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Is there a two dimensional surface like a cone but whose base is elliptic or any non circular but smooth closed curve ? The surface should be smooth everywhere except at the vertex.

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let $f(t)=(x(t),y(t),0), t\in[0,1], f(0)=f(1)$ be a planar curve in $\mathbb{R}^3$, and let $p=(a,b,c)$ be point in $\mathbb{R}^3$. then the surface $f(t)+s(p-f(t)), (s,t)\in[0,1]\times[0,1]$ is what you are looking for. this is differentiable if $f$ is. –  yoyo Apr 25 '11 at 21:17
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up vote 3 down vote accepted

Yes.

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Can the contour of the base be any other smooth planar curve other than an ellipse ? –  Rajesh D Apr 24 '11 at 3:41
    
Yes. Take any smooth planar curve $C$, embed this plane in ${\bf R}^3$ so that it doesn't contain the origin, and consider the surface consisting of points $t {\bf p}$ for ${\bf p} \in C$ and $t \in R$. –  Robert Israel Apr 24 '11 at 4:40
    
@Robert Israel : Please suggest me some reference/text book to look into. I am not able to understand the notation and notions in your comment. –  Rajesh D Apr 24 '11 at 6:18
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