After Effects, can I parent specific points on a path to a null object? Can I select two points on a rectangle and have them follow a null object's movement? I want one side of several shapes to stay put, and the other side to track an eased movement.

I believe stib has answered this, but as per tomh's comment I'll post a gif anyway to explain. There is probably an even simpler way to do what I'm trying to.



The grey box on the right is swiped right to left with a steep ease curve. The boat graphic needs to scale with the same curve as the X position of the grey box so they move together. Making the first ease is simple, but I had to 'bake' the second curve frame by frame, then smooth it as best I could, but the OCD in me wants the curve to be identical! It also might be useful for future projects.

I thought, if I could make a rectangle where the two left paths are stuck on the far left of the comp, and the two right paths track the movement of the grey box, that this would make my perfect, mask for the scaling boat layer.
Use the puppet tool to do this. Place puppet pins on the rectangle and they will allow you to distort the rectangle to follow them. You may need other puppet pins or starch to hold the rest of the rectangle in place. 

Then connect their position to the null using an expression:

thisComp.layer("&lt;the name of the null&gt;").transform.position


or if the nulls are parented to something else:

thisComp.layer("&lt;the name of the null&gt;").toWorld(thisComp.layer("&lt;the name of the null&gt;").transform.anchorPoint)


EDIT Are you asking whether you can attach specific points on a vector path to a null? Not natively, but there is a third party plugin that lets you animate individual path points: http://aescripts.com/bao-mask-avenger/ Fairly pricey though.