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So I watched that tutorial and I hate to say this but it's not very good and it isn't teaching you efficient or even proper techniques for doing what needs to be done to run the morphs. His terminology is also not consistent with the real AE terminology. He eventually stumbles into a solution but the technique isn't anything that you would want in your bag o' tricks.
Here's are some things you should consider with trying to animate paths. First, every closed path has a first vertex. Second, if you are animating the position of the vertices the points are going to move in a straight line from first vertex to first vertex and continue around the path so the direction you drew the path in and the position of the first vertex in the first shape and the second shape must make sense. Let's take changing a square or a circle to a triangle. The square and the circle have 4 points. The triangle has 3. To make things smoothly from square or circle to triangle you'll have to add an additional point to the triangle. The first vertex is the bigger one. Here's the circle as drawn with the Ellipse tool:
Here's the triangle:
Notice that the first vertex of both is at the top center of the shape. Now here's the square:
The First Vertex is in the top right corner.
If you try and convert the square to a triangle then the top right corner of the square is going to move to the point of the triangle, the bottom right is going to move up, the bottom left is going to move right to match the bottom right corner of the triangle, and the top left corner is going to move down resulting in a very odd distortion in the transition.
If you set the bottom left corner of the square as the first vertex then you'd get a different kind of motion:
A smoother transition that looked more normal can be achieved by setting the adding a new point to the square path at the center of the first line and setting that to first vertex:
Then adding a couple of points to the sides of the triangle (stroke removed so you can see the points better:
Then you would get a transition that looked like this half way through.
If the first vertex of the square was not reset to the middle of the top line then you would get this mess and wonder what happened:
The reason the triangle is moving is that you don't have your vertices lined up properly because the tutorial didn't teach you what you need to know. Be very suspicious of tutorials and try and vet the authors so you don't go down blind alleys.
First of all, thank you for the very fast and detailed answer. I searched a lot of tutorials, and they all basically told me to do the same thing.
I saw that one vertex was different and I did understood that it would shape the transition, but couldn't figure out how to change it.
Thank you very much, I'll try it right now, but it already makes a lot more sense.