Session: Human Performance Gives Us Fitts'
CHI 2012, May 5–10, 2012, Austin, Texas, USA
Extending Fitts’ Law to Account for the Effects of
Movement Direction on 2D Pointing
Xinyong Zhang♦ , Hongbin Zha† , Wenxin Feng♦
School of Information, Renmin University of China, Beijing, China
†
Key Lab of Machine Perception, MOE, Peking University, Beijing, China
x.y.zhang@ruc.edu.cn, zha@cis.pku.edu.cn, fengwenxin@gmail.com
♦
Nevertheless, pointing and selecting visual targets to activate
the corresponding commands is still the most dominant task
in current graphic user interfaces. Therefore, it is very important to deeply understand the user’s capabilities of pointing. Among the published models in the literature for this
daily task, the most prevalent one is Fitts’ law [10]. Regardless of its original expression, the widely-applied “standard”
form of Fitts’ law for movement time (M T ) predictions is
as follows [17, 21]:
ABSTRACT
Fitts’ law is the most widely applied model in the field of
HCI. However, this model and its existing extensions are
still limited for 2D pointing task especially when the effects
of movement direction (θ) remain in the task. In this paper,
we employ the concept of projection to account for the effects of target width (W ) and height (H) on movement time
so that we seamlessly integrate the four factors, i.e. θ, amplitude (A), W and H, into the new extension of Fitts’ law,
which can uncover not only the periodicity of the asymmetrical impacts of W and H with the variation of θ but also
their interrelation. Carrying out two experiments, we verify
that the vertical projection of W and the horizontal projection of H in the line of movement direction can be viewed
as the determinants of movement time. Finally, we offer recommendations for 2D pointing experiments and discuss the
implications for interface designs.
M T = a + b log2 (A/W + 1)
(1)
where a and b are two regression coefficients, and W and
A denote target width and movement distance of the cursor, respectively. The logarithmic term is defined as the index of difficulty (ID) of pointing tasks. This equation expresses a linear relationship between ID and M T that the
user takes when moving the cursor to acquire the desired
target as quickly and accurately as possible.
Author Keywords
With respect to the two-dimensional (2D) targets in conventional user interfaces, target position (including direction and distance) and size (including width and height) are
all relevant to the time of target acquisition [13, 23, 6, 22].
However, Fitts’ law and its existing extensions [16, 2, 4, 26]
do not take account of these factors at the same time and also
do not reveal the potential relationship among them. In this
paper, therefore, we endeavor to provide a more appropriate
extension of Fitts’ law. We simply view the projections of
target width and height (H) in the line of movement direction as the essentials to determine M T , and we find that the
different impacts of W and H on M T will periodically vary
with movement direction (θ). This periodicity can be expressed using a single weighted parameter in the new extension of Fitts’ law, directly exposing the impact of movement
direction on both of W and H.
Fitts’ law; two-dimensional pointing; movement direction.
ACM Classification Keywords
H.5.2. [Information interfaces and presentation]: User interfaces - evaluation, theory and methods; H.1.2 [Models and
principles]: User/machine systems - human factors
General Terms
Experimentation; Human Factors; Theory.
INTRODUCTION
In the field of human-computer interaction (HCI), predictive
performance models often serve as theoretical tools to evaluate the user’s capabilities of interacting with computers. On
the one hand, performance models can offer substantial evidence to indicate whether or not a new technique or user
interface is effective and efficient from a theoretical perspective, and, on the other hand, they can also provide inspiration
for the designs of HCI. Up to now, a number of quantitative
models have been proposed for different tasks, such as steering task [1], peephole pointing [8], menu selection [3] and
navigation in scrolling and hierarchical lists [9].
RELATED WORK
The validity and reliability of Fitts’ law for pointing task
were confirmed in many different situations. Recently, for
example, it was reported that Fitts’ law effectively governed
the performance under the conditions that the visual feedback was restricted [25] or that the motor movements included translational and rotational components [18].
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CHI’12, May 5-10, 2012, Austin, Texas, USA.
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In the field of HCI, Fitts’ law is the defacto standard of user
performance evaluation [21]. It can also serve as a theoretical foundation to deduce new models for different tasks,
such as steering [1] and peephole pointing [8]. More importantly, Fitts’ law itself can be extended for different sit-
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Session: Human Performance Gives Us Fitts'
CHI 2012, May 5–10, 2012, Austin, Texas, USA
2D pointing tasks. However, they did not evaluated the effectiveness of IDaz for the tasks performed in various movement directions but only in horizontal directions. Therefore,
the implication was uncertain in different directions.
uations, such as pointing with the need of manual obstacle
avoidance [14] and pointing at targets accompanied with distractors [5]. Especially, there were a number of studies on
extending Fitts’ law for 2D pointing [16, 19, 2, 4] and on
modeling it using different concepts [11, 26]. Next, we will
give a review on those studies, which are highly related to
our current work, as well as the investigations about the effects of movement direction on user performance.
The ISO9241-9 standard recommends a multidirectional tapping task to neutralize the effects of different directions, but
lots of studies (see the reference [4] for a brief review) took
account of this factor when evaluating the user’s capabilities of pointing. In this connection, Appert et al. treated
the factor of movement direction as an explicit variable in
their work [4]. According to the general form of IDaz , they
simply set ω = η = r = 1 due to the acceptable correlation coefficients (R2 > 0.89) that Accot and Zhai’s work presented
under this settings [4]. At the same time, they introduced
a special term related to movement direction to capture the
dominant effect of the smaller of W and H. They proposed
a new index definition as IDφ expresses as follows:
Taking Account of Target Width and Height Together
MacKenzie and Buxton made the first effort to seek more
accurate ID for 2D pointing tasks [16]. They tested each of
the terms: min(W, H), W + H, W × H and W ′ 1 , in the
place of W in Fitts’ law, to capture the effect of H. They examined the accuracies of the redefined indexes of difficulty
under different conditions of movement directions, including 0◦ , 45◦ and 90◦ . They found that both min(W, H) and
W ′ resulted in better data fitting than the others. W ′ appears
to be theoretically appealing, but it could lead to unreasonable predictions of M T in some special cases as pointed out
by themselves. In order to properly take account of the interaction effect between the two factors W and H, Accot and
Zhai employed the methodology of weighted ℓr -norm to extend the standard Fitts ID [2] and got the general definition
(denoted using IDaz in this paper) as follows:
!
r 1/r
r
A
A
IDaz = log2
+η
ω
+1
(2)
W
H
A
A
A
+
+ 0.6cos(φ)
+1
W
H
min(W, H)
(4)
where φ represents the absolute value of the angle between
the approach line and the vertical axis. As their experimental results indicated, IDφ appeared to be more accurate to
express the difficulties of 2D pointing than IDaz .
IDφ = log2
As can be seen from the related work above, it is inevitable
that the more the factors are taken into consideration, the
performance model will be more complex. Therefore, there
is another kind of efforts to address the issue of modeling 2D
hand pointing. Grossman and Balakrishnan [11] proposed
a generalized index of difficulty (IDP r ) only based on the
probabilities that targets are rapidly hit by users. They constructed a mapping relationship between the probabilities of
hitting targets and the values of IDP r . Intuitively, the difficulty of pointing will decrease with the increase of the probability of hitting. According to the 2D spread of hits [20], it
is possible to calculate the probabilities and then the values
of IDP r of acquiring targets in different directions as well as
targets with arbitrary shapes. The essence of this approach
is to calculate the probabilities of hitting targets, without the
need to explicitly interpret the impacts of different factors.
where ω and η denote two positive weights, and as a real
number of norm order, r ≥ 1. This ID definition has five
properties describing the effects of both W and H:
• Scale Independency. When the sizes of A, W and H are
changed at the same rate, theoretically, the values of M T
will be identical due to the unchanged IDaz .
• Limit Tasks. The 2D index of difficulty can converge to
the one-dimensional (1D) form of Fitts ID when one of
the variables W and H is increasingly close to infinity.
• Dominance Effect. When either W or H is gradually
getting small, it will dominate the difficulties, and consequently weaken the impact of the other.
• Duality of H and W. The definition of IDaz includes both
of W and H, indicating their similar contributions to M T
in terms of functional relation.
• Continuity. The factors W and H do not stepwise or segmentally but continuously affect M T .
More recently, Yang and Xu employed a novel concept to
interpret 2D pointing task [26]. They converted 2D pointing
task from a complex state to a simple one, which could be
described using fewer parameters without the loss of information. Concretely, for example, it is possible to describe a
directional 2D pointing task without including a parameter
of angle. Thus, in principle, the corresponding definition of
ID could be devoid of the component related to movement
direction. Regarding the different states of 2D pointing task,
Yang and Xu derived several IDs. These indexes were defined as the logarithm of the inverse of the probability that
the target was successfully hit. They picked out the one as
expressed in Equation 5 (IDconf ) as the superior due to its
accuracy and simplicity.
In order to remove unnecessary parameters, Accot and Zhai
examined the accuracies of Equation 2 under twelve different parameterization conditions. They pointed out that when
r = 2 and ω = 1, the Euclidean form of IDaz , as the following Equation 3 expresses, was the most appropriate one.
p
IDaz = log2
(A/W )2 + η(A/H)2 + 1
(3)
They reported that there was a strong linear correlation between IDaz and M T (R2 > 0.94) when 1/7 ≤ η ≤ 1/3.
The values of η implied that W had greater impact than H on
1
The symbol W ′ represents the width along the line where the cursor approaches the target.
IDconf = log2
3186
1p
(A/W )2 + (A/H)2 + 1
2
(5)
Session: Human Performance Gives Us Fitts'
CHI 2012, May 5–10, 2012, Austin, Texas, USA
The methodology used by Yang and Xu does not need to
introduce an arbitrary term into the definition of ID. Thus,
the risk of overfitting raised by such an unexplainable term,
like that in IDφ , could be reduced.
(a)
(b)
Wθ
Wθ
Hθ
Investigating the Effects of Movement Direction
Fitts’ law does not take the factor of movement direction (θ)
into account. In traditional 1D Fitts experiments, the tasks
are reciprocally performed in horizontal directions, while
ISO9241-9 recommends that pointing tasks should be performed in different directions. However, there is a common
view that the difficulties of acquiring a given target from different directions are likely different. A number of studies
[7, 16, 23, 22, 11, 4] have investigated the effects of movement direction on user performance or considered this factor in their experimental designs. For example, MacKenzie
and Buxton took account of movement direction in their 2D
pointing experiment and reported that target acquisition time
in horizontal and vertical axes were almost the same, faster
than that in diagonal axis [16]. Whisenand and Emurian
systematically investigated the effects of movement directions in every 45◦ angle from 0◦ to 360◦ , with the consideration of target shape (circular vs. square) and task type
(drag-drop vs. point-select) [23]. They reported that it was
more efficient to point at horizontal targets than at vertical
targets. Using circular targets and different control-display
gains, Thompson et al. tested the validity of Fitts’ law in
eight different movement directions as used in Whisenand
and Emurian’s work. They also observed the best performance in horizontal pointing and the worse in vertical pointing in general [22]. Grossman and Balakrishnan further confirmed this point when they employed five movement directions from 0◦ to 90◦ at an interval of 22.5◦ in their work
[11]. They found that when one of the two target dimensions (W and H) was parallel to movement direction, it was
more critical than the other to determine M T , and that W
and H would equally affect M T in the direction of 45◦ .
W
H
θ
W
Figure 1. Projections of target width and/or height in the line of movement direction in (a) Fitts tasks and (b) general 2D pointing tasks.
target based on a uniform distribution of hits when deducing
Equation 5. This pattern of hand pointing is inconsistent
with the findings in other work [20, 11]. Furthermore, if the
target only rotates ±90◦ at a given position (i.e. W and H
are replaced by each other), the corresponding M T should
almost be the same since the value of IDconf is constant
according to its definition. Unfortunately, as reported in the
studies [2, 11], the performance differences were notable in
this situations. That is to say, IDconf is unable to interpret
the usual cases of target transposition, which can cause significant differences of M T . Actually, IDconf in form is the
special case of Equation 2 when ω = η = 0.25 and r = 2.
Mathematically, the extension of Fitts’ law made by Accot
and Zhai for 2D pointing was sufficiently completed.
Recall the traditional 1D Fitts task, which is reciprocally performed in horizontal directions. As Figure 1a depicts, we
use Wθ to denote the vertical projection of target width W
in movement direction. When θ = 0◦ or 180◦ , W = Wθ .
Then, the traditional Fitts ID in Equation 1 can be replaced
with the form log2 (A/Wθ + 1). This means that the vertical
projection Wθ can be viewed as the real determinant capturing the effect of target width. Extending this point of view
to the situation of 2D pointing, we propose
Hypothesis 1: The vertical projection of W and the horizontal projection of H in movement direction are two essential target factors determining the difficulty of pointing.
Theoretically, Grossman and Balakrishnan’s model can accommodate the effect of movement direction (θ), but this
factor is “invisible” without directly exposing how user performance can be affected by θ. At the same time, their model
is difficult to be used to explicitly inform designers how user
interfaces can be improved. If there is a hidden truth about
θ in 2D pointing, how does it systematically affect M T ?
A general expectation is that this factor should be harmoniously integrated into the definition of ID unlike the form
of Equation 4, which results in an unreasonable implication
that only one of the two factors W and H could determine
the effect of movement direction. Equation 4 also implies
that IDφ is not a continuous index, i.e. IDφ does not have
the property of continuity. Next, we will propose a new extension of Fitts’ law to uncover the truth of 2D pointing, with
the advantage that it can be conveniently applied.
As shown in Figure 1b, the projections of target width and
height, Wθ and Hθ , in general can be given by
Wθ = W/ cos(θ), Hθ = H/ sin(θ)
(6)
According to Accot and Zhai’s work as well as that of Yang
and Xu, the ℓ2 -norm extension of Fitts’ law appears to be the
pretty-much optimal form for 2D pointing. Then, substituting Wθ and Hθ for W and H, we can directly define:
p
IDθ = log2 ( (A/Wθ )2 + (A/Hθ )2 + 1)
(7)
Replacing Wθ as well as Hθ based on Equation 6, we have:
!
r
A 2
A 2
2
2
IDθ = log2
cos (θ)( ) + sin (θ)( ) + 1 (8)
W
H
We can let ω = cos2 (θ) and η = sin2 (θ) for 2D pointing
task, leading to the constraint ω + η = 1. Now, the two
weighted parameters built an internal relation between the
two terms A/W and A/H in the binomial. This is unlike
the situation of Equation 2, where ω and η are independent
from each other. In addition, the mutually constrained two
parameters η and ω let IDθ has a new property of backward
A NEW EXTENSION OF FITTS’ LAW
We can find a common theme in the prior literature [2, 26]
that the two terms A/W and A/H are essential to the determination of ID for 2D hand pointing tasks. The effects of
W and H could be accumulated to determine ID so as to
ultimately determine M T . With respect to the work of Yang
and Xu, we find that they calculated the probability of hitting
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Session: Human Performance Gives Us Fitts'
CHI 2012, May 5–10, 2012, Austin, Texas, USA
Under the same conditions (A, W and H), the difficulties of
pointing task in different directions are not same, since the
freedoms of hand moving on a given plane, such as the horizontal desktop, range in different directions. Nevertheless,
the freedoms of the hand as well as the movement difficulties should be equivalent in any pair of reciprocal movement
directions. The point is supported by the periodicity of ω or
η (i.e. π) as indicated in Equation 9.
compatibility (as explained later in this section). Equation
8 expresses the ideal situation of projection. However, this
equation implies that M T would be relevant to only one of
the target dimensions in horizontal or vertical movement directions. This point is not the case reported by different studies [2, 26]. A reasonable explanation is that there should be a
constant component in the trigonometric functional relation
between θ and ω or η. Thus, we further propose
Hypothesis 2: There should be simple linear transformation in the trigonometric functional relations to ensure that
none of the target dimensions would be excluded to determine M T in any movement direction.
Therefore, we maintain the meanings of ω unchanged when
considering the following form of IDθ to study whether or
not there is another norm order that, mathematically, can result in better data fitting for 2D pointing.
!
1/r
A r
A r
IDθ = log2
ω( ) + (1 − ω)( )
+1
(11)
W
H
That is to say, the trigonometric functions that express the
weighted parameters ω and η should be generalized using
linear transformations. Therefore, we have:
ω = c1 + c0 cos2 (θ), η = c2 + c0 sin2 (θ)
(9)
Equation 11 in form appears to be the result of subjecting
Equation 2 to an additional constraint ω + η = 1. But now,
we have endowed ω with the ability to access the effects
of movement direction. When the norm order r is close to
infinity ∞, IDθ can be expressed as follows:
A
A
+1
(12)
IDθ = log2 max ω , (1 − ω)
W
H
where c0 , c1 and c2 ≥ 0. To maintain the backward compatibility, we still let ω + η = 1. So, we can extend Fitts ID
to express the effects of movement direction on 2D target
acquisition as follows:
!
r
A 2
A 2
IDθ = log2
ω( ) + (1 − ω)( ) + 1
(10)
W
H
This form means that W and H are mutually exclusive to determine IDθ , like the use of min(W, H) in MacKenzie and
Buxton’s work [16]. The purpose of introducing Equations
11 and 12 is to explore how the accuracy of IDθ depends on
r and ω via curve fitting so as to indicate that ℓ2 -norm extension is the most appropriate choice for 2D pointing. Next,
we present the results of two experiments to verify our hypotheses, especially the satisfying accuracy of Equation 10
under each of the twelve different θ conditions we tested.
where ω = c1 + c0 cos2 (θ) and 0 < ω < 1. In practice,
the estimate of ω rather than c0 or c1 is the major concern.
ω in form is a weighted parameter, but essentially it has inherent connection to movement direction, unlike the counterpart in Equation 2. The two components of ω imply that
not only target dimensions themselves but also their projections are the determinants of IDθ . We can look Equation
8 as a special case of Equation 10. Besides the aforementioned properties of IDaz , Equation 10 further emphasizes
the following points:
EXPERIMENT 1: 2D POINTING TASK WITH DIFFERENT
MOVEMENT DISTANCES
• Periodicity and Symmetry of movement direction’s effects
on both of H and W . The factor θ does not affect one of
W and H selectively, like Equation 4, but both of them simultaneously. More importantly, the trigonometric function expressing the relationship between ω and θ indicates
how movement direction periodically affects H and W so
as to determine the difficulties of 2D pointing.
• Backward Compatibility. Regardless of the concrete value
of ω, Equation 10 will be exactly equivalent to the standard form of Fitts’ law when W = H.
• Proportionality of H and W . The factors W and H proportionally contribute to ID via the quantities ω and (1ω). The relative proportion of one factor still remains even
when the other’s contribution is close to zero. This is unlike Equation 3, which appears to imply that ID would
be entirely dependent on the single factor W when H is
close to infinity. Unlike Equation 5, different proportions
also imply the inequality between W and H’s effects.
• Complementarity of H and W . There is no more target
property (e.g. aspect ratio, or area) than target width and
height used to compose the whole determinant of ID. Increasing the weight of W ’s contribution will cause the
equivalent decrease of H’s weight, and vice versa.
As an indirect pointing device, mouse separates motor space
from visual space without the possibility of causing target to
be obscured by hand in any movement direction. Thus, we
carried out the experiment using a traditional mouse.
Apparatus and Participants
The experiment was performed on a HP Compaq DC7700
computer, which featured an Intel Core 2 Duo CPU at 2.0
GHz, 2.00 GB RAM, and a 17-inch LCD at 1280 × 960
resolution. A PS/2 mouse in its default settings was used as
the pointing device.
Sixteen undergraduates (8 females and 8 males, with the average age of 20.3) participated in this experiment. All of
them are right-handed.
Task and Experimental Design
For each trial, the participant needed to click a trial-start button and then moved the cursor to acquire a rectangular target
as quickly and accurately as possible. The start button and
the target were alternatively displayed. The effective diameter of the start button was 100 pixels, but it was visually
rendered as a small 24-pixel-diameter circle. This form was
helpful to concentrate the hits closer on the center of the button so as to prevent significant biases of actual movement
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Session: Human Performance Gives Us Fitts'
CHI 2012, May 5–10, 2012, Austin, Texas, USA
900
694.9
784.4
665.6
(80,80)
(120,30)
(120,80)
(120,120)
633.9
771.3
753.1
804.2
(50,80)
856.5
821.7
(30,80)
(50,30)
(50,50)
567.8
(30,30)
(80,30)
621.4
691.3
713.6
(30,120)
(50,50)
(80,50)
748.5
(30,50)
(80,80)
(120,120)
753.1
(30,30)
657.4
489.4
772.9
(120,120)
612.2
559.2
(120,50)
526.6
523.2
(80,120)
(120,30)
150
(W, H)
pixel
(30,30)
0
Trial-start Button
596.1
300
(50,50)
θ
450
(80,80)
W
(c)
600
678.9
A
A=650 pixels
A=450 pixels (b)
(a)
A=250 pixels
750
644.8
H
Movement Time (ms)
The dashed dodecagon is invisible for subjects. It is just used to
demonstrate the layout of different trials. Movement distance
A is measured from the center of
the trial-start button to that of
the target. The two center points
are a pair of opposite vertexes.
(30,120)
Target
Figure 4. M T by the combination of W , H and A.
Figure 2. Experimental interface.
r
distance. As Figure 2 shows, the start button and the target
were respectively placed at a pair of opposite vertexes of an
invisible regular dodecagon, which was located at the center
of the screen to control the layout. The duration from the
start of trail by clicking the button to the moment the subject
made a click for target selection was measured as M T . If
the hit missed the target, an error trial was recorded. The
trial settings would not be changed to the next combination
of experimental factors until the success of the current one.
ω
0.5
*
0.5
2
*
0.5
∞
*
0.5
*
*
Equation 3
Equation 5
1
a
b
Est. Std. Err. Est. Std. Err.
277.6 13.6 129.2 4.3
278.3 11.3 129.2 3.6
276.4 11.7 127.4 3.6
277.2
8.2 127.5 2.5
359.1 18.3 128.8 7.1
343.7 13.0 132.3 4.9
276.1 11.6 127.8 3.7
277.1
8.2 127.6 2.6
250.6
9.1 123.9 2.4
311. 9 10.4 134.2 3.7
ω
Est. Std. Err.
—
—
0.584 0.027
—
—
0.616 0.023
—
—
0.563 0.013
—
—
0.614 0.026
0.62 0.061
—
—
r
Est. Std. Err.
—
—
—
—
—
—
—
—
—
—
—
—
1.81 0.353
1.95 0.281
for η
—
—
—
R2
.977
.984
.983
.992
.940
.972
.983
.992
.992
.983
Table 1. Results of model fitting. The asterisk denotes a free parameter.
There were four experimental factors: movement distance A
(250, 450, 650 pixels), target width W (30, 50, 80, 120 pixels), target height H (30, 50, 80, 120 pixels), and movement
direction θ (i.e. approach angle, with 12 levels equally distributed from 0◦ to 360◦ , as determined by the centerlines
of the vertex angles of the dodecagon, respectively). The
core goal of the experiment was not to investigate the effects
of the first three factors on user performance, like the studies [16, 2], but to test the validity of the new modification
of Fitts’ law. Therefore, the factors A, W and H were not
fully crossed but manipulated to generate 25 combinations
(as shown in Figure 4), with the requirement that the ID values calculated using Equation 5 were distributed throughout
the corresponding range as evenly as possible because ID
is the essential factor to determine M T [12]. A task block
included 300 trials, resulting from the design of 25 combinations × 12 directions × 1 trial. These trials were presented
in a predetermined random order. There were 6 blocks for
each of the participants, who successfully completed the experiment within one session lasting about one hour.
Using repeated measures ANOVA, we found a significant
learning effect among the task blocks (F5,75 = 9.63, p <
.001) because of the significant differences between the first
block and some of the others. Thus, we also excluded the
data of the first block. Figure 4 depicts the averages of M T
under the 25 different conditions of (A, W , and H). It indicates that for a given combination of A × W or A × H,
M T will decrease in general with the increase of H or W ,
and vice versa. This point directly reflects the duality of H
and W . The results also reveal the dominance effect. In Figure 4a, for example, when W = 30 pixels, the difference of
M T between the two H levels 30 and 120 pixels is obviously smaller than the difference when W = 120 pixels. In
Figure 4c, we can also find a similar result when H increases
from 30 to 80 pixels. Figure 4b illustrates a more direct evidence of the dominance effect that the difference between
the two H levels (50 and 120 pixels) when W = 30 pixels
is insignificant (F1,15 = 0.363, p = .556).
Results
Model Fitting to Entire Data
Movement Time
When fitting the model of Equation 11 to the grand means
of M T , we considered different parameterization schemes
of ω and r. Similar to the method in Accot and Zhai’s work
[2], we also let the norm order r ∈ {1, 2, ∞, *} to parameterize it, where the asterisk denotes being free for the best
estimate. The weight ω was parameterized by setting it free
or equal to 0.5, which denotes the equal impact of W and
H. Therefore, there were 8 models resulting from the parameterization. Table 1 lists the schemes and summarizes
the results of model fitting. As can be seen, the most suitable combination was that when r = 2, and ω was set as a
free parameter (i.e. Equation 10), which resulted in a higher
R2 up to .99. We further plotted R2 as a function of ω and r.
As Figure 3 shows, the convex curved surface confirms the
reliability of the estimates in Table 1. These results justified
the derivation from our hypotheses to Equation 10. Comparing IDθ with IDaz and IDconf , we found that they all could
accurately predict the grand means of M T . Next, we further
present their performances under different θ conditions.
This experiment generated more than 29 thousand trials. The
errors and outliers, which we at first excluded from our analyses, were about 3.69% of the whole data. This means that
the subjects properly maintained an approximate 4% error
rate required for Fitts’ law studies [17]. As recommended
by Soukoreff and MacKenzie [21], we used aggregated trial
data for the analyses.
R2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.0
0.2
1
0.4
2.5
r
0.6
4
ω
0.2
5.5
4
2.5
r
1
0.0
0.2
0.4
0.6
0.8
1.0
ω
0.8
5.5 1.0
Views from different perspectives
Figure 3. IDθ model’s fitness plotted as a function of ω and r.
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760
720
680
640
600
560
520
480
440
400
Experiment 1
Experiment 2
(a)
CHI 2012, May 5–10, 2012, Austin, Texas, USA
η
2.0
ω
(b)
η ≈ 0.170 + 1.342sin2θ, R2 = .924
1.5
When η > 1, H has greater
impact than W
1.0
When ω > 0.5, W has greater
impact than H
0.5
ω ≈ 0.362 + 0.487cos2θ, R2 = .958
0.0
0°
330° 0°
30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330°
30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330°
Figure 5. (a) The grand means of M T by θ in Experiments 1 and 2; (b) Estimates of ω as well as η by θ. It appears to be that there is a periodic
function relationship between the parameter ω and the factor θ, matching the use of ω as the replacement of c1 + c0 cos2 θ based on our hypotheses.
Within a period (π), the area above the dashed line where ω = 0.5 is obviously larger than that below (see the light green area and the gray one),
indicating that target width W , overall, has greater contribution than target height H to hand pointing tasks’ difficulties.
Effects of Movement Direction
θ
The factor θ had a significant main effect on M T (F11,165 =
11.79, p < .001). Figure 5a shows the averages of M T at
different θ levels. Post hoc pairwise comparisons indicated
that there were 40 pairs of directions that generated significant differences of M T , but in general there was no significant difference between two opposite or axisymmetric directions. Table 2 summarizes the results of model fitting.
As can be seen, there were some interesting results when the
data were differentiated by movement direction. The R2 values of the new modification and Equation 3 were very close
to each other under every θ condition, and they were higher,
in general, than those of Equation 5, especially under the
horizontal conditions. In other words, the new modification
and Equation 3 were able to effectively capture the effect
of movement direction on M T via the introduced parameter
ω or η, and they were better than Equation 5 when taking
account of movement direction.
330°
0°
30°
60°
90°
120°
150°
180°
210°
240°
270°
300°
a
288.9
277.4
284.2
267.2
258.6
297.8
284.9
264.8
280.3
246
278.1
289.2
Equation 12
b
ω
124.7 0.778
128.1 0.851
119.5 0.719
132.7 0.451
138.7 0.361
120.6 0.449
119.2 0.671
128.1 0.841
121.7 0.769
139
0.44
136.6 0.402
127.9 0.531
R2
.981
.975
.969
.963
.936
.972
.963
.971
.983
.965
.957
.956
Equation 3
η
R2
0.284
.981
0.174
.975
0.386
.969
1.195
.961
1.772
.935
1.224
.972
0.487
.962
0.189
.971
0.299
.984
1.267
.964
1.491
.957
0.888
.957
Equation 5
R2
.925
.880
.933
.963
.925
.970
.941
.872
.931
.963
.952
.953
Table 2. Summary of model fitting to the data under each of the θ
conditions in Experiment 1.
• R2 has similar dependence on ω when θ is in axisymmetric or opposite directions. As shown by the convex curves
in each of the subfigures, ω apparently has similar impacts on the accuracies of IDθ in each pair of axisymmetric or opposite directions due to the similar summits, with
the curves themselves closely matched with each other.
A reasonable explanation about this point is that the user
performances in axisymmetric or opposite directions are
similar to each other. Furthermore, the convexities of all
the curves also confirm the reliabilities of the optimal estimates of ω under different θ conditions (see Table 2).
More importantly, it was implied that both ω and η were dependent on θ. As Table 2 indicates, the estimates of ω as
well as η could be divided into two clusters. They consistently indicated that when θ leaned to the horizontal directions, W had greater weight affecting M T than H (ω > 0.5,
η < 1.0), otherwise the weight of H was greater (η > 1.0,
ω < 0.5). In general, the closer the movement direction
to the horizontal axis is, the greater the weight of W for
M T will be (compared to that of H and the weight of W
under a different θ condition as well). The relationship between the estimates of ω and θ could be formulated using
ω = 0.362 + 0.487 cos2 θ (R2 = .958), correctly matching
the meanings of ω based on our hypotheses, while those of η
could be expressed as η = 0.170+1.342 sin2 θ (R2 = .924).
Figure 5b plots the estimates and the calculated values of ω
and η by θ. It seemed that the parameter ω was more accurate than η to reveal the symmetry and periodicity of the
effect of movement direction on M T .
• R2 is more sensitive to W . Comparing the curvatures near
the summits, it is also clearly indicated that the accuracies
are more sensitive to the variations of ω (ω > 0.5) under
the horizontal θ conditions than under the vertical conditions (ω < 0.5). That is to say, W is prior to determine
IDθ for horizontal pointing tasks, but the corresponding
priority of H is not stronger for vertical pointing tasks
because the differences of R2 are not obvious when ω is
changed from the best estimate to 0.5 or beyond.
• W is superior to H for the accuracies of IDθ . Comparing the differences between the R2 values when ω = 0.5
and the best estimate under different movement direction
conditions, we can find that the improvements of the accuracies of IDθ via increasing the weight of H under the
vertical θ conditions are not as great as those via increasing the weight of W under the horizontal θ conditions.
Figure 5b also indicates that the balanced directions where
W and H theoretically have equal impacts (ω = 0.5) are not
the bisectors of the quadrants but biased to the vertical axis.
Thus, within a period, e.g. from 60◦ to 240◦ (π), the area
(i.e. the probability) of the θ conditions where ω > 0.5 is
obviously larger than the area where ω < 0.5. That is to say,
overall, target width W had greater contribution than H to
M T as implied by the estimates of ω in Table 1 (ω > 0.6).
Recently, Guiard argued that the linear relationship between
M T and ID could be “contaminated” when using a crossed
design of A × W [12]. In other words, the regression coefficients would be different under different movement distance (A) or target width (W ) conditions. Accordingly, we
calculated the regression coefficients for different A levels
We also plotted R2 by ω under each of the θ conditions. As
Figure 6 shows, it further reveals the following three points:
3190
0.5
0°
ω = 0.5
0.3
2
1.0 R
(a)
(b)
(d)
(c)
ω = 0.5
0.8
θ
0.7
0.6
90°
60°
240°
120°
300°
ω = 0.5
.5 .6
.7
270°
330°
0°
30°
60°
90°
120°
150°
180°
210°
240°
270°
300°
Total
0.4
0.3
.0
.1
.2
.3
.4
ω = 0.5
.5 .6
.7
.9
.8
1. .0
.1
.2
.3
.4
.8
.9
1.
Figure 6. R2 plotted as a function of ω under every different θ condition in Experiment 1.
Movement Time (ms)
950
457.7
480.2
511.2
490.6
537.0
549.9
565.2
630.4
30 50 80 120 30 50 80 120 30 50 80 120 30 50 80 120
Target Height (pixel)
Figure 8. M T by the combination of W and H when A = 550 pixels.
0.9
0.5
527.4
150°
330°
W=120
583.1
30°
210°
180°
0.4
W=80
597.2
0.6
W=50
609.6
0.7
W=30
626.0
0.8
700
600
500
400
300
200
100
0
632.4
0.9
650.8
2
Movement Time (ms)
1.0 R
CHI 2012, May 5–10, 2012, Austin, Texas, USA
560.7
Session: Human Performance Gives Us Fitts'
A (pixels):
250 450 650
850
750
a
169.2
208.9
198.9
226.9
176.1
187.2
163.7
138.1
168.3
176.2
158.9
174.7
182.3
Equation 12
b
ω
109.3 0.771
100.5 0.877
103.2 0.791
97.70 0.484
117.2 0.386
109.9 0.536
115.5 0.786
122.0 0.857
111.0 0.785
113.6 0.419
125.7 0.396
111.3 0.489
110.0 0.644
R2
.969
.976
.952
.943
.966
.975
.930
.988
.958
.942
.939
.971
.994
Equation 3
η
R2
0.296
.969
0.139
.976
0.263
.953
1.067
.942
1.600
.967
0.866
.975
0.272
.930
0.166
.988
0.274
.958
1.387
.941
1.530
.940
1.045
.971
0.552
.994
Equation 5
R2
.817
.676
.785
.942
.939
.974
.765
.734
.785
.931
.918
.970
.952
Table 3. Summary of model fitting to the data under each of the twelve
θ conditions and the entire data in Experiment 2.
650
blocks, and each of them included 192 trials, resulting from
a fully crossed design (4 W × 4 H × 12 θ × 1 trial). Seventeen right-handed participants (8 females and 9 males, with
the average age of 22) successfully finished this experiment.
Only five of them took part in the first experiment.
550
θ=270°
(a)
450
1
1.5
2
2.5
3
ID
3.5
4
4.5
θ=150°
(b)
51
1.5
2
2.5
3 3.5
ID
4
4.5
5
Figure 7. Regression lines at different A levels when θ = 270◦ , 150◦
Results
and further found that the contamination extents appeared to
be relevant to the direction factor θ. As illustrated by the
regression lines of the cases when θ = 270◦ and 150◦ in
Figure 7, the contamination was noticeable in vertical directions, but, on the contrary, it was negligible in horizontal (or
near-horizontal) directions.
We recorded more than 23 thousand trials in this experiment,
and when analyzing the results we left out the errors and
outliers (3.49%) as well as the data of the first block due to
the significant learning effect (F6,96 = 19.99, p < .001).
We found that the factor θ had a significant main effect on
M T (F11,176 = 10.25, p < .001). Similar to the results of
the first experiment, post hoc pairwise comparisons revealed
that there was also no significant difference among almost
all pairs of opposite directions but 150◦ and 330◦ (p < .05)
as well as in all pairs of axisymmetric directions. This result clearly supported our explanation about the similarity
of the relationships between R2 and ω in opposite and axisymmetric directions. Consequently, the data under these θ
conditions could be grouped together to construct a common
model as shown in Figure 10. Furthermore, it was also indicated that the vertical directions generated significant differences compared with all the other non-vertical directions
but 240◦ compared with 90◦ (p = .108). Figure 8 shows
the means of M T by the combination of W and A, while
the means by θ are also shown in Figure 5a. It indicates that
the user performance would gradually deteriorate if θ varied
from horizontal to vertical directions. This result is consistent with that presented in prior studies [23, 22, 11].
Overall, this experiment suggests that the parameter ω can
effectively capture the unequal effects of W and H in different movement directions, and that the parameter ω is a
periodic function of θ. To substantiate these findings, we
further carried out the following experiment.
EXPERIMENT 2: 2D POINTING TASK WITH A CONSTANT
MOVEMENT DISTANCE
As Figure 7a indicates, the different slopes of the regression
lines under different movement distance conditions also imply that using different A levels in 2D pointing experiments
could probably bias the estimations of ω in some movement
directions. In order to purify and highlight the effects of W
and H, we employed a constant distance (550 pixels) in this
experiment as Guiard recommended [12].
The apparatus was the same as in Experiment 1 but a 22-inch
LCD at 1680 × 1050 resolution in place of the 17-inch LCD.
The task as well as the procedure was also similar. The main
difference was that the start button and the target, which was
disabled at first, were both presented before the beginning
of each trial. This was helpful for subjects to reduce the
possible (maybe negligible) delay of target locating. Once
clicked, the start button would disappear, with the target enabled immediately. Both of target width and height varied at
four levels of 30, 50, 80 and 120 pixels. There were 7 task
Table 3 summarizes the results of model fitting. We can
see that IDθ and IDaz almost had the same accuracies either for the entire data or the data grouped by θ, and that
both of them were better than Equation 5, whose accuracies obviously deteriorated under the conditions that movement was in or near the horizontal directions. We rechecked
the accuracy of Equation 5 under each of the three A con-
3191
Session: Human Performance Gives Us Fitts'
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
CHI 2012, May 5–10, 2012, Austin, Texas, USA
R2
0°
ω = 0.5
R2
90°
.0
.1
.2
.3
.4
ω = 0.5
.5 .6
(a)
(b)
(d)
(c)
ω = 0.5
.8
.9
120°
300°
60°
240°
270°
.7
150°
330°
30°
210°
180°
1. .0
.1
Second, excluding the possible contamination of movement
distance, Experiment 2 successfully reproduced and further
strengthened the findings about the parameter ω. These findings included the good model fit to the data differentiated
by movement direction and the entire data as well, the periodic variation of ω in the whole range of θ and the clear
relationship between ω and the accuracy of IDθ in each
of the different movement directions. These repeated results substantially confirmed the validity of our hypotheses.
Comparing the two approximate expressions of ω (i.e. ω =
0.362 + 0.487 cos2 θ vs. ω = 0.372 + 0.518 cos2 θ) resulted
from the empirical data of the two experiments, we further
found that the parameters c1 and c0 , calculated based on the
estimates of ω, had little variations between the two experiments. It probably means that these two parameters are not
dependent on subjects but on pointing devices. This property is desirable for the definition of ID as it can provide a
new criterion for device comparisons.
.2
.3
.4
ω = 0.5
.5 .6
.7
.8
.9
1.
Figure 9. R2 plotted as a function of ω under every different θ condition in Experiment 2.
ditions in Experiment 1 and found similar situations. These
results substantially implied that W had greater impact on
M T especially in the directions close to horizontal. The
estimates of ω could be approximately expressed as ω =
0.372 + 0.518 cos2 θ (R2 = .969). Regarding the parameter η, the estimates could be approximately calculated using
η = 0.130 + 1.311 sin2 θ (R2 = .913). This means that η
could capture the effects of movement direction on M T as
accurate as ω, but its periodicity was not as strong as that
of ω. We plotted R2 as a function of ω as shown in Figure
9, which clearly demonstrates the superiority of W to determine M T , the stronger W -sensitivity of R2 and the similarities that ω impacts R2 in opposite or axisymmetric directions. As shown in Figure 9a&b, for example, if only use H
(ω = 0) to define IDθ , there is almost no linear correlation,
while it is still obvious if only use W (ω = 1). Overall, this
experiment strengthened our findings as we anticipated.
Third, the weighted parameter η also exhibited an apparent
periodicity with the variation of θ. In Equation 3, η is the
only adjustable parameter so that it highlights the variations
of the impact of target height as implied by the larger amplitude of the sine square wave in Figure 5b. The same change
pattern of η and 1 − ω (i.e. c2 + c0 sin2 θ) directly confirms
the potential connection between the impacts of target width
and height as indicated in our hypotheses.
As can be seen in Tables 2 and 3, IDθ and IDaz almost had
the same power to model 2D pointing. The results clarified
that IDaz did not only work for horizontal directions but
also for every different direction. However, we need to point
out that IDaz is incompatible with the standard Fitts ID.
In Experiment 2, for example, when pointing at the rectangular target of 50 × 80 pixels along the movement direction
θ = 0◦ and 270◦ , the corresponding difficulties should be
between the difficulties of pointing at two different square
targets 80 × 80 and 50 × 50 pixels. As Figure 10a shows,
however, the values of IDaz (3.62 and 3.90) are not in the
550
Fitts ID range (log2 ( 550
80 +1), log2 ( 50 +1)) but larger than
the upper bound. This means that the extension of Equation
3 leads to a distortion of IDs, making IDaz incomparable
with Fitts ID and even with itself due to the different distortion caused by different η. On the contrary, it is easy to
prove that for any rectangular target, its IDθ value will always fall into a specific Fitts ID range, whose bounds are
determined by the two dimensions of the rectangular target.
In other words, we can find an equivalent square target for
any rectangular target, whose IDθ value is equal to the Fitts
ID value of the square target, in a specific size range.
DISCUSSION
In this paper, we aim to uncover how movement direction
affects user performance, although some other researchers
[11, 26] prefer to accommodate this factor without directly
interpreting it. With the following three points, the results of
our study definitely justified our hypotheses.
First, the findings consistent with those in prior studies implied the reliability of the data. For example, our experiments indicated that 2D pointing could achieve the best performance in horizontal directions or their near directions and
the performance would become increasingly degraded, as reported in the studies [22, 11], when θ was gradually transferred to vertical directions. Our experiments also indicated
that the performance change pattern, especially in Experiment 2, regularly took place in the range from 0◦ to 360◦
like the result in Whisenand and Emurian’s work [23]. Regarding the results of model fitting, the parameter η got similar estimates to those in the Accot and Zhai’s work [2] when
Equation 3 was used to model horizontal 2D pointing. The
fitness of Equation 5 under the horizontal conditions in Experiment 1 was also as low as Yang and Xu themselves reported [26] and even lower in Experiment 2, which eliminated the effect of movement distance. These consistencies
indirectly indicated that our experiments properly captured
the performance features of 2D pointing.
The incomparability of IDaz with itself under different η
conditions can greatly bias or even reverse the results of user
performance evaluation. As presented in Figure 10b, for
example, the regression lines wrongly imply that the user
performance in horizontal directions would get worse than
that in vertical directions at the “same” level of IDaz . In
addition, regardless of the effectiveness of data fitting using IDaz , this problem might bias and invalidate the dependent measure of throughput, which has been accepted
by ISO9241-9 for the evaluation of pointing devices. We
can find two different points of view about the definition of
3192
Session: Human Performance Gives Us Fitts'
Index of Difficulty
4.2
4.0
log 2 (
550
+ 1)
W
700
(a)
θ=270°: IDaz=3.90
3.4
Movement Time (ms)
(b)
(d)
(c)
650
600
3.8
3.6
CHI 2012, May 5–10, 2012, Austin, Texas, USA
θ=0°: IDaz=3.62
θ=0°: IDθ=3.53
θ=270°: IDθ=3.28
550
500
3.2
450
3.0
400
2.8
350
35 40 45 50 55 60 65 70 75 80
W (pixels)
θ= 0°, 180°
θ=90°, 270°
2.2
2.7
3.2
3.7
IDaz
θ=30°, 150°, 210°, 330°
θ=60°, 120°, 240°, 300°
4.2
4.7
5.2 2.2
2.7
3.2
3.7
4.2
4.7
Buttons
IDθ
Figure 10. (a) Width of the square target whose Fitts ID is equal to a rectangular (50 × 80 pixel) target’s ID calculated using IDaz or IDθ ; (b) and
(c) Regression lines for the data in Experiment 2 based on IDaz and IDθ , respectively; (d) Button groups arranged in the same limited area.
(e.g. 135◦ ). The similarity between the estimate of ω for
the entire data and the calculated value using the approximate expression (0.616 vs. 0.610 in Experiment 1; 0.644
vs. 0.631 in Experiment 2) supports this point.
throughput [21, 27], but, anyhow, this measure is not independent from ID. Therefore, it is also necessary to maintain
the consistency of different definitions of ID, like IDθ , with
the standard Fitts ID for the comparability of experimental
evaluations [21]. Figure 10c clearly shows the benefit that
the regression lines based on IDθ can correctly represent
the ranks of the user performance in different directions.
Design Implications for User Interfaces
The findings of our study can benefit the development of user
interfaces. First of all, it is more efficient to facilitate the acquisition of targets by elongating them in horizontal directions than in vertical directions unless the targets are usually
acquired via vertical pointing. Comparing the impact of H
in vertical directions and that of W in horizontal directions
(e.g. 0.628 vs. 0.890 in Experiment 2), we can find that the
user cannot make use of target height as fully as of width.
Overall, W has greater impact (ω > 0.6) than H. Secondly,
the frequently used buttons should be placed in the left or
right side of the workspace in user interfaces due to the better user performance in (or near) the horizontal directions.
Thirdly, with respect to a group of icons or buttons that need
to be arranged in a limited area (such as the case in Figure
10d), it is better to make them as wide as possible.
Design Recommendations for 2D Pointing Experiments
We have presented the reliability and advantages of IDθ .
Now, we make the following design recommendations for
Fitts’ law experiments that take account of 2D targets.
• Use rectangular targets with various aspect ratios instead
of circular targets. ISO9241-9 recommends using circular or square targets so as to match the variables of the 1D
Fitts’ law. Our study indicates that it is no longer necessary to restrict the shape of targets. Accounting for the
difficulties of pointing at general rectangular targets, IDθ
with an empirical constant ω to capture the different effects of W and H can offer more valuable information for
interface designs and pointing device evaluations as Accot
and Zhai previously mentioned [2].
• Avoid the risk of contamination caused by the factor of
movement direction. The results of Experiment 1 confirmed that the use of different A levels could indeed result
in the contamination of Fitts’ law parameters [12]. Unfortunately, the use of different θ values could also cause
contamination, as Figure 10c shows, even though the contamination caused by A were excluded. This means that
multidirectional tapping tasks may be unsafe for model
fitting. A simple way to prevent this potential risk is to
use only a group of opposite or axisymmetric directions.
• Use horizontal or vertical directions to highlight the asymmetrical impact of W and H on movement time. Our findings about the asymmetry of W and H are consistent with
those of the prior studies [2, 11]. Our work further reveal that the asymmetry, quantified using ω and 1-ω for
W and H, respectively, is periodically varied with movement direction (i.e. ω = c1 + c0 cos2 θ). W achieves the
maximum impact in horizontal directions, but so does H
in vertical directions. When one of the target dimensions
gets the maximum impact, the other gets the minimum.
• Use diagonal directions to reveal the overall impacts of
W and H. If a target can be equally accessed from different directions (0◦ − 360◦ ), the mathematical expectation
of ω, which could be interpreted as the overall impact of
W , will be equal to the value of ω in diagonal directions
In addition, the results of our study indicate that when θ ≈
60◦ , 120◦ , 240◦ or 300◦ , instead of 45◦ as Grossman and
Balakrishnan reported [11], the two target factors W and H
have equal impact on M T . This implies that increasing the
widths of the targets placed at the corners of a window can
still obtain greater benefits, especially when wide LCD monitors increasingly get popular today. The inconsistency between this implication and that in previous work [11] maybe
resulted from the use of different pointing devices.
CONCLUSIONS AND FUTURE WORK
The experimental results we have presented above indicate
that the projections of target width and height (W and H) on
the line of movement can properly express their asymmetrical impacts on movement time; that the projection method
can effectively uncover the inherent dependence of the impacts of W and H on movement direction and the internal relation between them. To summarize, the difficulty of pointing at 2D targets can be defined as:
p
IDθ = log2
ω(A/W )2 + (1 − ω)(A/H)2 + 1
where 0 < ω < 1, and it depends on movement direction
θ, i.e. ω = c1 + c0 cos2 θ. In practice, it only needs several different angles to calibrate c1 and c0 , and we believe
that they will be consistent in different experiments when the
same pointing device is used. Thus, the parameter ω can be
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Session: Human Performance Gives Us Fitts'
CHI 2012, May 5–10, 2012, Austin, Texas, USA
12. Guiard, Y. The problem of consistency in the design of
Fitts’ law experiments: Consider either target distance
and width or movement form and scale. In Proc. CHI
2009, ACM Press (2009), 1809–1818.
13. Hoffmann, E. R. and Sheikh, I. H. Effect of varying
target height in a Fitts’ movement task. Ergonomics,
Vol. 37, No. 6 (1994), 1071–1088.
14. Jax, S.A. and Rosenbaum, D. A. Extending Fitts’ law
to manual obstacle avoidance. Experimental Brain
Research, Vol. 180 (2007), 775–779.
15. MacKenzie, I. S. A note on the information-theoretic
basis for Fitts’ law. Journal of Motor Behavior, Vol.21
(1989), 323–330.
16. MacKenzie, I.S. and Buxton, W. Extending Fitts’ law
to two-dimensional tasks. In Proc. CHI’92, ACM Press
(1992), 219–226.
17. MacKenzie, I.S. Fitts’ law as a research and design tool
in human-computer interaction. Human-Computer
Interaction, Vol.7 (1992), 91–139.
18. Martin, F. S. Assessment of Fitts’ Law for Quantifying
Combined Rotational and Translational Movements.
Human Factors, Vol. 52, No. 1 (2010), 63–77.
19. Murata, A. Extending effective target width in Fitts’
law to a two-dimensional pointing task. Int. J.
Human-Computer Interaction, Vol. 11, No. 2 (1999),
137–152.
20. Schmidt, R.A., Zalaznik, H.N., Hawkins, B., Frank,
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used as a new measure for pointing device evaluation. The
simplicity of IDθ in form, the easy use and its compatibility with the standard Fitts ID can benefit the field of HCI
as we have discussed. In the future, we plan to investigate
the benefits of using effective width and height based on 2D
joint probability distribution [19] or endpoint deviation [24],
confirm the effectiveness of IDθ for other pointing devices,
evaluate target expansion in two dimensions and study the
acquisition of rotated 2D targets.
ACKNOWLEDGMENTS
The authors thank the anonymous CHI reviewers for their
detailed and valuable commentaries, especially the constructive recommendations to revise this paper. This work was
supported by the National Natural Science Foundation of
China (Grant No. 61070144).
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