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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.

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?

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 curves for this (if it exists) developable surface? Finally, and independently, is such a surface algebraic.

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.

Consider two conic sections located in non-parallel planes 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 curves for this (if it exists) developable surface. Finally, is such a surface algebraic.

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 curves for this (if it exists) developable surface? Finally, and independently, is such a surface algebraic.