# Time to Capture a Moving Target Travelling along a Circular Trajectory
## 1 Introduction  
The time to move a physical stylus from a home position to a stationary target depends on the size of the target and the distance from the home position to the target. The longer the distance between the home position and the target, the longer the movement time. On the other hand, the smaller the size of the target, the longer the time required for the movement. The proposed law based on this principle is Fitts' law, which is expressed in Equation (1) [1]. It was found that the time to move the physical stylus from the home position to the stationary target is linearly proportional to \(log_{2}(2A/W)\), called the difficulty index (ID), where \(W\) is the width of the target and \(A\) is the distance from the home position to the target. MT is the average time to complete the movement. a and b are constants that depend on the choice of input device and are usually determined empirically. The parameter a defines the intersection on the \(y\) axis and is often interpreted as a delay. The b parameter is a slope and describes an acceleration. \[\text{MT}=\text{a}+\text{b}\;log_{2}\left(2A/W\right) \tag{1}\]  
[PERSON] law is applicable to the actual physical movement of the hand to a target, such as the movement of a cursor to a screen target [2]. It is also applicable to movement as coarse as a foot reaching for a pedal and as fine as manipulation under a microscope [3, 4]. This generality provides the law great value in allowing designers to determine keyboard layouts and target sizes in a wide variety of circumstances [5, 6, 7]. The theoretical basis for Fitts' law was developed and expended. Several modifications were also proposed such as a model considering the length of probe [8], a model fittable for the low index of difficulty [9] and models not using a logarithmic function [10, 11]. However, most of the studies on [PERSON] law were performed on the movement time for approaching a stationary target. There have been just few studies of Fitts' law considering moving targets. [PERSON] et al., (1980) first studied the time it takes to catch a moving target based on the Fitts' law [12]. They found that even with a moving target, the size of the target is inversely proportional to the time taken, and the distance between the target and the home position is proportional to the time taken. A time prediction model considering the speed of a moving target were proposed as in Equation (2). \[\mathrm{MT}=\mathrm{c}+\mathrm{d}A+\mathrm{e}(V+1)(1/W-1) \tag{2}\]  
In Equation (2), \(\mathrm{c}\), \(\mathrm{d}\), and \(\mathrm{e}\) are constants, which are determined according to the characteristics of the work or worker's skill, and are derived through experiments. The velocity variable \(V\) at which the target moves, which does not exist in Equation (1), is additionally considered. The number of empirical constants were increased and the logarithmic function was not used  
To follow the form of the traditional Fitts' law, [PERSON] (1991) proposed a new modified model by reanalyzing [PERSON] et al. 's experimental data without a separate experiment. Equation (3) is a mathematical model proposed by [PERSON] (1991) [13]. MT, \(\mathrm{a}\), \(\mathrm{b}\), \(A\), and \(W\) in Equation (3), are the same as the variables and constants of the Equation (1) and \(V\) is the variable representing the moving speed of the target. \[\mathrm{MT}=\mathrm{a}+\mathrm{b}log_{2}\left((A\pm V/K)/(W/2-V/K)\right) \tag{3}\]  
This model is a one-dimensional model and describes the time it takes to capture the target by the physical styles when the target approaches or moves away from the home position of the physical styles. The numerator of the logarithmic function of this model, \(A\pm V/K\), indicates the effect of the moving speed of the target on \(A\). The A is the distance between the target and home position before the physical stylus movement begins. Actual \(A\) is shortened as the target approaches the origin at a constant speed. The shortened distance was assumed to be \(A-V/K\).