TITLE: How can I find the average of two 2D curves?
QUESTION [10 upvotes]: I have a curve interpolation problem. 
I have two closed curves that are defined on an X,Y plane.  How can I define a 3rd curve that is the average of those two?  Programmatically, I have a list of points for each curve, let's say N1 for curve 1 and N2 for curve 2, where N1 != N2 (most likely).
When I say 'average', initially, I would like the contribution of each curve to the final curve to be identical.  Eventually, I would like to be able to weight the contributions of each curve (i.e., have my interpolated curve be 'closer' to one curve than another).
How can I go about doing this?  
In a 1D case, I believe that the problem is somewhat easier, like I could solve it using some kind of projection (although my linear algebra is really rusty at this point).  Is that intuition somewhat correct, and therefore can be extended to the 2D case?

REPLY [2 votes]: Excellent question! The answer was not known last year.
The following recent paper may have the answer you need:
"Average curve of n smooth planar curves" by
Sati, Rossignac, Seidel, Wyvill, Musuvathy. 
Computer-Aided Design.
Volume 70 Issue C, January 2016,
Pages 46-55.
http://dl.acm.org/citation.cfm?id=2831700
It presents a fast, linear cost, construction that works very well when the curves are "compatible" (reasonably parallel). The paper also talks about how to use weighted averaging for blending and morphing between several curves.
The solution is independent of the order of the input curves and does not assume any given parameterization (like the timing that one of the comments mentions).

Jarek