Originally posted by THUDandBLUNDERThat doesn't work.
Non-mathematical definition: the geometric centroid of a 2D shape is the point on which it would balance if placed on a needle.
Originally posted by AThousandYoungGood point. OK, the axis should be 'external'.
That doesn't work.
If you take a circle and rotate it about a line that passes through a diameter of the circle, then the distance from centroid to axis of revolution is zero. Therefore, you get a volume of zero for any sphere.
Originally posted by THUDandBLUNDERThis is nice, but it doesn't answer the original question, which wants formulas for spheres, cones, etc.
Good point. OK, the axis should be 'external'.
In fact, the second theorem of Pappus states that the volume V of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area A of the lamina and the distance d traveled by the lamina's geometric centroid.
Originally posted by fearlessleaderI don't think his original question is well er...formed.
is there a universal function for the volume of such forms in terms of the rotated shape
Originally posted by AThousandYoung
This is nice, but it doesn't answer the original question, which wants formulas for spheres, cones, etc.
Originally posted by THUDandBLUNDERWell, a 2-manifold embedded in 3-space has a shape operator S, which is the differential of the normal to the surface. The shape operator is a 2x2 matrix which tells you exactly how the surface is curved at any point. For example, a sphere has shape operator I/r, where I is the identity and r is the radius. It'd be interesting to see a concise formula linking the shape operator with the volume enclosed by a compact manifold (if such a formula exists).
I don't think his original question is well er...formed.
'Shape' is a function of what?