What is the corresponding term for center of mass for a two-dimensional shape?

Question

What is the term for the point on closed surface with no holes which would correspond to the point on that surface directly above the center of mass for a 3-dimensional figure of constant density and constant thickness projected outward at 90 degrees from such a surface? To give a simple example, for a symmetrical shape, such as an ellipse, it would be the point half way between the two foci. Additionally, is there a simpler way to define this term than the way it is done here? If the shape is constrained to be convex, does that simplify the definition?

Answer 1

If you were to call it a center of mass, others would understand you. Sometimes it is called a "centroid."

If you know calculus, it is pretty easy to define a centroid for well-behaved regions.

Generally, if a reasonable (in the sense that it can be integrated over) $n$-dimensional volume $V$ has mass $\rho(\vec x)$ at each point $\vec x$, then we can define the center of mass to be $$ \vec{R} = \frac{1}{M} \int_V \rho(\vec r) \vec r dV,$$ where $M$ is the total mass of the volume.

Answer 2

It's just called the center of mass. There's nothing in the definition of the center of mass that requires any particular dimension of space.