2.1.1 Implicit Representation

Figure 2.2: Implicit geometries

Let ${\mathcal{G}}$ be a geometry. A function $F: {\mathbb{R}}^n \rightarrow {\mathbb{R}}$ with ${\mathcal{G}}= \{\bm{x} \in {\mathbb{R}}^n \vert F(\bm{x}) \geq 0\}$ and ${\operatorname{bnd}}^\star ({\mathcal{G}}) = \{\bm{x} \in {\mathbb{R}}^n \vert F(\bm{x}) = 0\}$ is called the implicit representation of ${\mathcal{G}}$. The function used for this representation is not unique. However, there is a guarantee that for every geometry ${\mathcal{G}}$ there exists an implicit representation. The existence of such a function $F$ for a given geometry ${\mathcal{G}}$ can easily be shown by setting

$$F(\bm{x}) := \begin{cases}\phantom{-} \min_{\bm{y} \in {\operator... ...x}-\bm{y}\right\rVert_2} ) & \mbox{if } \bm{x} \notin {\mathcal{G}} \end{cases}$$ (2.2)

For example, the closed $n$-ball with radius $r$ and center $\bm{0}$ can be represented using the function $F(\bm{x}) = {\left\lVert\bm{x}\right\rVert_2} - r$. Another popular example of objects represented by implicit functions are metaballs [70]. Metaballs are organic-looking ball-like objects fulfilling an implicit function which is based on distances to ball centers. These two examples of geometries represented with implicit functions are visualized in Figure 2.2. Implicit geometries can also be used to represent multi-region geometries with one implicit function $F_i$ for each region. Although geometries represented by mathematical expressions are robust with respect to modeling, they rarely are used in engineering applications, because finding a function in closed form is challenging for general objects.