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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