# Movement Time for Pointing Tasks in Real and Augmented Reality Environments
## 1 Introduction
VR users may see the virtual image displayed and interact with virtual objects using one or two handheld remote controllers, which may encompass push buttons, touch pads, joysticks, and so on. They may also manipulate one or two virtual hands and/or raycasting to \"contact\" virtual targets. Augmented reality (AR), on the other hand, deals with the combination of virtual objects and real environments [29; 30; 31; 32; 33; 34]. AR users may see the real world with some virtual, unreal objects in their vision. They may also interact with virtual objects using similar ways as in a VR system. Alternatively, they may \"contact\" virtual objects using hands directly, such as touching a virtual button using their finger to activate an input command [35; 36]. Human interaction with virtual objects is, apparently, quite different from a human interacting with real objects [37]. When studying the MT for VR and AR users, the original [PERSON]'s law and its variants may no longer be valid because human-object interactions are replaced by the more complicated human-virtual object interactions. One of the major issues in interacting with a virtual object is the depth perception deficiencies of stereoscopic displays [38; 39; 40]. The literature [41] has shown that the accuracy rate for localizing targets in virtual environments using stereoscopic displays is relatively lower than that in the real world. It was also found that participants tend to overestimate the depth of targets in virtual environments [42; 43]. If users cannot locate an object accurately, they need more time to perform a task requiring contact of their hands with objects. This will lead to inferior performance [40; 44; 45]. Investigations of the MT in virtual environments and human-virtual object interaction tasks have been reported. [PERSON] and [PERSON] [46] performed an MT study of 3D object manipulation techniques based on a virtual hand metaphor in a tangible augmented reality environment. [PERSON] et al. [38] reported that, when handling a virtual object in an immersive virtual environment, both ID and interaction of target position and movement distance were significant on the MT. They found high selection error along the view axis as compared to those in the other axes. [PERSON] and [PERSON] [45] found that their participants needed significantly more time to select virtual 3D targets than to select identical physical targets. [PERSON] et al. [47] indicated that the MT in operating a handheld controller to perform a target positioning task in a virtual environment may be split into the acceleration, deceleration, and correction phases of hand movement. They included amplitude, object size, and target tolerance in their MT equations and suggested calculating the MT in different phases using different equations. [PERSON] et al. [48] proposed an extended Fitts's law model to consider the effects of the inclined angle of the virtual targets. Even though there were studies investigating the MT dealing with hand-virtual object interactions, there are still many untouched issues. For example, users have no haptic feedback when \"contacting\" a virtual object. Lacking haptic feedback could make fine adjustment of the hand/finger before touching the virtual target difficult and thus lead to longer MT than that of contacting a real target. This has not been discussed in the literature and is the main issue considered in the current study. We believe that this could be attributed to a tactile (T) factor. This factor is determined by the presence of tactile feedback. This factor should be incorporated in [PERSON]'s law (Equation (2)) so as to provide good fitness of the MT when the pointing tasks are performed on virtual targets. Equations (3) and (4) show the inclusion of T in the function estimating the MT:  
\[\text{MT}=f(\text{distance},\text{W},\text{T}) \tag{3}\]  
\[\text{MT}=a+b\ \log_{2}\left[\frac{2\times\text{distance}}{W}\right]+f(T), \tag{4}\]  
where \(f(T)\) is a function of the tactile factor \(T\).