What is the expression for the centroid of an arbitrary parameterized space curve?

Question

Let $\gamma:t\in[a,b]\rightarrow (x(t),y(t),z(t))\in \mathbb{R}^3$ be a parametrized curve

I am looking for the expression of the centroid of the curve $\gamma$ in a good reference. (I didn't find a good one.)

Answer 1

You can use the standard definition of center-of-mass: $$ r={\int_a^b\gamma(t)\,|\dot\gamma(t)|\,dt\over \int_a^b|\dot\gamma(t)|\,dt}, $$ where: $\dot\gamma(t)=(\dot x(t), \dot y(t), \dot z(t))$ and $|\dot\gamma(t)|=\sqrt{\dot x^2(t)+\dot y^2(t)+\dot z^2(t)}$.