A cone in 3 dimensions has a vertex and a base. The contour of the base is a circle which is a smooth closed planar curve. Can there be a more general cone which can have any smooth closed planar curve as the contour of its base ?
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Certainly; if you have some plane curve represented parametrically as $(f(u)\quad g(u))^T$, the surface represented parametrically by
is one simple parametrization for a generalized (right) cone whose cross sections are scaled versions of the given plane curve.