One day the question arose: how to determine centroid of annulus to be in the specified place?
Of course, it can't be usual centroid as a mass center.
We can use symmetry, radus and concentric circle for geometric definition.
But this is not suitable for more general cases:
Intuitively, the center is supposed to be in a given location, but is it possible to give it a meaningful definition?
I experimented a lot with Mathematica. It became clear that:
- The center must be inside the figure (kind of medoid)
- The distances used to define it $d(p,q)$ are the shortest (in the Euclidean sense) inside the figure, as shown:
The classic definition of a medoid (using distance summation) yields undesirable point. As a result, I have formulated the following simple definition:
- Find the boundary of the figure $\partial F$
- For each point we find the maximum and minimum distance to the boundary (inside figure)
- The point that minimizes the difference between them is the desired centroid
$$\arg \min _{p \in F} (\max _{q \in \partial F} d(p,q) - \min _{q \in \partial F}d(p,q) )$$ You can use variance instead of difference, but it doesn’t change the essence. To put it mildly, this centroid is in the most equal position with respect to the boundary of the figure.
This formulation is retained for figures with holes and easily generalizes to any dimensions and several centroids.
I wonder if something like this is already being used in ordinary or computational mathematics, and what other properties of this definition can be explored?