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 ?
Can any smooth planar curve which is closed, be a base for a 3 dimensional cone?
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real-analysis
differential-geometry
analytic-geometry
1 Answers
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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
$\begin{align*}x&=f(u)(1-v)\\y&=g(u)(1-v)\\z&=v\end{align*}$
is one simple parametrization for a generalized (right) cone whose cross sections are scaled versions of the given plane curve.
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0Of course, if what you need is a *skew* cone (axis not perpendicular to a coordinate plane), a different approach is needed... – 2011-04-24