Proof of Symmetry for a Geometrical Object

[Trifis]

I was wondering, if there is a general way of finding the axis of symmetry of a geometrical object and also if there is a proof for the fact that the centre of mass must lie on that axis.

Let us consider a trivial structure, the cone. We know by intuition that the zz' axis is the axis of symmetry. Then with a simple subtitution (x'=-x and y'=-y) one can verify that the implicit cartesian equation remains invariant.

My question is, what happens when we have no a priori knowledge of the symmetry of a more complex geometry. Is there a certain procedure to "unravel" it or we just make approximately educated guesses?

As far as the second part of my question is considered. Let us examine the cone once again. One can also find the centre of mass based on the assumption that its x and y components are zero. But how can we prove this? Specifically, why $\int_{cone}(x,y,z)^{T}rdrdφdz$ = $\int_{cone}(0,0,z)^{T}rdrdφdz$ ?

 

[lippyka]

Unfortunately, there is no general way to find the axis of symmetry of a geometrical object. The best one can do is make educated guesses based on experience and observation. Regarding the proof of the fact that the centre of mass must lie on the axis of symmetry, it can be proven using the theorem of moments. This theorem states that for any given object, if the moment around any point is zero, then that point is the centre of mass. Therefore, since the moment around any point on the axis of symmetry is zero, it follows that the centre of mass must lie on the axis of symmetry.