Definition

Consider a domain $\Omega \subseteq \mathbb{R}^3$ and three differentiable maps $f_i:\Omega \rightarrow \mathbb{R}$, $i = 1, 2, 3$. If at every point $x \in \Omega$, $\nabla f_i(x) \cdot \nabla f_j(x) \ne = 0$ only if whenever $i=j$, i \ne j$, then the$f_i$are triply orthogonal coordinates on$\Omega$. Context Dupin's theorem states that orthogonal coordinate surfaces (i.e., level sets of orthogonal coordinate functions) intersect along lines of principal curvature. A classic example is Monge's ellipsoid -- see Jorge Sotomayor, "Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in \mathbb{R}^3." Question Suppose we start with a compact, connected, orientable smooth surface$\mathcal{M}$embedded in$\mathbb{R}^3$, and let$\Omega$be a band of uniform thickness around$\mathcal{M}$(i.e., a union of$\epsilon$-balls centered at each point on$\mathcal{M}$). \mathcal{M}$) small enough so that at each point $x \in \Omega$ there is a unique closest point on $\mathcal{M}$. Further, suppose $f_1:\Omega \rightarrow \mathbb{R}$ gives the signed distance to $\mathcal{M}$, so that $f^{-1}(0) = \mathcal{M}$.

Can we always find (nontrivial) functions $f_2, f_3$ such that $(f_1,f_2,f_3)$ are triply orthogonal coordinates?

Or less formally, can we always find orthogonal coordinates "around" a given surface?

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Definition

Consider a domain $\Omega \subseteq \mathbb{R}^3$ and three differentiable maps $f_i:\Omega \rightarrow \mathbb{R}$, $i = 1, 2, 3$. If at every point $x \in \Omega$, $\nabla f_i(x) \cdot \nabla f_j(x) \ne 0$ only if $i=j$, then the $f_i$ are triply orthogonal coordinates on $\Omega$.

Context

Dupin's theorem states that orthogonal coordinate surfaces (i.e., level sets of orthogonal coordinate functions) intersect along lines of principal curvature. A classic example is Monge's ellipsoid -- see Jorge Sotomayor, "Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in \mathbb{R}^3."

Question

Suppose we start with a compact, connected, orientable smooth surface $\mathcal{M}$ embedded in $\mathbb{R}^3$, and let $\Omega$ be a band of uniform thickness around $\mathcal{M}$ (i.e., a union of $\epsilon$-balls centered at each point on $\mathcal{M}$). Further, suppose $f_1:\Omega \rightarrow \mathbb{R}$ gives the signed distance to $\mathcal{M}$, so that $f^{-1}(0) = \mathcal{M}$.

Can we always find (nontrivial) functions $f_2, f_3$ such that $(f_1,f_2,f_3)$ are triply orthogonal coordinates?

Or less formally, can we always find orthogonal coordinates "around" a given surface?

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Definition

Consider a domain $\Omega \subseteq \mathbb{R}^3$ and three differentiable maps $f_i:\Omega \rightarrow \mathbb{R}$, $i = 1, 2, 3$. If at every point $x \in \Omega$, $\nabla f_i(x) \cdot \nabla f_j(x) \ne 0$ if and only if $i=j$ (in other words: all coordinate surfaces intersect at right angles), i=j$, then the$f_i$are triply orthogonal coordinates on$\Omega$. Context Dupin's theorem states that orthogonal coordinate surfaces (i.e., level sets of orthogonal coordinate functions) intersect along lines of principal curvature. A classic example is Monge's ellipsoid -- see Jorge Sotomayor, "Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in \mathbb{R}^3." Question Suppose we start with a compact, connected, orientable smooth surface$\mathcal{M}$embedded in$\mathbb{R}^3$, and let$\Omega$be a band of uniform thickness around$\mathcal{M}$(i.e., a union of$\epsilon$-balls centered at each point on$\mathcal{M}$). Further, suppose$f_1:\Omega \rightarrow \mathbb{R}$gives the signed distance to$\mathcal{M}$, so that$f^{-1}(0) = \mathcal{M}$. Can we always find functions$f_2, f_3$such that$(f_1,f_2,f_3)\$ are triply orthogonal coordinates?

Or less formally, can we always find orthogonal coordinates "around" a given surface?

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