Continuous Donut-Pie $k$ CNN Queries

In section , we presented our technique to continuously monitor the $k$ CNN queries where the region is constrained by some angle range and in section  we discussed how we can use CircularTrip to answer continuous $k$ CNN queries with the region constrained by some distance bound. In this section we discuss the $k$ CNN queries in the region which is constrained by both the angle range and the distance.

Figure: Donut-Pie $k$ CNN Query

The user specifies an angle range $\langle \theta_1,\theta_2 \rangle$ and the distance range $d_L$ and $d_U$ where $d_L$ is the lower bound and $d_U$ is the upper bound. The query $q$ is to continuously monitor the $k$ NNs in the region bounded by the defined distance and angle range constraints. The constrained region is shown shaded in Fig. .