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deterministic physics engine

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In isolation, all of them should be deterministic, as long as you use fixed time steps (e.g., physics always steps at 0.01 seconds rather than at the frame rate). And I think all of the popular engines support fixed time steps. Even in game, as long as you can guarantee deterministic inputs and used fixed time steps, the physics engine should produce deterministic results.

You can get into trouble if you have nondeterministic things going on that affect physics. For example, on a single player game, if there are kinematic objects being animated via a traditional keyframe animation system and the keyframe animation is updated with the frame rate, then all of a sudden your kinematic objects have nondeterministic motions, and so collisions with these objects become nondeterministic resulting in nondeterministic physics. Player inputs are by their very nature nondeterministic since humans aren't capable of deterministic motion. Networking stuff for multiplayer also tends to be heavily nondeterministic and dead-reckoning algorithms will predict different things each time, even if the player happens, somehow, to play the game in exactly the same way, precisely, each time.

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