Classes of curves closed under Minkowsky sum

Question

Let $C$ be a class of plane curves, regarded as subsets of $\mathbb{R}^2$ (parametrization won't matter), I'm thinking for example of splines or algebraic subsets. Let $D$ be a class of topological discs bounded by plane curves from a possibly different class, for example circles, ellipses, closed splines.

Now if I take a $c \in C$ and a $d \in D$ I can form the Minkowski sum $c + d$ (simply all sums of points from $c$ and $d$). Its boundary will again be a curve (possibly not connected). My question is:

Are there "interesting" classes $C$ and $D$ that ensure that the boundary of $c + d$ will again lie in $C$? Is this kind of closure behavior ever considered?

By "interesting" I mean that the classes should be small enough, so that each curve can be described by finite data, but not so small that the statement becomes trivial. For instance taking both classes to be $C^1$ is too large and taking $D$ to consist just of points is too small.

The origin for this question is the following: the sum $c + d$ models tracing the curve $c$ with a pen whose shape is $d$. At some point I learned that Knuth's Computer Modern typeface was turned into OpenType by first producing high resolution pixel versions and then vectorizing these. I was surprised at first because to me Metafont was vector graphics as was OpenType. But the point is that Knuth drew Computer Modern tracing curves with pens (so the letters are described as $c + d$) while OpenType uses the outlines (the boundary of $c + d$). The fact that the transition is not purely formal suggests that splines do not have the above closure property. On the other hand, some vector graphics software can transform a traced curve into an outline (perhaps inexactly?). I'm not interested practical solutions but in whether there is a class of curves that works theoretically.

Answer 1

There are lots of finite dimensional curve families that are closed under Minkowski sum with circular disks. Here's a way to construct examples:

First, choose a finite dimensional space $\mathcal{C}$ of smooth, functions that contains the constants. Then, for any $f\in\mathcal{C}$, consider the curve $X_f$ defined by $$ X_f(\theta) = \int_0^\theta \begin{pmatrix}\cos t\\\sin t\end{pmatrix}\,f(t)\,\mathrm{d}t. $$ Let $C$ be the family of curves plust their translations in the plane. (If one wants the family to be also invariant under rotation, one should require that $\mathcal{C}$ be invariant under translation.)

Then the Minkowski sum of $X_f$ and the disk of radius $r$ will be bounded by the curves $$ X_{f\pm r}(\theta) = p+ \int_0^\theta \begin{pmatrix}\cos t\\\sin t\end{pmatrix}\,(f(t)\pm r)\,\mathrm{d}t, $$ which, by construction, also belong to the family $\mathcal{C}$.

Of course, these curves may be singular, and if one wants to avoid that one might want to discard the locus in $X_f$ where $f$ vanishes or where $f\pm r$ vanishes. This can also be done by restricting the range of $r$ and restricting the allowable $f$ to an appropriate open set in $\mathcal{C}$.

This can be generalized considerably: Let $\mathcal{D}$ be a finite dimensional vector space of $2\pi$-periodic functions that contains the constants with the property that, for all $f\in\mathcal{D}$ we have $$ \int_0^{2\pi} \begin{pmatrix}\cos\theta\\\sin\theta\end{pmatrix}\,f(\theta)\,\mathrm{d}\theta=0. $$ For example, $\mathcal{D}$ might be the span of $\{1,\cos2\theta,\sin2\theta,\ldots,\cos m\theta,\sin m\theta\}$ for some integer $m\ge 2$. Now let $\mathcal{D}^+\subset\mathcal{D}$ denote the set of those elements of $\mathcal{D}$ that are positive. Then, for $f\in\mathcal{D}^+$, the curve $$ Y_f(\theta) = \int_0^\theta \begin{pmatrix}\cos t\\\sin t\end{pmatrix}\,f(t)\,\mathrm{d}t $$ will be $2\pi$-periodic and will bound a convex region $R_f$ in the plane. Let $D$ be the (finite dimensional) set of such convex regions $R_f$. Now assume, as before, that $\mathcal{C}$ is a finite dimensional subspace of smooth functions that contains $\mathcal{D}$. Then the Minkowski sum of a curve $X_f$ for $f\in\mathcal{C}$ and a region $R_g$ for $g\in\mathcal{D}^+$ will be bounded by the curves $X_{f\pm g}$, which, since $\mathcal{D}\subset\mathcal{C}$, will still be a curve in $C$.