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In this thesis we applied powerful mathematical tools§such as §interval arithmetic for applications in computational§geometry, §visualization and computer graphics, leading to§robust, general and §e cient algorithms. We presented a completely novel§approach for §computing the arrangement of arbitrary implicit§planar curves and §performed ray casting of arbitrary implicit functions§by jointly §achieving, for the rst time, robustness, e ciency§and exibility. §Indeed we were able to render even the most di cult§implicits in §real-time with guaranteed topology and at high§resolution. We used §subdivision and interval arithmetic as§key-ingredients to guarantee §robustness. The presented framework is also§well-suited for §applications to large and unstructured data sets due§to the inherent §adaptivity of the techniques that are used. We also§approached the §topic of tensors by collaborating with mechanical§engineers on §comparative tensor visualization and provided them§with helpful §visualization paradigms to interpret the data.
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