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I have my camera constrained to a curve with Follow Path and it stays with its origin on the path exactly. But when I add a strong curve to the path the camera starts going off the path. It is cutting corners for some reason. Has anyone had experience with this problem?

I found I can make the path I want and then select all the points in edit mode and subdivide them until the camera comes back to the path on strong turns. But this is not a functional solution if I want to quickly edit the path later.

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Perhaps you have the curve resolution set to low? The curve will render at the preview resolution when it is set to zero. This will make any objects tracking its length cut corners as they evaluate the curve in a parametric manner.

curve low res path

vs

curve high res path

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