\section{Discussion}
\label{discussion}

We evaluated six visual hand renderings, as described in \secref{hands}, displayed on top of the real hand, in two virtual object manipulation tasks in \AR.

During the \level{Push} task, the \level{Skeleton} hand rendering was the fastest (\figref{results/Push-CompletionTime}), as participants employed fewer and longer contacts to adjust the cube inside the target volume (\figref{results/Push-ContactsCount} and \figref{results/Push-MeanContactTime}).
Participants consistently used few and continuous contacts for all visual hand renderings (Fig. 3b), with only less than ten trials, carried out by two participants, quickly completed with multiple discrete touches.
However, during the \level{Grasp} task, despite no difference in \response{Completion Time}, providing no visible hand rendering (\level{None} and \level{Occlusion} renderings) led to more failed grasps or cube drops (\figref{results/Grasp-ContactsCount} and \figref{results/Grasp-MeanContactTime}).
Indeed, participants found the \level{None} and \level{Occlusion} renderings less effective (\figref{results/Ranks-Grasp}) and less precise (\figref{results_questions}).
To understand whether the participants' previous experience might have played a role, we also carried out an additional statistical analysis considering \VR experience as an additional between-subjects factor, \ie \VR novices vs. \VR experts (\enquote{I use it every week}, see \secref{participants}).
We found no statistically significant differences when comparing the considered metrics between \VR novices and experts.

All visual hand renderings showed \response{Grip Apertures} close to the size of the virtual cube, except for the \level{None} rendering (\figref{results/Grasp-GripAperture}), with which participants applied stronger grasps, \ie less distance between the fingertips.
Having no visual hand rendering, but only the reaction of the cube to the interaction as feedback, made participants less confident in their grip.
This result contrasts with the wrongly estimated grip apertures observed by \textcite{al-kalbani2016analysis} in an exocentric VST-AR setup.
Also, while some participants found the absence of visual hand rendering more natural, many of them commented on the importance of having feedback on the tracking of their hands, as observed by \textcite{xiao2018mrtouch} in a similar immersive OST-AR setup.

Yet, participants' opinions of the visual hand renderings were mixed on many questions, except for the \level{Occlusion} one, which was perceived less effective than more \enquote{complete} visual hands such as \level{Contour}, \level{Skeleton}, and \level{Mesh} hands (\figref{results_questions}).
However, due to the latency of the hand tracking and the visual hand reacting to the cube, almost all participants thought that the \level{Occlusion} rendering to be a \enquote{shadow} of the real hand on the cube.

The \level{Tips} rendering, which showed the contacts made on the virtual cube, was controversial as it received the minimum and the maximum score on every question.
Many participants reported difficulties in seeing the orientation of the visual fingers,
while others found that it gave them a better sense of the contact points and improved their concentration on the task.
This result is consistent with \textcite{saito2021contact}, who found that displaying the points of contacts was beneficial for grasping a virtual object over an opaque visual hand overlay.

To summarize, when employing a visual hand rendering overlaying the real hand, participants were more performant and confident in manipulating virtual objects with bare hands in \AR.
These results contrast with similar manipulation studies, but in non-immersive, on-screen \AR, where the presence of a visual hand rendering was found by participants to improve the usability of the interaction, but not their performance \cite{blaga2017usability,maisto2017evaluation,meli2018combining}.
Our results show the most effective visual hand rendering to be the \level{Skeleton} one.
Participants appreciated that it provided a detailed and precise view of the tracking of the real hand, without hiding or masking it.
Although the \level{Contour} and \level{Mesh} hand renderings were also highly rated, some participants felt that they were too visible and masked the real hand.
This result is in line with the results of virtual object manipulation in \VR of \textcite{prachyabrued2014visual}, who found that the most effective visual hand rendering was a double representation of both the real tracked hand and a visual hand physically constrained by the \VE.
This type of \level{Skeleton} rendering was also the one that provided the best sense of agency (control) in \VR \cite{argelaguet2016role,schwind2018touch}.
