#### 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 $$S$$.