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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 [itex]\int_{cone}(x,y,z)^{T}rdrdφdz[/itex] = [itex]\int_{cone}(0,0,z)^{T}rdrdφdz[/itex] ?

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 [itex]\int_{cone}(x,y,z)^{T}rdrdφdz[/itex] = [itex]\int_{cone}(0,0,z)^{T}rdrdφdz[/itex] ?

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