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| '''Fitts's law''' (often cited as '''Fitts' law''') is a model of human movement primarily used in [[human–computer interaction]] and [[ergonomics]] that predicts that the time required to rapidly move to a target area is a function of the distance to the target and the size of the target. Fitts's law is used to [[Conceptual model|model]] the act of ''pointing'', either by physically touching an object with a hand or finger, or virtually, by pointing to an object on a [[computer monitor]] using a [[pointing device]]. It was proposed by [[Paul Fitts]] in 1954.
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| == Model ==
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| Fitts's law has been formulated mathematically in several different ways. One common form is the Shannon formulation (proposed by [http://www.yorku.ca/mack Scott MacKenzie], professor at [[York University]], and named for its resemblance to the [[Shannon–Hartley theorem]]) for movement along a single dimension:
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| : <math>T = a + b \log_2 \Bigg(1+\frac{D}{W}\Bigg)</math>
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| where:
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| * ''T'' is the average time taken to complete the movement. (Traditionally, researchers have used the symbol ''MT'' for this, to mean ''movement time''.)
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| * ''a'' represents the start/stop time of the device (intercept) and
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| * ''b'' stands for the inherent 1/speed of the device (slope). These constants can be determined experimentally by fitting a straight line to measured data.
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| * ''D'' is the distance from the starting point to the center of the target. (Traditionally, researchers have used the symbol ''A'' for this, to mean the ''amplitude'' of the movement.)
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| * ''W'' is the width of the target measured along the axis of motion. ''W'' can also be thought of as the allowed error tolerance in the final position, since the final point of the motion must fall within ±{{frac|''W''|2}} of the target's center.
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| From the equation, we see a ''speed–accuracy'' trade off associated with pointing, whereby targets that are smaller and/or further away require more time to acquire.
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| == Success and implications ==
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| Fitts's law is an unusually successful and well-studied model. Experiments that reproduce Fitts's results and/or that demonstrate the applicability of Fitts's law in somewhat different situations are not difficult to perform. The measured data in such experiments often fit a straight line with a [[Pearson product-moment correlation coefficient|correlation coefficient]] of .95 or higher, a sign that the model is very accurate.
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| Although Fitts only published two articles on his law (Fitts 1954, Fitts and Peterson 1964), there are hundreds of subsequent studies related to it in the [[human–computer interaction]] (HCI) literature, and quite possibly thousands of studies published in the larger psychomovement literature. The first HCI application of Fitts's law was by Card, English, and Burr (1978), who used the index of performance (''IP''), defined as {{frac|1|''b''}}, to compare performance of different [[input device]]s, with the [[mouse (computing)|mouse]] coming out on top. This early work, according to [[Stuart Card]]'s biography, "was a major factor leading to the mouse's commercial introduction by [[Xerox]]".<ref>{{cite web |url=http://www2.parc.com/istl/groups/uir/people/stuart/stuart.htm |title=Stuart Card |website=[[PARC (company)|PARC]] |archiveurl=//web.archive.org/web/20120711210730/http://www2.parc.com/istl/groups/uir/people/stuart/stuart.htm |archivedate=2012-07-11}}</ref>
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| Fitts's law has been shown to apply under a variety of conditions, with many different limbs (hands, feet,<ref>{{cite journal |first=Errol R. |last=Hoffmann |date=1991 |title=A comparison of hand and foot movement times |journal=Ergonomics |volume=34 |issue=4 |pages=397-406}}</ref> head-mounted sights,<ref>{{cite journal |last=So |first=R. H. Y. |last2=Griffin |first2=M. J. |date=2000 |title=Effects of target movement direction cue on head-tracking performance |journal=Ergonomics |volume=43 |issue=3 |pages=360–376}}</ref> eye gaze), manipulanda (input devices), physical environments (including underwater), and user populations (young, old, special educational needs, and drugged participants).
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| Note that the constants ''a'', ''b'', ''IP'' have different values under each of these conditions.
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| Since the advent of [[graphical user interface]]s, Fitts's law has been applied to tasks where the user must position a mouse [[cursor (computers)|cursor]] over an on-screen target, such as a button or other [[GUI widget|widget]]. Fitts's law models both [[point-and-click]] and [[drag-and-drop]] actions. Dragging has a lower ''IP'' associated with it, because the increased muscle tension makes pointing more difficult. Fitts's law has also been shown to model target-directed hand and head motions in a virtual environment.<ref>{{cite journal |last=So |first=R. H. Y. |last2=Cheung |first2=K. M. |date=2002 |title=Combined and interacting effects of hand and head movement lags on discrete manual performance in a virtual environment |journal=Ergonomics |volume=45 |pages=105–123}}</ref><ref>{{cite journal |last=So |first=R. H. Y. |last2=Cheung |first2=K. M. |last3=Goonetilleke |first2=R. S. |date=1999 |title=Target-directed head movements in a head-coupled virtual environment: predicting the effects of lags using Fitts' law |journal=Human Factors |volume=41 |issue=3 |pages=474–486}}</ref>
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| In its original and strictest form:
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| * It applies only to movement in a single dimension and not to movement in two dimensions (though it is successfully extended to two dimensions in the [[Steering law|Accot-Zhai steering law]]);
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| * It describes simple motor response of, say, the human hand, failing to account for software acceleration usually implemented for a mouse cursor;
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| * It describes untrained movements, not movements that are executed after months or years of practice (though some argue that Fitts's law models behaviour that is so low level that extensive training doesn't make much difference).
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| If, as generally claimed, the law does hold true for pointing with the mouse, some consequences for [[user interface]] design include:{{Citation needed|date=May 2007}}
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| * Buttons and other GUI controls should be a reasonable size; it is relatively difficult to click on small ones.
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| * Edges and corners of the [[computer monitor]] (e.g., the location of the [[Start menu|Start button]], [[Taskbar]] and a maximized window's Close button in [[Microsoft Windows]]; or the menus and [[Dock (computing)|Dock]] of Mac OS X) are particularly easy to acquire with a mouse, touchpad or trackball. Because the pointer remains at the screen edge regardless of how much further the mouse is moved, they can be considered as having infinite width.<ref>{{cite web |url=http://particletree.com/features/visualizing-fittss-law/ |title=Visualizing Fitts' Law |first=Kevin |last=Hale |website=Particle Tree |date=3 October 2007}}</ref><ref>{{cite web |url=http://www.codinghorror.com/blog/2006/08/fitts-law-and-infinite-width.html |title=Fitts' Law and Infinite Width |first=Jeff |last=Atwood |authorlink=Jeff Atwood |website=Coding Horror |date=9 August 2006}}</ref> This doesn't apply to touchscreens, though.
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| * Similarly, top-of-screen menus (e.g., Mac OS) are sometimes easier to acquire than top-of-window menus (e.g., Windows OS).
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| * Pop-up [[menu (computing)|menu]]s can usually be opened faster than pull-down menus, since the user avoids travel: the pop-up appears at the current cursor position.
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| * [[Pie menu]] items typically are selected faster and have a lower error rate than linear [[menu (computing)|menu]] items, for two reasons: because pie menu items are all the same, small distance from the centre of the menu; and because their wedge-shaped target areas (which usually extend to the edge of the screen) are very large.<ref>{{cite journal |last=Hopkins |first=Don |title=The Design and Implementation of Pie Menus |url=http://www.drdobbs.com/database/the-design-and-implementation-of-pie-men/184408667 |journal=[[Dr. Dobb's Journal]] |date=1 December 1991}}</ref>
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| Fitts's law remains one of the few hard, reliable human–computer interaction predictive models, joined more recently by the [[Accot-Zhai steering law]], which is derived from Fitts's law.
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| == Mathematical details ==
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| The [[logarithm]] in Fitts's law is called the index of difficulty ''ID'' for the target, and has units of bits. We can rewrite the law as
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| : <math>T = a + b ID = a + \frac{ID}{IP} \,</math>
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| where
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| : <math>ID = \log_2 \left(\frac{D}{W}+1\right).</math>
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| Thus, the units for ''b'' are time/bit; e.g., ms/bit. The constant ''a'' can be thought of as incorporating reaction time and/or the time required to click a button.
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| The values for ''a'' and ''b'' change as the conditions under which pointing is done are changed. For example, a mouse and [[stylus]] may both be used for pointing, but have different constants ''a'' and ''b'' associated with them.
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| An index of performance ''IP'' (also called throughput ''TP''), in bits/time, can be defined to characterize how quickly pointing can be done, independent of the particular targets involved. There are two conventions for defining ''IP'': one is ''IP'' = 1/''b'' (which has the disadvantage of ignoring the effect of ''a''), the other is ''IP'' = ''ID''<sub>avg</sub>/''MT''<sub>avg</sub> (which has the disadvantage of depending on an arbitrarily chosen "average" ''ID''). For a discussion of these two conventions, see Zhai (2002). Whatever definition is used, measuring the ''IP'' of different input devices allows the devices to be compared with respect to their pointing capability.
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| Slightly different from the Shannon formulation is the original formulation by Fitts:
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| : <math>ID = \log_2 \left(\frac{2D}{W}\right).</math>
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| The factor of 2 here is not particularly important; this form of the ''ID'' can be rewritten with the factor of 2 absorbed as changes in the constant ''a''. The "+1" in the Shannon form, however, does make it different from Fitts's original form, especially for low values of the ratio ''D''/''W''. The Shannon form has the advantage that the ''ID'' is always non-negative, and has been shown{{cn|date=February 2013}} to better fit measured data.
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| == Derivation ==
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| Fitts's law can be derived from various models of motion. A very simple model, involving discrete, deterministic responses, is considered here. Although this model is overly simple, it provides some intuition for Fitts's law.
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| Assume that the user moves toward the target in a sequence of submovements. Each submovement requires a constant time ''t'' to execute, and moves a constant fraction 1-''r'' of the remaining distance to the centre of the target, where {{nobr|0 < ''r'' < 1}}. Thus, if the user is initially at a distance ''D'' from the target, the remaining distance after the first submovement is ''rD'', and the remaining distance after the ''n''th submovement is ''r''<sup>''n''</sup>''D''. (In other words, the distance left to the target's centre is a function that [[exponential decay|decays exponentially]] over time.) Let ''N'' be the (possibly fractional) number of submovements required to fall within the target. Then,
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| : <math>r^N D = \frac{W}{2}.</math>
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| Solving for ''N'':
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| : <math>
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| \begin{align}
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| N & = \log_r \frac{W}{2D} \\
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| & = \frac{1}{\log_2 r} \log_2 \frac{W}{2D}\quad(\text{since } \log_x y = (\log_z y)/(\log_z x)) \\
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| & = \frac{1}{\log_2 1/r} \log_2 \frac{2D}{W}\quad(\text{since } \log_x y = - \log_x(1/y)).
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| \end{align}
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| </math>
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| The time required for all submovements is:
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| : <math>
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| \begin{align}
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| T = Nt & = \frac{t}{\log_2 1/r} \log_2 \frac{2D}{W} \\
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| & = \frac{t}{\log_2 1/r} + \frac{t}{\log_2 1/r} \log_2 \frac{D}{W}.
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| \end{align}
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| </math>
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| By defining appropriate constants ''a'' and ''b'', this can be rewritten as
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| : <math>T = a + b \log_2 \frac{D}{W}.</math>
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| The above derivation is similar to one given in Card, Moran, and Newell (1983). For a critique of the deterministic iterative-corrections model, see Meyer et al. (1990).<ref>{{cite web |last=Meyer |first=D. E. |last2=Smith |first2=J. E. K. |last3=Kornblum |first3=S. |last4=Abrams |first4=R. A. |last5=Wright |first5=C. E. |date=1990 |chapter=Speed-accuracy tradeoffs in aimed movements: Toward a theory of rapid voluntary action. |editor-first=M. |editor-last=Jeannerod |title=Attention and performance XIII |pages=173–226 |location=Hillsdale, NJ |publisher=Lawrence Erlbaum |url=http://www.umich.edu/~bcalab/documents/MeyerSmithKornblumAW1990.pdf |format=pdf}}</ref>
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| == See also ==
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| * [[Hick's law]]
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| * [[Point-and-click]]
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| * [[Accot–Zhai steering law]]
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| * [[Crossing-based interface]]
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| == Notes ==
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| {{Reflist|2}}
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| == References ==
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| {{More footnotes|date=November 2010}}
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| {{refbegin|2}}
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| * Paul M. Fitts (1954). The information capacity of the human motor system in controlling the amplitude of movement. ''Journal of Experimental Psychology'', volume 47, number 6, June 1954, pp. 381–391. (Reprinted in ''Journal of Experimental Psychology: General'', 121(3):262–269, 1992).
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| * Paul M. Fitts and James R. Peterson (1964). Information capacity of discrete motor responses. ''Journal of Experimental Psychology'', 67(2):103–112, February 1964.
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| * Stuart K. Card, William K. English, and Betty J. Burr (1978). Evaluation of mouse, rate-controlled isometric joystick, step keys, and text keys for text selection on a CRT. ''Ergonomics'', 21(8):601–613, 1978.
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| * I. Scott MacKenzie and William A. S. Buxton (1992). Extending Fitts' law to two-dimensional tasks. Proceedings of [[Association for Computing Machinery|ACM]] CHI 1992 Conference on Human Factors in Computing Systems, pp. 219–226. http://doi.acm.org/10.1145/142750.142794
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| * A. Murata. Extending effective target width in Fitts' law to a two-dimensional pointing task. International Journal of Human–Computer Interaction, 11(2):137–152, 1999. http://www.leaonline.com/doi/abs/10.1207/S153275901102_4
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| * Johnny Accot and Shumin Zhai (2003). Refining Fitts' law models for bivariate pointing. Proceedings of ACM CHI 2003 Conference on Human Factors in Computing Systems, pp. 193–200. http://doi.acm.org/10.1145/642611.642646
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| * Johnny Accot and Shumin Zhai (2002). More than dotting the i's—foundations for crossing-based interfaces. Proceedings of ACM CHI 2002 Conference on Human Factors in Computing Systems, pp. 73–80. http://doi.acm.org/10.1145/503376.503390
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| * Stuart K. Card, Thomas P. Moran, [[Allen Newell]] (1983). The Psychology of Human–Computer Interaction.
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| * I. Scott MacKenzie (1992). Fitts' law as a research and design tool in human–computer interaction. ''Human–Computer Interaction'', volume 7, 1992, pp. 91–139. http://www.yorku.ca/mack/hci1992.pdf
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| * Meyer, D. E., Smith, J. E. K., Kornblum, S., Abrams, R. A., & Wright, C. E. (1990). Speed-accuracy tradeoffs in aimed movements: Toward a theory of rapid voluntary action. In M. Jeannerod (Ed.), Attention and performance XIII (pp. 173–226). Hillsdale, NJ: Lawrence Erlbaum. http://www.umich.edu/~bcalab/Meyer_Bibliography.html
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| * A. T. Welford (1968). Fundamentals of Skill. Methuen, 1968.
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| * Shumin Zhai (2002). On the Validity of Throughput as a Characteristic of Computer Input, IBM Research Report RJ 10253, 2002, Almaden Research Center, San Jose, California. http://www.almaden.ibm.com/u/zhai/papers/ZhaiIBMReporRJ10253.pdf
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| {{refend}}
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| == External links ==
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| * [http://www.interaction-design.org/encyclopedia/fitts_law.html Peer-reviewed Encyclopedia entry on Fitts' Law] by Mehmet Göktürk
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| * [http://www.asktog.com/basics/firstPrinciples.html#fittsLaw Fitts' Law] at AskTog
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| * [http://www.asktog.com/columns/022DesignedToGiveFitts.html A Quiz Designed to Give You Fitts] at AskTog
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| * [http://ei.cs.vt.edu/~cs5724/g1/ Fitts' Law] at CS Dept. NSF-Supported Education Infrastructure Project
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| * [http://www.cs.umd.edu/class/fall2002/cmsc838s/tichi/fitts.html Fitts’ Law: Modeling Movement Time in HCI]
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| * [http://www.yorku.ca/mack/RN-Fitts_bib.htm Bibliography of Fitts’ Law Research] compiled by I. Scott MacKenzie
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| * [http://blogs.msdn.com/jensenh/archive/2006/08/22/711808.aspx Fitts' Law in Microsoft Office User Interface] by Jensen Harris
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| * [http://particletree.com/features/visualizing-fittss-law/ Visualizing Fitts's Law ] by Kevin Hale
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| {{DEFAULTSORT:Fitts's Law}}
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| [[Category:Human–computer interaction]]
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| [[Category:Motor control]]
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