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4.4 Boolean Operations

If geometries are represented as level sets, Boolean operations can be expressed as simple operations on the corresponding LS functions [89,110]. If one considers the sets $ {\mathcal{M}}_A$ , $ {\mathcal{M}}_B$ , and $ {\mathcal{M}}_C$ and the corresponding boundaries $ {\mathcal{S}}_A=\partial{\mathcal{M}}_A$ , $ {\mathcal{S}}_B=\partial{\mathcal{M}}_B$ , and $ {\mathcal{S}}_C=\partial{\mathcal{M}}_C$ , which are represented by the LS functions $ {\Phi}_A$ , $ {\Phi}_B$ , and $ {\Phi}_C$ , the LS counterparts of Boolean operations are listed in the following.

Union:   $\displaystyle {\mathcal{M}}_A$ $\displaystyle ={\mathcal{M}}_B \cup {\mathcal{M}}_C$ $\displaystyle \Leftrightarrow$   $\displaystyle {\Phi}_A$ $\displaystyle =\min({\Phi}_B,{\Phi}_C)$ (4.10)
Intersection:   $\displaystyle {\mathcal{M}}_A$ $\displaystyle ={\mathcal{M}}_B \cap {\mathcal{M}}_C$ $\displaystyle \Leftrightarrow$   $\displaystyle {\Phi}_A$ $\displaystyle =\max({\Phi}_B,{\Phi}_C)$ (4.11)
Complement:   $\displaystyle {\mathcal{M}}_A$ $\displaystyle =\mathbb{R}^{D}\setminus{\mathcal{M}}_B$ $\displaystyle \Leftrightarrow$   $\displaystyle {\Phi}_A$ $\displaystyle =-{\Phi}_B$ (4.12)
Relative complement:   $\displaystyle {\mathcal{M}}_A$ $\displaystyle ={\mathcal{M}}_B\setminus {\mathcal{M}}_C$ $\displaystyle \Leftrightarrow$   $\displaystyle {\Phi}_A$ $\displaystyle =\max({\Phi}_B,-{\Phi}_C)$ (4.13)

Here the convention is used that the LS function $ {\Phi}_X$ is negative for all $ {\vec{x}}\in{\mathcal{M}}_X$ .

Boolean operations are very useful for more general topography simulations, if consecutive process steps, like etching and/or deposition processes, should be simulated or several materials are involved.



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Next: 4.4.1 Implementation Up: 4. A Fast Level Previous: 4.3.3 Dilation

Otmar Ertl: Numerical Methods for Topography Simulation