# A Study of Model Iterations of Fitts' Law and Its Application to Human-Computer Interactions
## 1 Introduction  
[PERSON] research in 1954 extended information theory to human motor systems. Based on information theory, he formulated the original model of Fitts' law, which quantifies the difficulty of human motor tasks. The difficulty index (ID) of a motor task is defined as follows:  
\[ID=-\log_{2}\frac{W}{2A}=\log_{2}\frac{2A}{W} \tag{1}\]  
W represents the width of the motor task's acquisition goal, and A denotes the magnitude of the task. The difficulty index (ID) is measured in bits. Essentially, the formula can be succinctly described as follows: the farther and smaller the target, the more challenging it is to acquire it. Additionally, [PERSON] defines a performance metric for a motor task completed within time t seconds, formulated as follows below in Equation (2). \[IP=-\frac{1}{t}\log_{2}\frac{W}{2A} \tag{2}\]  
The index of performance (IP), measured in bits/s, is the full name of the term \"IP\". Some later HCI researchers refer to it as throughput (TP) [1, 2]. [PERSON] hypothesized that, for a motor task, the IP remains constant over a wide range of values for A and W. In his experiments involving three different motor tasks, the IP values were not exactly constant but exhibited minimal variation [3]. By substituting Equation (1) into Equation (2), the following form can be written:  
\[t=\frac{ID}{IP} \tag{3}\]  
The formula suggests that the time needed to complete a motor task is proportional to the motor difficulty index (ID). When applying [PERSON] law to HCIs, we need to find ways toreduce the value of the ID. We need to make the target as large as possible and as close as possible to the limb with which we are performing the interactive behavior. In [PERSON] 1964 article, he proposed a formula for predicting the time required to complete a sport based on the difficulty index:  
\[MT=a+b\cdot ID \tag{4}\]  
The formula referred to is the well-established Fitts' law model, where MT represents movement time. The parameters a and b denote the intercept and slope, respectively, which vary depending on the specific application [4]. [PERSON] law asserts a linear correlation between task complexity and movement time (MT), and it has evolved into a fundamental principle of interaction design in contemporary practice. Subsequent researchers have built upon and refined [PERSON] ideas and findings, introducing modified versions of [PERSON] law to meet the increasing demands of real-world interactions [5; 6; 7]. Initially, the vectors defining motion from the starting point to the target were confined to a two-dimensional plane, limiting the applicability of the original Fitts' law model to simpler movements. In everyday scenarios, translational motion within a plane is less common, with rotations and other forms of movements predominating, necessitating models in three-dimensional space to address these complexities. Regrettably, the diversity of variables and scenarios in three-dimensional motion precludes the existence of a universally accepted model capable of perfectly simulating motion behaviors in such environments. Current adaptations of [PERSON] law to the three-dimensional domain are tailored to specific applications where they can effectively resolve the challenges. Nevertheless, the foundational principles of the original Fitts' law persist in these variants, shaping ongoing research and application in this field. Since [PERSON] law was first proposed about 70 years ago, researchers have worked hard to connect it to everyday interactions and HCIs. This fundamental idea is widely used in simulating the interaction between people and various input devices in HCIs. Scholars have investigated how to apply [PERSON] law to various user demographics and interaction patterns, with a focus on examining its applicability to handheld devices, treadle devices and a range of HCI applications. Fitts' law's implications for exceptional groups have also received recent attention, indicating a developing area of research interest.