Game:

Doom II: Hell on Earth (1994)

Duration: 0:44

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2

This is not too hard to set up provided there is enough space but I find it quite fun when it occasionally happens on its own!

Revenants randomly choose between two kinds of projectiles: A standard missile that acts the same as all other monster projectiles or a homing missile (distinguished by a smoke trail).
The homing missile moves forward at constant velocity, but also tries to correct its trajectory deviation every 2 gametics (1 gametic = roughly 1/35 of a second) to aim directly at the target's origin. However, the angle correction cannot exceed 16.875 degrees (0.09375π radians for pedants) per 2 gametics in which case it corrects the angle by that much only.
On long and short distances this is not a problem, but when shot from mid-range, sometimes it can happen (when the target is appropriately positioned or gets in a particular position) that the homing missile during its constant movement deviates exactly by the amount it tries to then correct to home in on the target and thus starts "orbiting" it.

Nitrogolem's projectiles in Heretic have homing capabilities too but I have never been able to do this with them since their homing properties seem to be especially set for long range annoyance.

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