Consider two conic sections located in non-parallel planes in $\mathfrak{R}^3$ and parameterized as $\mathbf{P}_1(t)$ and $\mathbf{P}_2(t)$. Consider the ruled surface: $\mathbf{R}(s,t)=(1-s)\mathbf{P}_1(t)+s\mathbf{P}_2(t)$ that "interpolates" the two curves. Are there always reparameterizations of $\mathbf{P}_1(t)$ and $\mathbf{P}_2(t)$ such that the ruled surface $\mathbf{R}(s,t)$ becomes a developable surface? In addition, are there corresponding rational parameterizations of the conic sections for this (if it exists) developable surface? Finally, and independently, is such a surface algebraic?
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2$\begingroup$ Consider the graph of the function $f(x,y)=e^x$. Your question is more appropriate for math stack exchange. $\endgroup$– Moishe KohanCommented Jan 16, 2021 at 18:07
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8$\begingroup$ @MoisheKohan I think an asker, especially a new user, deserves more chances to clarify their question than that. $\endgroup$– Will SawinCommented Jan 16, 2021 at 20:11
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2$\begingroup$ @Jap88 Is a cylinder, viewed as the convex hull of two circles in two planes, an example of what you mean? Then I guess you want to take the boundary of the convex hull, less any components contained in the planes. Such a set will always be semialgebraic, at least - defined by polynomial equations and inequalities. One proof of this is to use the quantifier elimination for real closed fields, but that's a bit of a sledgehammer. $\endgroup$– Will SawinCommented Jan 16, 2021 at 20:15
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2$\begingroup$ You might look at this paper: Ranestad, Kristian, and Bernd Sturmfels. "On the convex hull of a space curve." arXiv:0912.2986 (2009). "The edge surface is in general the tangent developable of a curve." "The union of all stationary bisecant lines is the edge surface of the curve." $\endgroup$– Joseph O'RourkeCommented Jan 16, 2021 at 20:47
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1$\begingroup$ This was a question related to a stack-exchange question. For some applied algebraic geometry and a partial answer to this question, see: math.stackexchange.com/questions/3974536/… $\endgroup$– Jap88Commented Jan 23, 2021 at 23:46
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