I am interested in approximating the time evolution of 2D curves. Here's an illustration:
An issue that arises when naively making this approximation as illustrated above, is that as one increases the number of "line segments" used to approximate the curve, then one is also introducing extra "flexibility". That is, curves with less approximating line segments are stiffer than curves with many approximating line segments.
I am interested in knowing what the theory that handles analytical solutions of simple continuous cases is called. I realize it would be a topic under solid mechanics/elasticity, but are there any more specific subject headings I might use to look up in textbooks? (an idea I have from my novice intuition is that this is going to be some sort of a PDE problem -- the elasticity is very simple: so I don't think it needs a tensor description?)
I am also interested in knowing if there are any textbooks that explore the sort of numerical approximation I illustrated? (an idea I have from my novice intuition is that it is not finite element method)