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?
Best Answer
I think the keyword you need to find literature on your problem is morphing. There is an extensive computer graphics literature on this. Below is a figure from one paper selected almost at random: "Morphing Using Curves and Shape Interpolation Techniques" Johan, H.; Koiso, Y.; Nishita, T. in Computer Graphics and Applications, pp. 348 - 454, 2000. The 4th figure in the sequence could serve as the average you seek.
Another reference is "Multiresolution Morphing for Planar Curves," Hahmann, S. and Bonneau, G.-P. and Caramiaux, B. and Cornillac, M., Computing 79 (2007) 197-209. Using the key search terms, and Google Scholar with these two references, should bring you to a wealth of relevant literature.
See also this related MO question on the distance between two curves.