# A Study of Model Iterations of Fitts' Law and Its Application to Human-Computer Interactions
## 3 The Origin of the Fitts' Law Formula and How It Was Developed
### Research in 3D Space and VR
#### 3.3.1 Research in Three-Dimensional Fields of Varying Forms  
Target thickness could have a major influence on prediction time, which led to a move from limited two-dimensional studies to three-dimensional research. [PERSON]'s seminal work provided the basis for the three-dimensional formulation of [PERSON] law in this expanded area [6]. Due to the greatly increased complexity of human motion in three dimensions, modifications to the Fitts formula are not only required for the coefficient of difficulty but also for the motion time formula as well. Building on this foundation, studies conducted in the past ten years have explored [PERSON] law's applications in three dimensions, yielding important new understandings and findings. To provide an overview of the state of Fitts' law research in the field of three-dimensional studies, a representative summary of the most important discoveries is assembled in Table 2.  
[PERSON]'s team proposed an early formulation of [PERSON] law in three dimensions, which involved similar modifications to the coefficient of difficulty (ID). In [PERSON]'s formula, a sine function was introduced to account for the angle of motion (\(\theta\)), where \(\theta\) represents the angle pointing towards the motion. Additionally, an arbitrary constant c was incorporated through linear regression. This formula outperformed the traditional Fitts formula in predicting motion time, although the improvement in fit was not significant [23]. [PERSON]'s team also presented their formula for predicting motion time by [PERSON] law in three dimensions. This formula considered the inclination angle (\(\alpha\)), which refers to the angle between the target and the horizontal plane, as well as the orientation angle (\(\beta\)), representing the angle of position after establishing a coordinate system centered on one's hand. By enhancing the spherical coordinate system, this formula aimed to more accurately model motion in all directions within three dimensions. However, the addition of Sin(\(\theta\)) to the formula did not significantly contribute to movement time prediction [7]. [PERSON]'s team conducted research using VR equipment and discovered that display defects had unexpected effects, specifically related to the change of target depth (CTD). It was observed that prediction time was not solely influenced by the difficulty factor but also by display defects during virtual presentations. Their formula exhibited greater effectiveness in predicting motion time. This work holds significant implications for VR and augmented reality technologies [24]. There is no complete system for comparing the advantages and disadvantages of each 3D model, and each formula has good experimental results within its experimental range. Despite the continuous efforts of researchers, it is still difficult to create a generalized formula to describe reaction time. Many scholars have proposed various ideas and models for different research questions [25; 26; 27]. Nowadays, as virtual devices become more commonplace, there is an increasing interest in studying 3D Fitts' law further. Researchers are using these cutting-edge technologies to learn more about how humans interact with computers [28; 29; 30]. To properly traverse this changing landscape, it becomes critical to recognize and understand the particular circumstances in which these new gadgets are employed. Researchers can effectively meet the varied issues offered by 3D Fitts' law by tailoring their findings and applying pertinent formulas by recognizing the subtleties of usage circumstances [31; 32; 33].