Point Clouds of Shapes

Imagine a subspace \(S\) of our environment, denoted by \(X\), is densely covered with icing sugar. Further we assume we can see the icing sugar but not the space \(S\) itself. Now let \(f \colon X \rightarrow {\mathbb{R}}\) be the function that assigns to each point in \(X\) it’s distance to \(S\) and let \(g \colon X \rightarrow {\mathbb{R}}\) assign to each point the distance to a nearest grain of icing sugar. Then \(S\) is the zero locus of \(f\). Moreover \(g\) can be as close to \(f\) as we like, if we only spread the icing sugar densely enough. The icing sugar is also referred to as point-cloud data for \(S\).