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I am trying to let my camera move along a path. The method to first select the camera, then the path and then use Ctrl P > Follow Path works fine.

However, if I select the object, go to Object Constraints and add a Follow Path constraint, the behavior is not the same.

Mainly, in the second procedure my object jumps to a different position once I added the constraint. Why is that and what is the difference between both methods?

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CtrlP > Follow Path parents the selected objects to the active object (=curve), enables Path Animation in the curve's Object data tab in the Properties Editor and adds an F-Curve for the Evaluation Time property. An F-Curve Modifier is added to that F-Curve of type Generator (first order expanded polynomial).

Adding a Follow Path constraint to an object and target it at a curve only relocates the object (apparently to one of the end points of the curve), you need to click Animate Path to add an F-Curve with modifier (1st order exp. polynomial generator again) to the curve object in order to let the object move along the curve.

If you don't want that offset when using a Follow Path constraint, select your object and a curve,
CtrlP > Path Constraint. It adds a Follow Path constraint, creates the F-Curve with modifier for the the curve and does not translate your object.

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    $\begingroup$ When adding a "Path Constraint", in my case however it also jumps my object to a different position. The only option is to really use "Follow Path", which you helped me immensly. $\endgroup$ – Henrique de Sousa Jan 22 '15 at 12:21
  • $\begingroup$ Basically, the constraint assumes the object is at world origin and jumps it to the curve. Translation off the origin translates it around the curve. If you want to use the constraint, don't parent the object, leave it at world origin, and it works fine. Personally, I find it more intuitive to use the constraint and animate the offset of the constraint than to use path animation. $\endgroup$ – Nathan Mar 4 at 17:08

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