Algorithm 3 Strategy 3: Predictions with head gaze and threshold on eye-gaze angle.

Input: Hand position $P_h\in {\mathbb{R}}^3$, Gaze direction vector $G_d\in {\mathbb{R}}^3$, hand threshold ${\lambda }_d$, head gaze threshold $\alpha$.

Output: Best point $P\ast$ in the set of $P_i$, $i=1\dots 16$.

function ST3($P_h,G_d,P_{prev}$)

$p_{bh}\leftarrow$ best point position from the KD tree.

3:     if ($\left|\right|\left((P_{bh},P_h)\right||<{\lambda }_d)$ then

$P\ast \leftarrow P_{bh}$

else

6:         build a list of points P in the view frustum of HMD with ${\alpha }_i<\alpha$.

if $P=\varnothing$ then

$P\ast \leftarrow P_{prev}$

9:         else

$p_{next}\leftarrow$ point with $min\left({\alpha }_i\right)$ from the list P.

$P\ast \leftarrow P_{next}$

12:         end if

end if

$P_{prev}\leftarrow P\ast$

15: end function