Centroids describe the average of all the discrete points on a curve (in 2d space), but why is it that the center of a circle or ellipse is what it is? Moreover, is it possible to determine the center (not centroid) of more irregular closed paths (in 2d for simplicity's sake)? Is it analogous to a center of mass? Is it possible to calculate it given a set of functions that describe a curve?
Any guiding info on this issue would be very much appreciated.
A curve is represented by a continuous map $$\gamma : [a,b] \to \Bbb R^3 : t \mapsto (\gamma_x(t), \gamma_y(t), \gamma_z(t))$$ for some interval $[a,b]$. The curve is closed if $\gamma(a) = \gamma(b)$. The centroid of the curve is given by $$C_c = \dfrac {\int_a^b \gamma(t)\, dt}{b-a}$$
If you define some mass density $m(t) : [a,b] \to \Bbb R$ along the curve, the "center of mass" is given by $$C_m = \dfrac {\int_a^b m(t)\gamma(t)\, dt}{\int_a^bm(t)\,dt}$$
As you can see, the centroid is just the center of mass when the mass density is constant: $m(t) = M$ for all $t$.