How do paired animations work? Like I’ve heard that even getting to characters to pass an apple can be tough, so how do games manage to make throws and grapples and stuff look good/right?

Each participant in a synchronized animation acts using angles and positioning relative to some common point and performs all actions within a specific space around that common point. If one participant is off-angle or off-position, things won't look right. For the animation to look right, all participants must keep the exact proper distance and orientations while the synchronized animations all play at the same time.

This means that, as long as the environment and conditions are conducive (e.g. the ground is flat, there's enough space, there are no objects to clip through, etc.) to choose an in-game location to assign as the synchronized animation's common point, we can play the synchronized animation on the participants and things should line up. Any mismatch and things will look weird. If the ground is not flat, for example, we could see our characters standing in midair. If the positions are not correct, a punch could visibly miss by a mile. If the angles are not correct, it would look like a phantom judo throw. If the environment is not clear of other objects, we would see our participants walking/flying/falling through seemingly-solid objects. Only when all of the conditions are met will the synchronized animation look right.

The TLDR is that every synchronized animation was created with very specific dimensions, angles, and positions in mind. If we can recreate an environment with those dimensions, angles, and positions, we can play the synchronized animations on the participants and they'll line up. In games where you can perform synchronized animations in arbitrary locations (e.g. executions in Call of Duty), the engineering team has built a system to find good places near the position and orientation of the action that meet all of the necessary criteria in order to play the synchronized animation.

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