communication. Our numerical simulations confirm the effectiveness of the proposed method by showing its high success rate and low costs in various placement patterns. These advantages particularly improve with the increase in the number of steering agents. We also confirm the robustness and resilience properties of the proposed method via numerical simulations.
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A noise elimination method based on an improved particle swarm algorithm is applied to direct absorption spectroscopy. The algorithm combines the theory of spectral line shape to calculate a fitness function according to the original spectra. Comparing the particles and the fitness function to calculate the updating direction, and position of particles, the iterative update finally finds the optimal solution. The algorithm is applied to direct absorption spectroscopy to measure methane; compared with the signal without algorithm processing, the signal-to-noise ratio (SNR) is improved by 4.17 times, and the minimum detection limit in the experiment is 15.3 ppb. R2 = 0.9999 is calculated in the calibration experiment, and the error is less than 0.1 ppm in the repeatability experiment of constant methane at 2 ppm concentration.
A collaborative unmanned aerial vehicle (UAV) swarm can support various applications, including aerial surveillance and emergency communication. Owing to the high mobility, limited energy of UAVs, and frequent link breakages, data routing from remote UAVs to a base station may produce retransmissions, loops, energy holes, and long delay. In a UAV swarm network, therefore, relative mobility, path stability, and delay should be jointly taken into consideration to improve routing performance because they are highly coupled with each other. However, they are not properly exploited in the existing literature. To address these issues, in this paper, a Q-learning (QL)-based routing protocol inspired by adaptive flocking control (QRIFC) is proposed. The proposed adaptive flocking control algorithm generates optimal mobility with fairness in travel distance for each UAV to control the optimal node density. It also addresses the trade-off between aerial coverage and quality of service in connectivity by imposing constraints on the minimum separation distance and maximum allowable inter-UAV spacing using two-hop neighbor information. Additionally, it provides a stable link duration (LD) between neighboring UAVs and optimizes the control overhead. Furthermore, QL performs multi-objective optimization by utilizing a new state exploration and exploitation strategy to select an optimal routing path in terms of delay, stable path selection defined by predictive LD, and energy consumption. According to an extensive performance study, the proposed QRIFC outperforms existing routing protocols by 21-40 percent of average end-to-end delay and 9-23 percent of average packet delivery ratio, with less retransmissions.
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The vast majority of work to date concerned with obstacle avoidance for manipulators has dealt with task descriptions in the form ofpick-and-place movements. The added flexibil ity in motion control for manipulators possessing redundant degrees offreedom permits the consideration of obstacle avoidance in the context of a specified end-effector trajectory as the task description. Such a task definition is a more accurate model for such tasks as spray painting or arc weld ing. The approach presented here is to determine the re quired joint angle rates for the manipulator under the con straints of multiple goals, the primary goal described by the specified end-effector trajectory and secondary goals describ ing the obstacle avoidance criteria. The decomposition of the solution into a particular and a homogeneous component effectively illustrates the priority of the multiple goals that is exact end-effector control with redundant degrees of freedom maximizing the distance to obstacles. An efficient numerical implementation of the technique permits sufficiently fast cycle times to deal with dynamic environments.
This paper introduces particle systems--a method for modeling fuzzy objects such as fire, clouds, and water. Particle systems model an object as a cloud of primitive particles that define its volume. Over a period of time, particles are generated into the system, move and change form within the system, and die from the system. The resulting model is able to represent motion, changes of form, and dynamics that are not possible with classical surface-based representations. The particles can easily be motion blurred, and therefore do not exhibit temporal aliasing or strobing. Stochastic processes are used to generate and control the many particles within a particle system. The application of particle systems to the wall of fire element from the Genesis Demo sequence of the film Star Trek H: The Wrath of Khan [10] is presented.
Thousands of birds flying together at high speeds are able to execute abrupt manoeuvres with such precise coordination that some investigators have postulated that `thought transference'1 or electromagnetic communication2 must be taking place. Recently, Davis proposed that coordination is achieved by a `threshold' number of birds executing `preliminary movements' which signal to the flock that a turn is imminent3. Here I show by analysis of film of dunlin (Calidris alpina) flocks that a single bird may initiate a manoeuvre which spreads through the flock in a wave. The propagation of this `manoeuvre wave' begins relatively slowly but reaches mean speeds three times higher than would be possible if birds were simply reacting to their immediate neighbours. These propagation speeds appear to be achieved in much the same way as they are in a human chorus line: individuals observe the approaching manoeuvre wave and time their own execution to coincide with its arrival.