Visualization of Overtwisted Disks in R³

This thesis is about the numerical extraction of overtwisted disks in contact structures in R³. Overtwisted disks play an important role in contact geometry as their existence distinguishes two different kinds of contact structures, namely the overtwisted and the tight contact structures. On the other side they are also useful for the visualization of overtwisted contact structures, especially in combination with further extracted data, like the region of the contact structure that consists of overtwisted disk center points.

We will define and extract pairs of distinguished tangential directions in the plane field that are extremal with respect to certainmeasures. These directions then serve as a strategy to integrate Legendrian curves on a contact structure which are used for the extraction of overtwisted disks. The associated anisotropy fields measure the quality of the extracted vector fields and will be also used to determine possible center points of overtwisted disks. As the nonintegrability is a key aspect of contact structures, we will derive a formula to efficiently calculate a measure for the local nonintegrability of a plane fields.