How does one derive the centroid formula for multiple shapes?

Question

I know how to derive the formula for the centroid of n sets of finite points, each with $m_k$ points and centroid $C_k$. The formula is: $$C=\frac{\sum_{k=1}^{n}C_km_k}{\sum_{k=1}^{n}m_k}$$

However, when the sets of points become shapes, I'm unsure of what to do. An intuitive way of thinking about it would be that the ratio of areas is equal to the ratio of points, thus yielding the new formula: $$C=\frac{\sum_{k=1}^{n}C_kA_k}{\sum_{k=1}^{n}A_k}$$ Where $A_k$ is the area of the kth shape. However, I'm not entirely comfortable due to the presence of infinities despite knowing single-variable calculus and limits. Is there another, more rigorous way of deriving the formula above, or at least make my intuitive idea rigorous?

Also, I've noticed that quite a few questions relating to centroids have received answers using the concept of a centre of mass. If you can, please refrain from using some ideas in your answers, as I'm looking for a purely mathematical one. However, if this is impossible, then I suppose it's fine.

Answer 1

The term "Centroid" specifically means the "center of geometry". That is

• For $0$-dimensional objects - that is, a countable set of discrete points, the average of the point locations: $C = \lim_{n \to N} \frac 1n\sum_{i=1}^n x_i$. If $N$ is finite, the limit is not needed, but it is required when $N = \infty$.
• For $1$-dimensional objects - curves, the centroid is the center of length: Given a curve $\gamma(t)$ defined on $t \in [a,b]$, the centroid is given by $$C = \dfrac{\int_\gamma \gamma\,ds}{\int_\gamma 1\,ds} = \dfrac{\int_a^b \gamma(t)|\gamma'(t)|\,dt}{\int_a^b|\gamma'(t)|\,dt}$$
• For $2$-dimensional objects - surfaces, the centroid is the center of area: Given a surface $\alpha(u,v)$ defined for $(u,v) \in D$ for some $D \subseteq \Bbb R^2$, then $$C = \dfrac{\int_\alpha \alpha\,dA}{\int_\alpha 1\,dA} = \dfrac{\int_D \alpha(u,v)|J_\alpha(u,v)|\,dudv}{\int_D |J_\alpha(u,v)|\,dudv}$$where $J$ is the Jacobian of $\alpha$.
• For $3$-dimensional objects - regions in space, the centroid is the center of volume: Given a region $S$ of space, and letting $\vec r$ be the position vector, $$C = \dfrac{\int_S \vec r\,dV}{\int_S 1\,dV} = \dfrac{\int_S (x,y,z)\,dxdydz}{\int_S 1\,dxdydz}$$

These should be taken as definitions rather than anything that must be proved. Indeed, as there is no other definition for Centroid, there would be no way to "prove" it. Though, the similarities between the definitions hint that it should be possible to come up with a universal definition that applies to all dimensions, of which these are the results of applying that definition to various dimensions. This is true, but is very abstract to accomplish.

"Center of Mass" is used for weighted summing. If there is a function $w$ on the appropriate domain, we get the following definitions for center of mass, depending on the dimension: $$C_m = \lim_{n \to N} \dfrac{\sum_{i=1}^n w(x_i)x_i}{\sum_{i=1}^n w(x_i)}$$ $$C_m = \dfrac{\int_a^b w(\gamma(t))\gamma(t)|\gamma'(t)|\,dt}{\int_a^bw(\gamma(t))|\gamma'(t)|\,dt}$$ $$C_m = \dfrac{\int_D w(\alpha(u,v))\alpha(u,v)|J_\alpha(u,v)|\,dudv}{\int_D w(\alpha(u,v))|J_\alpha(u,v)|\,dudv}$$ $$C_m = \dfrac{\int_S w(x,y,z)(x,y,z)\,dxdydz}{\int_S w(x,y,z)\,dxdydz}$$