Is the circling pattern flown by law enforcement helicopters tracking potential suspects a tactical decision?
Answer by Thomas Zerbarini:
Actually, it takes more effort, energy and concentration to hover a helicopter than to keep it moving. Some more advanced helicopters have auto hover autopilots that may or may not couple to a mounted stabilized camera system.
Here is a great explanation of Translational Lift that is the main factor in the benefit of a helicopter to keep moving instead of stay in a hover:
Here are two images showing how airflow acts through a rotor of a helicopter in slow flight and forward flight:
Here is an image of how air moves in a hover:
Here is a video from a Helicopter CFI explaining it a little bit: