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Mathematically, this piece of paper is called a hyperplane.

Many safety check algorithms work by generating this hyperplane at a single point in time.

However, each time the robot moves, a new hyperplane needs to be recomputed to perform the safety check.

Instead, this new technique generates a hyperplane function that moves with the robot, so it can prove that an entire trajectory is collision-free rather than working one hyperplane at a time.

This function is an equation that describes where the hyperplane needs to be at each point in the planned trajectory so it remains collision-free.