Derivation of The Model From Fitts's Law
This derivation is only meant as a high level sketch. It lacks the illustrations of, and may differ in detail from, the derivation given by Accot and Zhai (1997).
Assume that the time required for goal passing (i.e. passing a pointer through a goal at distance A and of width W, oriented perpendicular to the axis of motion) can be modeled with this form of Fitts's law:
Then, a straight tunnel of length A and constant width W can be approximated as a sequence of N evenly spaced goals, each separated from its neighbours by a distance of A/N. We can let N grow arbitrarily large, making the distance between successive goals become infinitesimal. The total time to navigative through all the goals, and thus through the tunnel, is
| Tstraight tunnel | ||
| (applying L'Hôpital's rule ...) | ||
Next, consider a curved tunnel of total length A, parameterized by s varying from 0 to A. Let W(s) be the variable width of the tunnel. The tunnel can be approximated as a sequence of N straight tunnels, numbered 1 through N, each located at si where i = 1 to N, and each of length si+1 − si and of width W(si). We can let N grow arbitrarily large, making the length of successive straight tunnels become infinitesimal. The total time to navigative through the curved tunnel is
| Tcurved tunnel | ||
| (... by the definition of the definite integral) |
yielding the general form of the steering law.
Read more about this topic: Steering Law
Famous quotes containing the words model and/or law:
“Research shows clearly that parents who have modeled nurturant, reassuring responses to infants’ fears and distress by soothing words and stroking gentleness have toddlers who already can stroke a crying child’s hair. Toddlers whose special adults model kindliness will even pick up a cookie dropped from a peer’s high chair and return it to the crying peer rather than eat it themselves!”
—Alice Sterling Honig (20th century)
“The law of humanity ought to be composed of the past, the present, and the future, that we bear within us; whoever possesses but one of these terms, has but a fragment of the law of the moral world.”
—Edgar Quinet (1803–1875)