I am working with a complex 2-dimensional shape (similar to this image), and would like to pass a single, continuous line through it to aid future analyses. I have distinct data points on each pixel of this object and need to keep them independent and ordered.
After a while of trying to fit a line to it with different compromises/unfolding, I remembered that in linear regression, you map data points to higher dimensions, turning data that cannot be fitted by a line, into data that can be fitted by a line.
Is it possible to do a similar thing in this case? Would it be reasonable for me to try to map this object into 3-4+ dimensional space to get a line through it/ make it linear?
Does the fact that there are several "end points" mean that it would not be possible, at all, to linearize my object?
Edit (to add details):
I have data points on an object surface that looks like the above linked image. They are in a complex 2-dimensional shape, and analyzing them effectively is not practical while in this shape.
I want to transfer my data onto a line. On this line, I would like to keep the order of my data, and relationships between points.
I would like to do this while losing as little data as possible.
An example of what I was thinking of:
Circle and 5 lines drawn in X,Y.
Cut the circle at one point and unfold it (circle + 5lines would turn into line + 5 lines).
Rotating my object in Z, and align my 5 lines to the 1 main line, as to have all lines in the same vertical plane as my object.
Collapse my points.
This method is very inoptimal, and I would be:
1) losing a lot of data from it
2) disrupting some of the data on my main line (or circle).
Another example:
"Blow up" my structure so as to have each line represented in two ways (center to edge, and edge back to center).
Collapse the structure onto a line by disconnecting the circle at all line points
What I would have is: Arc segment 1 > line1 up > line1 down > Arc segment 2 > line2 up > line2 down > Arc segment 3... etc
This method would make me lose the proximity relationship between the points on my arc.
I was wondering if mapping this structure onto higher dimensions before transforming it into a line would yield better results.
I don't know if I understood your problem, but if you have a sequence of 2-D points $$ (x_1,y_1), (x_2,y_2), \ldots, (x_n,y_n), $$ then you can turn them into 3-D points like that: $$ (x_1,y_1,h), (x_2,y_2,2h), \ldots, (x_n,y_n,nh), $$ where $h$ is any non-zero quantity. Could that help you?