Does top spin and bottom spin affect the cue ball's path after hitting a cushion?

Given your expertise I always must defer to you in situations such as this. So that means I'm wrong when I say the CB only curves once.
In general the cue ball will have some curve immediately after it hits any object. If you play a 3-cushion shot with follow off a ball, the cue ball will curve after it hits the object ball, and then it will curve after hitting each cushion. I think "the CB curves only once" is correct only if you have a lot of other explanation.

Does top spin and bottom spin affect the cue ball's path after hitting a cushion?

The normal state of the cue ball in motion is to be rolling smoothly on the cloth.

If the cue ball has spin that makes it curve, it is not rolling smoothly on the cloth. It will curve until it is rolling smoothly on the cloth.

While the cue ball is curving, that curve is a parabola.

There is no straight part of the path along the tangent line. The cue ball starts curving immediately after contact with an object ball.
Given your expertise I always must defer to you in situations such as this. So that means I'm wrong when I say the CB only curves once.

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