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In this paper: https://www.cs.cmu.edu/~kmcrane/Projects/RepulsiveSurfaces/RepulsiveSurfaces.pdf The term nested envelopes is used.

The paper is not a math paper, however I am fairly sure there's a proper mathematical definition of what an envelope of some 3D geometry might be, or whatever suitable generalization the would boil down to the same meaning in case we have a compact submanifold of $\mathbb{R}^3$ without boundary.

Can anyone point me to the right definition?

user8469759
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1 Answers1

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Just as the envelope of a $1$-parameter family of curves in the plane is defined, now define the envelope of a $2$-parameter family of surfaces in $\Bbb R^3$. The envelope should be a surface so that each member of the family is tangent to the surface at some point.

Analytically, if $U\subset\Bbb R^2$ is an open subset and we're given a $C^2$ function $f\colon \Bbb R^3\times U\to\Bbb R$ with $\nabla_{\mathbf x} f(\mathbf x,u,v)\ne \mathbf 0$ at each point of the level surface $S_{(u,v)} = \{\mathbf x: f(\mathbf x,u,v)=0\}$, then you can check that the envelope comes from solving the system $$f(\mathbf x,u,v)= \frac{\partial f}{\partial u}(\mathbf x,u,v) = \frac{\partial f}{\partial v}(\mathbf x,u,v) = 0.$$ There is a nondegeneracy hypothesis$^*$ that allows you to deduce from the Implicit Function Theorem that you'll locally have a smooth surface defined by these equations.

If on some open subset of $U$ we have $\mathbf x = \Phi(u,v)$ given by these equations, then we claim that this equation defines a parametric surface $S$. We now check that at the point $\Phi(u,v)$ it is tangent to the surface $S_{(u,v)}$. Set $g(u,v) = f(\Phi(u,v),u,v)$ and note that since $g$ is identically $0$, we have $$\frac{\partial g}{\partial u} = \nabla_{\mathbf x} f\cdot\frac{\partial\Phi}{\partial u} + \frac{\partial f}{\partial u} = 0, \tag{$\dagger$}$$ and similarly for the $v$ partial derivative. Thus, we conclude that $$\nabla_{\mathbf x} f\cdot\frac{\partial\Phi}{\partial u} = \nabla_{\mathbf x} f\cdot\frac{\partial\Phi}{\partial v} = 0.$$ Since $\nabla_{\mathbf x}f(\mathbf x,u,v)$ is the normal to the level surface $S_{(u,v)}$ and since $\frac{\partial\Phi}{\partial u}$ and $\frac{\partial\Phi}{\partial v}$ span the tangent plane of $S$, we are done.

$^*$For those who are interested, here's the hypothesis. Now we write $\mathbf x = (x,y,z)$. Assume that the matrix $$\left[\begin{matrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} \\ \frac{\partial^2 f}{\partial x\partial u} & \frac{\partial^2 f}{\partial y\partial u} & \frac{\partial^2 f}{\partial z\partial u} \\ \frac{\partial^2 f}{\partial x\partial v} & \frac{\partial^2 f}{\partial y\partial v} & \frac{\partial^2 f}{\partial z\partial v} \end{matrix}\right]$$ is nonsingular. This will give you a local representation of the solution of ($\dagger$) as a $C^1$ function of $(u,v)$.

Ted Shifrin
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  • A reference for this definition? So I can look at the properties of envelopes. – user8469759 May 30 '22 at 21:26
  • I don’t know a reference, although I’m sure you can find one. The 2-D version is an exercise in my Multivariable Mathematics text. – Ted Shifrin May 30 '22 at 22:20
  • This is a fine answer but the authors of the linked paper appear to attach a very different meaning to the word "envelope," they mean a "neighborhood." – Moishe Kohan May 30 '22 at 23:55
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    @MoisheKohan Oy. Nice way to make up terminology. – Ted Shifrin May 31 '22 at 00:07
  • @MoisheKohan how did you figure? – user8469759 May 31 '22 at 04:52
  • @user8469759: By looking at their text and pictures. But to know for sure, that's not enough. Consider checking other papers by the same authors, if they used this terminology. – Moishe Kohan May 31 '22 at 12:48
  • If you tell me what you mean by "neighborhood" maybe I can read this paper with that mindset. You mean the interior of the manifold? – user8469759 May 31 '22 at 12:50
  • @user8469759 If $A$ is a subset of a topological space $X$ (say, $X= {\mathbb R}^3$), then a neighborhood of $A$ in $X$ is any open subset $U$ of $X$ which contains $A$. It is possible, of course, that the authors have more restrictions in mind, such as, requiring $U$ to be the interior of a compact submanifold with boundary. Or, even more, maybe they assume that $X$ is a metric space with the metric $d$ and $U$ is the $r$-neighborhood of $A$ in $X$ for some $r>0$: ${x\in X: d(x,A)<r}$. – Moishe Kohan May 31 '22 at 16:10
  • After @MoisheKohan's comment, I went and searched through the paper. The term appears in 9.2.2 and, earlier, in one figure. It is not the classical use of the term "envelope" at all. They are using it in the sense of an envelope "containing" a letter inside. They do refer to Sacht et al., so I suppose you could check that reference. But my answer might help someone who encounters the geometric notion of envelopes ... who knows. – Ted Shifrin May 31 '22 at 16:37