Before revision:  This work initiates the study of the computation and maintenance of the \mathcal{O}-{\rm Kernel } of a polygon P as we rotate the set \mathcal{O} by an angle \theta, denoted \mathcal{O}-{\rm Kernel }_{\theta}(P).
After revision:  This work initiates the study of the computation and maintenance of the \mathcal{O}-{\rm Kernel } (P) as we rotate the set \mathcal{O} by an angle \theta, denoted \mathcal{O}-{\rm Kernel }_{\theta}(P).