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How can I animate the position of a Bézier curve segment as shown in the images below?

Beginning of animation End of animation

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You can 'Hook' the control point of the curve by pressing Ctrl+H while in edit mode, with the point of the curve you want to move selected and choosing 'Hook to new object'. Adding the hook via this shortcut is just a quick way of setting up the hook modifier, which will be present on the curve after the shortcut is pressed.

This will create an 'Empty' object which can be animated. The control point of the curve will be parented to the position of the empty and will follow the empty wherever you animate it to.

enter image description here

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    $\begingroup$ it might be obvious to some but you can add multiple hooks to a single curve and then animate each independently. Amazingly powerful. $\endgroup$ – rob Mar 27 '17 at 12:34
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To directly animate a curve's segment, without hooks or shapekeys, you can enable the AninAll addon. It allows you to create keyframes for each control point.

enter image description here

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Another way you could do this is with a shape key:

Demonstration of using a shape key to create animation

Steps:

  1. In Object Mode, select the Data tab of the Properties panel. Under Shape Keys, create two keys: the basis and a key for your animation.
  2. With the second key selected, enter Edit Mode and perform your transformations.
  3. Animate the Value of the shape key as desired.

Neither this method nor the hook method is inherently superior to the other. Here are some of the advantages and disadvantages of this method:

  • Advantage: You can create more complex transformations. For example, you could turn a sphere into a gear by scaling out every other vertex, and then animate a single shape key.
  • Disadvantage: You have less control. With a hooked empty, you can control the empty's location with all the standard animation tools—including linking it to a path, using constraints, etc. With a shape key, you can only animate a factor value that linearly interpolates "to" and "from."
  • Advantage: You can easily combine multiple shape keys. For example, on a circular path, you could have a "Gear" shape key (as described above) and also a "Pinch" shape key that squishes the path along some axis. It's easy to combine these with shape keys, and less so with hooks (you would have to fiddle around with the Force values).

As usual, you should know your options, consider the situation, and decide which method is best in that particular case.

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  • $\begingroup$ This no longer seems to work in 2.8 final. Shapekeys no longer seem to care about curves as far as I was able to tell. (Hopefully I am wrong, and am just doing it incorrectly.) $\endgroup$ – Martin S. Stoller Aug 24 at 16:22

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