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I would like to be able to take a plane, and animate one or more parts of it extending (getting longer over time. I include an image of what I want to happen over time. enter image description here

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You can create this extrusion, then keep the edge selected and CtrlH > Hook to New Object in order to hook these vertices to an empty:

enter image description here

Back to Object mode, select the empty and move it:

enter image description here

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  • $\begingroup$ A thousand thanks, moonboot: it worked! If anyone is interested in how I animated the plane moving, I began by following moonboot's advice above. Then, in Object mode, I set a key frame at frame 1, choosing the "location, rotation, scale and custom properties" option. Then, still in Object mode, I moved the empty, which made the plane grow longer. Then at key frame 30, I added another key frame, choosing the "location, rotation, scale and custom properties" option again. When I played back the animation, it looks as though the plane is growing longer along one of it's edges. FABULOUS! $\endgroup$
    – putter
    Jun 17, 2022 at 4:12

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