THE ARC-SIZE ILLUSION AS APPLIED TO PLANAR DISPLAYS

OF AIRCRAFT TRAFFIC INFORAMTION

Doreen Comerford, NASA Ames Research Center/Kansas State University

John Uhlarik, Kansas State University

Shorter arcs are perceived as having less curvature (i.e., they are
"perceptually flattened"), and this phenomenon has been labeled the
*Arc-size Illusion*. This illusion was explored as it relates to planar
displays of traffic information. Such displays often represent intent
and/or history information with lines. Participants were presented with
aircraft traveling on curvilinear paths, and their estimations of future
aircraft location were examined. There were three major findings from
the study. *First*, the data suggest that "perceptual flattening" and
the Arc-size Illusion do indeed affect predictive judgments about
aircraft on curvilinear paths. *Second*, and most surprising, the data
suggest that increasing the size of the arc with a history line does NOT
lessen "perceptual flattening." *Third*, the data suggest that
"perceptual flattening" is especially pronounced when predictor length
is relatively short and the judgment location is relatively far.
However, the effect of the predictor length essentially disappears when
the judgment location is at a

distance that is proportional to the predictor length.

**INTRODUCTION**

The *Arc-size Illusion* is a perceptual phenomenon in which smaller arcs
are perceived as flatter than longer arcs that have the same amount of
curvature. For example, each of the three arcs in Figure 1 originate
from a circle of the same radius, but arcs of smaller degrees appear
flatter than arcs of larger degrees (e.g., Arc C appears flatter than
Arc A and Arc B, or put another way, Arc C appears as if it originates
from a circle with a larger radius).

**Figure 1. An illustration of the Arc-size Illusion**

Virsu conducted a total of three studies in which he introduced the
illusion and examined various conditions as they relate to the illusion.
In the *first* study (Virsu, 1971a), he found that arcs of lesser
degrees were perceived as flatter, despite the fact that they originated
from circles of the same radius. In addition, Virsu found that, at
greater judgment distances, the errors associated with shorter arcs were
especially pronounced. In a *second* study, Virsu (1971b) used the
forced choice method. Again, the Arc-size Illusion was demonstrated. The
smallest arc size (9 deg) yielded a 35.2% difference between the point
of subjective equality and the true radius size, while the largest arc
size (342 deg) yielded a difference of only 3%. In a *final* study,
Virsu and Weintraub (1971) again found that shorter arcs are perceived
as flatter than longer arcs. In addition, when the arcs originated from
circles of smaller radii, the errors associated with shorter arcs were
especially pronounced.

While this illusion is interesting at the level of basic research, it
also may have an important impact in the applied domain. Planar displays
of traffic information often represent intent and/or history information
with lines. For example, intent information is available on the cockpit
display of traffic information (CDTI) (c.f., Johnson, Battiste, Deizell,
Holland, Belcher, & Jordan, 1997; Johnson, Battiste, & Holland Bochow,
1999), and history information is available on the air traffic
controller's radarscope. Generally speaking, *predictor lines* are lines
that represent a "best guess" at the future location of a vehicle or
object, and *history lines* are lines that represent the previous
location of the vehicle or object. When aircraft have been or are
currently on a *curvi*linear path, judgments about future aircraft
location may be affected by the Arc-size Illusion. An example of this
potential problem is illustrated in Figure 2.

**Aircraft A Aircraft B**

**Figure 2. Predictor lengths for a constant time period (e.g., five
minutes), a constant rate of turn, but a variation in aircraft
velocity.**

Aircraft A is traveling slower than Aircraft B. Therefore, in the same
amount of time, Aircraft A will travel through less airspace than
Aircraft B. Because predictor lines usually represent a constant time
period, the predictor line for Aircraft A is shorter. Both aircraft in
Figure 2 have the same rate of turn. However, due to the Arc-size
Illusion, the graphical representation of the prediction *appears* to
suggest that Aircraft A will follow a straighter trajectory than
Aircraft B.

The present experiment was aimed at exploring the extent to which the
Arc-size Illusion may affect judgments on instruments such as the CDTI.
Another goal was to explore whether increasing the length of an arc, by
the addition of a history line, serves to decrease illusory
"flattening." In the context of traffic displays, the history line
certainly is relevant information. If it serves to offset the illusion,
it is a convenient and reasonable method for doing so. Finally, this
experiment allowed for an investigation of other factors (i.e., circle
size and judgment location) that might affect the perceived curvature of
predictor and history lines.

###### EXPERIMENT ONE

####### Method

*Independent variables.* Six levels of *Predictor* *Length* (25, 32.5,
40, 47.5, 55, and 62.5 deg) were combined with two levels of *History
Information* (Absent and Present) to create twelve conditions. For
illustrative purposes, the stimuli from four of these conditions are
presented in Figure 3.

**Figure 3. Illustration of the stimuli in the largest and smallest
Predictor Length conditions as well as an illustration of the History
Information conditions (Absent and Present). Stimuli are not drawn to
scale.**

Two additional independent variables were included in the study. The
physical size of an arc depends not only on its degrees of arc, but it
also depends on the size of the circle from which the arc originates.
Therefore, *Circle Size* was included as a third independent variable,
and its two levels were Large (radius of 19 mm) and Small (14 mm).

The fourth, and final, independent variable was *Judgment Location*. For
illustrative purposes, Figure 4 presents the three levels of Judgment
Location (Constant Proportion, Constant Distance, and Constant Location)
for two of the six levels of Predictor Length.

The three levels of Judgment Location represent different locations of
the response line. The Constant Proportion condition was created by
taking the degrees of the predictor and dividing them in half. The
resulting degrees of arc were then used to determine the distance of the
response line from the end of the predictor. For example, in the
Constant Proportion condition, the response line was placed 12.5 degrees
from the end of a 25-degree predictor. This condition represents
judgments over a constant time frame. In practical terms, an aircraft
with a relatively long predictor is traveling faster than an aircraft
with a relatively short predictor. Therefore, given a constant amount of
time, judgments for the faster moving aircraft will be relatively
farther from the end of the predictor than judgments for the slower
moving aircraft. In the Constant Distance condition, the response line
was always located 30 deg from the end of the predictor. Finally, unlike
the former two conditions, the position of the response line did not
vary in the Constant Location condition.

The three conditions of Judgment Location also can be understood as
variations in the amount of empty space between the end of the predictor
and the response line. In the Constant Proportion and Constant Location
conditions, the empty space changes relative to degrees of arc.
Specifically, when the arc is small (as in the 25 deg condition), the
response line is relatively close to the predictor in the Constant
Proportion condition and relatively far in the Constant Location
condition. However, when the arc is large (as in the 62.5 deg
condition), the relation is exactly the opposite**;** the response line
is relatively far from the predictor in the Constant Proportion
condition and relatively close in the Constant Location condition. In
the Constant Distance condition, the space between the predictor and the
response line is simply always the same.

**Figure 4. Illustration of stimuli in the three Judgment Location
conditions, with two levels of Predictor Length shown for illustrative
purposes. Stimuli are not drawn to scale.**

*Experimental Design.* The experimental design was a 6 X 2 X 2 X 3
(Predictor Length X History Information X Circle Size X Judgment
Location). This design resulted in a total of 72 stimuli, and because
all independent variables were within-participants variables, each
participant responded to all 72 stimuli.

######### *Participants.* Eight volunteers participated in the experiment. Ages of the participants ranged from approximately 20 to 40 years, and none of the participants had a background in aviation.

*Procedure.* Each participant was given a paper-and-pencil packet
containing the stimuli. Each stimulus appeared on a separate sheet of
paper and was positioned in the center of the sheet. The participants
were told to imagine being an air traffic controller or a pilot
monitoring traffic. The concepts of both a predictor line and a history
line, as they relate to aviation, were explained to the participants
(i.e., lines that represent a "best guess" at the future location of an
aircraft and lines that represent the previous location of an aircraft,
respectively). They were asked to determine the location at which each
aircraft would intersect with the dotted line and to place a hash mark
on a dot or between two dots to signify their judgment. The experimenter
explained that a pilot or an air traffic controller sometimes must make
quick, intuitive judgments regarding the future location of aircraft.
Therefore, they were asked not to use their pencil for any purposes
other than drawing the hash mark (i.e., they were asked not to "fill in"
the rest of the circle with their pencil). The participants viewed only
one stimulus at a time, and the experimenter used a sample stimulus to
insure each participant understood the task.

*Dependent Variable.* The data were coded in terms of *Error*.
Specifically, the veridical radius was subtracted from the perceived
radius. Therefore, a positive number represents "flattening, " zero
suggests the curvature was perceived perfectly, and a negative number
suggests that the arc was perceived as having too much curvature (i.e.,
the opposite of "flattening").

####### Results

In terms of overall performance, the radius of the circle from which the
arcs originated was overestimated by 2.15 mm, and this grand mean was
significantly different from zero, [F]{.underline}(1, 7) = 46.02,
[p]{.underline} \< .001. Therefore, arcs were perceived as too flat in
general. Figure 5 illustrates that shorter arcs were perceived as
flatter than longer arcs, [F]{.underline}(5, 35) = 8.55, [p]{.underline}
\< .001. This finding suggests that the Arc-size Illusion indeed affects
tasks like those found in the aviation domain (e.g., the pilot using the
CDTI or the air traffic controller).

![](media/image2.wmf)

**Figure 5. Mean Error as a function of Predictor Length. Intervals
around each mean are confidence intervals of 95%.**

The main effect of History Information was not significant,
[F]{.underline}(1, 7) = 0.04, [p]{.underline} \> .05. The Predictor
Length X History Information interaction was significant,
[F]{.underline}(5, 35) = 3.62, [p]{.underline} \< .01. This interaction
is illustrated in Figure 6.

Four specific comparisons are especially interesting in Figure 6.
*First*, in the 25 deg Predictor Length condition, the History
Information Absent and History Information Present conditions are *not*
significantly different. This lack of significant difference is
surprising, since the total degrees of arc in the History Information
Present condition are twice that of the History Information Absent
condition. *Second*, the condition in which the predictor was 25 deg and
the history line was present is significantly different than the
condition in which the predictor was 47.5 deg and the history line was
absent. This significant difference is surprising, since the total
degrees of arc are almost identical (i.e., 50 and 47.5 deg of arc
respectively). *Third*, in the 32.5 deg Predictor Length condition, the
History Information Absent and History Information Present conditions
are *not* significantly different. Like the first observation, this lack
of significant difference is surprising, since the degrees of arc in the
History Information Present condition are twice that of the History
Information Absent condition. *Fourth*, when the predictor was 32.5 deg
and the history line was present, responses were significantly different
from the condition in which the predictor was 62.5 deg and the history
line was absent. Similar to the second observation, this significant
difference is surprising, since the total degrees of arc are almost
identical (i.e., 65 and 62.5 deg of arc respectively). Taken together,
these four observations suggest that participants made no use of the
history information.

![](media/image4.wmf)

**Figure 6. Mean Error as a function of Predictor Length and History
Information. Intervals around each mean are confidence intervals of
95%.**

The mean error was 2.41 mm for small circles and 1.90 mm for large
circles. This main effect was not significant, [F]{.underline}(1, 7) =
3.68, [p]{.underline} \> .05. However, the direction of the means is
consistent with the findings of Virsu and Weintraub (1971). The main
effect was significant when responses were coded in terms of relative
error, [F]{.underline}(1, 7) = 12.34, [p]{.underline} \< .01. Relative
Error was calculated by dividing the raw error by the actual radius of
the circle from which the arc originated. In other words, the Relative
Error measure reflects the fact that the same amount of raw error is
*relatively* greater for a smaller circle size. Of course, in a
practical sense, raw error is probably of most interest. Virsu and
Weintraub did find a difference between circle sizes when they examined
raw error. It is possible that the present study examined circle sizes
that were too similar to yield differences when raw error was examined.

The main effect of Judgment Location was significant, [F]{.underline}(2,
14) = 17.80, [p]{.underline} \< .001. However, the Predictor Length X
Judgment Location interaction also was significant, [F]{.underline}(10,
70) = 6.01, [p]{.underline} \< .001. This interaction is illustrated in
Figure 7.

![](media/image5.wmf)

**Figure 7. Mean Error as a function of Predictor Length and Judgment
Location. Intervals around each mean are confidence intervals of 95%.**

As illustrated in Figure 7, the Constant Distance and Constant Location
conditions were significantly different from the Constant Proportion
condition when the predictor length was relatively small (i.e., 25 or
32.5 deg). A more intuitive description of these results is to say that
only the Constant Distance and Constant Location conditions demonstrated
the Arc-size illusion. While the Constant Proportion condition did yield
consistent "flattening" overall, this condition did not demonstrate the
Arc-size Illusion.

This interaction is explained easily when the stimuli are considered. In
the Constant Proportion condition, the response line was relatively
close to the small predictors. By not requiring participants to
extrapolate over great distances, the inflated error normally associated
with smaller degrees of arc was offset. Although a different task was
required of participants, Virsu (1971a) also found that arcs of
relatively few degrees resulted in more "flattening" as judgments were
moved to relatively farther locations.

###### EXPERIMENT TWO

####### Method

Experiment Two was identical to Experiment One with only two exceptions.
The *first*, and most important, difference was that the task in
Experiment Two involved two aircraft. Both aircraft in each pair
originated from a circle of the same radius and were identical in
degrees of arc. Figure 8 provides an example of the stimuli used in
Experiment Two.

**Figure 8. Example of the stimuli used in Experiment Two. Stimuli are
not drawn to scale.**

Participants received the same instructions as those described for
Experiment One. The only difference was that they were asked to
determine the location at which the two aircraft would intersect the
dotted line (i.e., they were asked to determine where these two aircraft
would collide).

The *second* difference was that judgment location was not varied in
Experiment Two. The response line remained in the same location (i.e.,
as in the Constant Location condition of Experiment One). All other
variables from Experiment One were examined in Experiment Two.
Therefore, the design was a 6 X 2 X 2 (Length of Predictor X History X
Circle Size).

####### 

####### Results

In terms of overall performance, the radius of the circle from which the
arcs originated was overestimated by 2.58 mm, and this grand mean was
significantly different from zero, [F]{.underline}(1, 7) = 20.59,
[p]{.underline} \< .005. Therefore, as in Experiment One, arcs were
perceived as too flat in general. In Experiment 2, shorter arcs were
perceived as flatter than longer arcs, [F]{.underline}(5, 35) = 12.04,
[p]{.underline} \< .001. This finding again suggests that the Arc-size
Illusion indeed affects tasks like those found in the aviation domain
(e.g., the pilot using the CDTI or the air traffic controller).

Again, the main effect of History Information was not significant,
[F]{.underline}(1, 7) = 0.49, [p]{.underline} \> .05. In Experiment Two,
the Predictor Length X History Information interaction was not
significant, [F]{.underline}(5, 35) = 1.50, [p]{.underline} \> .05.
However, four specific comparisons again are interesting. *First*, the
History Information Absent and History Information Present conditions
were *not* significantly different in the 25 deg Predictor Length
condition. *Second*, the condition in which the predictor was 25 deg and
the history line was present was significantly different than the
condition in which the predictor was 47.5 deg and the history line was
absent. *Third*, the History Information Absent and History Information
Present conditions were *not* significantly different in the 32.5 deg
Predictor Length condition. *Fourth*, when the predictor was 32.5 deg
and the history line was present, responses were significantly different
from the condition in which the predictor was 62.5 deg and the history
line was absent. These four relations are consistent with those found in
Experiment One, and they suggest that participants were not making use
of history information.

The mean error was 2.55 mm for small circles and 2.63 mm for large
circles. This main effect was not significant, [F]{.underline}(1, 7) =
0.06, [p]{.underline} \> .05. Unlike Experiment One, the differences in
these means were not in the expected direction.

###### SUMMARY AND DISCUSSION

There were three major findings in the current study. *First*, the data
suggest that curvilinear paths are perceived as too flat and that the
Arc-size Illusion has an important impact on judgments of traffic
trajectories. These results are consistent with the research performed
by Virsu. However, in all three of Virsu's studies (Virsu,1971a;
Virsu,1971b; Virsu and Weintraub, 1971), participants were provided with
no context and merely were asked to make judgments about arcs. The
current study asked participants to imagine they were performing a
real-world task, and the stimuli differed in that they contained a
triangle representing an aircraft. Therefore, these results are
important, because they suggest that this illusion warrants examination
beyond pure, academic curiosity.

*Second*, and most surprising, the data suggest that increasing the size
of the arc with a history line does NOT lessen the effects of the
Arc-size Illusion. This finding is contrary to Virsu's findings (Virsu,
1971a; Virsu, 1971b; Virsu & Weintraub, 1971), in which the curvature of
generic arcs was perceived more accurately by merely increasing the
degrees of arc. However, the current findings are consistent with the
findings of Palmer, Jago, Baty, and O'Connor (1980). They performed a
series of studies in which they examined history and predictor lines.
Palmer et al. had pilots view a CDTI and asked them to determine if an
intruder would pass in front of or behind ownship. Pilots made these
judgments when only the current positions of the aircraft were presented
or when history lines were presented along with the current position.
The addition of a history line did not yield a significant increase in
the accuracy of pilots' judgments. Palmer et al. also asked a group of
pilots to report the manner in which they would develop a CDTI. Four of
six pilots suggested that they would not like a history line to appear
on a CDTI. These pilots felt the history information would be redundant
with the turn rate information provided by curvilinear predictors. The
two pilots who indicated they would like a history line on a CDTI
suggested that they would like either no predictor line or a predictor
that did not include turn rate information. The Palmer et al. research
and the current research suggest that an increase in arc size does not
necessarily improve judgments. The particular context and/or particular
stimulus features appear to be important.

*Third*, the data suggest that "flattening" is especially pronounced
when the judgment location is relatively far and the predictor length is
relatively short (e.g., the Constant Location and Constant Distance
conditions for the 25 deg Predictor Length produced much "flattening").
However, the effect of the predictor length essentially disappears when
the judgment location is at a distance that is proportional to the
predictor length. Because traffic avoidance judgments typically involve
judgments over a constant time period (e.g., "Where will my aircraft be
in relation to Aircraft X in 5 minutes?"), this finding suggests that,
in some cases, predictors for slower aircraft will cause no more
problems than predictors for quicker aircraft.

Research is currently in progress, in an attempt to replicate these
findings. The research in progress explores a wider range of predictor
lengths than the present study. The aircraft icon also is being
manipulated (i.e., chevron, circle, and no icon conditions are being
examined). Such a manipulation will assist in determining if and how the
icon might be responsible for participants' ignoring the history
information. Finally, pulsing is being explored. Pulsing is currently
utilized on the NASA Ames CDTI (c.f., Johnson, Battiste, & Holland
Bochow, 1999). When pulsing is activated on the display, portions of the
predictor line are bolded in succession. This bolding provides a sense
that an object is moving along the predictor line and creates an
illusion much like the phi phenomenon (i.e., apparent movement when two
or more lights are flashed in succession). Virsu (1971b) suggests that
the Arc-size Illusion may be a result of the human tendency to make
saccadic, rectilinear eye movements (i.e., staircase-like eye
movements). Because of this tendency, Virsu suggests that smaller arcs
provide less corrective feedback to the brain, and as a result, are
perceived as flatter. In other words, smaller arcs presumably yield
fewer staircase-like eye movements. Therefore, the brain may interpret a
smaller arc as having less curvature. Only in the presence of moving
objects are smooth, pursuit eye movements possible (Sekuler & Blake,
1990). Such movements may serve to decrease "flattening" in general and
may also lessen the Arc-size Illusion. A moving pulse should encourage
smooth, pursuit eye movements and serve to lessen "perceptual
flattening" as well as the Arc-size Illusion.

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Johnson, W., Battiste, V. Deizell, S., Holland, S., Belcher, S., &
Jordan, K. (1997). *Development and Demonstration of a Prototype Free
Flight Cockpit Display of Traffic Information.* (SAE Technical Paper No.
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Palmer, E., Jago, S.J., Baty, D.L., & O'Connor, S.L (1980). Perception
of horizontal aircraft separation on a Cockpit Display of Traffic
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Sekuler, R. & Blake, R. (1990). *Perception*. New York: McGraw-Hill

Virsu, V. (1971a). Tendencies to eye movement, and misperception of
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Virsu, V. (1971b). Underestimation of curvature and task dependence in
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Virsu, V., & Weintraub, D.J. (1971). Perceived curvature of arcs and dot
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