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?
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I'm pretty sure the term you're looking for is still "center of mass"--you're just considering a 2-dimensional lamina with a surface density instead of a solid with volume density. – Glare Aug 19 '15 at 01:06
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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.
Chris Culter
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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.
davidlowryduda
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