# Movement Time for Pointing Tasks in Real and Augmented Reality Environments
## 1 Introduction  
One of the most commonly adopted laws for discussing the behaviors of a human interacting with an object is [PERSON]'s law. This law estimates the time required to touch a target of a certain size over a certain distance. It suggests that the movement time (MT) of a body part from one location to a target depends on the distance of movement and the size of the target. A term, index of difficulty (ID) of the movement, was defined, incorporating the movement distance of the body segment and target size (W), and MT can be estimated using the following equations [1, 2]. \[\text{MT}=a+b\text{ ID} \tag{1}\]  
\[\text{ID}=\log_{2}\left[\frac{2\times distance}{W}\right] \tag{2}\]  
where \(a\) and \(b\) are constants to be determined. ID itself is unitless, and a unit of bit is given that may be regarded as the amount of information transferred via the movement. This implies that the difficulty of a target pointing or tapping task may be quantified using the information metric bit. [PERSON]'s law was originally applied to study the MT of hand and finger movements. These movements may include tapping, pointing, tracking, positioning, and other hand-object contact tasks. They are required in human-machine and human-object interactions. In addition to hand and finger movements, scientists have applied this law to study the MT of other body segments such as the head [3; 4; 5; 6], leg [7; 8], foot [9; 10; 11; 12], and trunk [13; 14]. [PERSON]'s law has been applied not only in ordinary environments but also in special environments such as underwater [15; 16; 17]. In addition to assessing the MT of body parts for healthy people, [PERSON]'s law has been adopted to assess the body functions of patients for medical and healthcare purposes [18; 19]. Since [PERSON]'s law was announced, many variants have been proposed to accurately quantify ID and thus MT. Both [PERSON] [20] and [PERSON] [21] proposed revised equations to predict the MT of moving the right index finger from the starting point to a target. These authors suggested using the terms log2[(A/W) + 0.5] and log2[(A/W) + 1], respectively, to replace the original ID definition. They mentioned that such revision was necessary considering the deviation of the original definition of ID from Theorem 17 of Shannon's information theory [22]. [PERSON] and [PERSON] [23] found that MT is linearly related to the square root of the movement amplitude for ballistic movements when the ID is small. This implies that considering the target size is no longer necessary in estimating the MT under such circumstances. [PERSON] [24] investigated the use of an effective target tolerance in an inverted Fitts task. Considering the finite width of the finger probe, he believed that the original [PERSON]'s law could overestimate the ID of the movement. He introduced a finger index pad size into the original ID equation. [PERSON] and [PERSON] [25] found that the MT of the 3D pointing task was not satisfactorily explained by the original [PERSON]'s law because the duration of the 3D movements was affected by the direction of the target. They introduced a directional parameter (\(\theta\)) of the pointing movements into the original [PERSON]'s law. Similar results were reported by [PERSON] and [PERSON] [26]. They added a second directional parameter (\(\theta_{2}\)), in addition to the \(\theta\) in [PERSON] and [PERSON] [25], to delineate the 3D movement of finger touch. [PERSON] et al. [27] conducted a study to test the effects of probe length on the MT of pointing tasks. They suggested introducing a function of probe length plus a logarithm of the amplitude into the original Fitts's law. They claimed that such additions allow for better prediction of the MT for a worker using a long tool such as a screwdriver or a crowbar. Virtual reality (VR) devices have been widely used for both entertainment and commercial purposes. VR strives to immerse users completely within a designed scenario so as to provide an intuitive link between a computer and the user [28].