Problem 83 - Algebra and Trigonometry

Question

Difficulty of a Task The difficulty in "acquiring a target" (such as using your mouse to click on an icon on your computer screen) depends on the distance to the target and the size of the target. According to Fitts's Law, the index of difficulty (ID) is given by $$\mathrm{ID}=\frac{\log (2 A / W)}{\log 2}$$ where \(W\) is the width of the target and \(A\) is the distance to the center of the target. Compare the difficulty of clicking on an icon that is 5 \(\mathrm{mm}\) wide to one that is 10 \(\mathrm{mm}\) wide. In each case, assume the mouse is 100 \(\mathrm{mm}\) from the icon.

Step-by-Step Solution

Answer

The 5 mm wide icon is harder to click than the 10 mm wide icon.

Step 1: Understanding Fitts's Law Formula

The formula given in the exercise is Fitts's Law: \( \mathrm{ID} = \frac{\log (2A / W)}{\log 2} \), which calculates the index of difficulty (ID) for hitting a target. Here, \( A \) is the distance to the target, and \( W \) is the width of the target.

Step 2: Identify Variables for the First Icon

For the first icon, we have a target width \( W_1 = 5 \) mm and a distance \( A = 100 \) mm.

Step 3: Calculate ID for the First Icon

Using the variables for the first icon in Fitts's Law: \[ \mathrm{ID}_1 = \frac{\log (2 \times 100 / 5)}{\log 2} \]Simplify: \[\mathrm{ID}_1 = \frac{\log (40)}{\log 2} \approx \frac{1.60206}{0.30103} \approx 5.32\]Therefore, the index of difficulty for the first icon is approximately 5.32.

Step 4: Identify Variables for the Second Icon

For the second icon, we have a target width \( W_2 = 10 \) mm, while the distance \( A \) remains 100 mm.

Step 5: Calculate ID for the Second Icon

Using the variables for the second icon in Fitts's Law: \[ \mathrm{ID}_2 = \frac{\log (2 \times 100 / 10)}{\log 2} \]Simplify: \[\mathrm{ID}_2 = \frac{\log (20)}{\log 2} \approx \frac{1.30103}{0.30103} \approx 4.32\]Therefore, the index of difficulty for the second icon is approximately 4.32.

Step 6: Compare the Two Difficulties

Comparing both indices of difficulty, we find that \(\mathrm{ID}_1 = 5.32\) for the first icon and \(\mathrm{ID}_2 = 4.32\) for the second icon. This indicates that the first icon, which is 5 mm wide, is more difficult to click compared to the second icon, which is 10 mm wide.

Key Concepts

Index of Difficulty

In the context of Fitts's Law, the "index of difficulty" (ID) is a quantifiable measure of how challenging it is to acquire a target with a pointing device, like a mouse. This index is calculated based on two primary factors:

• The distance to the target, denoted as \( A \)
• The width of the target, denoted as \( W \)

The concept is to model the difficulty level by considering how far the target is and how large it is, making it a more comprehensive measure. The formula, \[ \mathrm{ID} = \frac{\log (2A / W)}{\log 2} \]allows us to calculate a numerical value that represents this difficulty. The higher the ID, the more difficult it is to acquire the target. Compare this to everyday tasks like hitting a smaller button on a zoomed-out screen; the smaller button represents a higher ID due to the smaller target width \( W \).

Target Acquisition

"Target acquisition" refers to the task of successfully selecting or reaching a specified target using a pointing device. It is not just about moving towards a point but also about hitting it accurately. In physical terms, it's like aiming for a dartboard and hitting the bullseye. The ability to quickly and accurately acquire a target involves:

• Movement time to reach the target
• The precision required to select the target

Fitts's Law helps predict this movement time and accuracy by providing a model that states movement time will increase as the distance to the target \( A \) increases and as the target's size \( W \) decreases. This makes intuitive sense — think about the manual dexterity required to click on a very tiny link far away from your original cursor position.

Logarithmic Calculation

The "logarithmic calculation" within Fitts's Law is central to understanding how difficulty scales with changes in target size and distance. The computation hinges on the logarithm function, denoted as \( \log \). This function inherently manages large variations in input numbers due to its compressing effect. In Fitts's Law, the formula \[ \mathrm{ID} = \frac{\log (2A / W)}{\log 2} \]uses the base-10 logarithm to normalize differences in \( A \) and \( W \), turning them into a more manageable linear scale. This can be thought of as transforming multiplicative changes in target size and distance into additive changes, allowing for clearer comparison of target difficulty. For instance, doubling a target's width or reducing its distance by half results in the same calculated decrease in difficulty, simply because the logarithmic scale levels out these kinds of multiplicative shifts.