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Britta Denner-Broser:

Complex Tracing


Dynamic Geometry is the field of interactively doing geometric constructions
using a computer. Usually, the classical ruler-and-compass constructions are
considered. The available tools are simulated by the computer. A Dynamic Geometry
System is a system to do geometric constructions that has a drag mode.
In the drag mode, geometric elements with at least one degree of freedom can
be moved, and the remaining part of the geometric construction adjusts automatically.
Thus, the computer has to trace the paths of the involved geometric
objects during the motion.

In this thesis, we focus on the beautiful model by Kortenkamp and Richter-Gebert
that is the foundation of the geometry software Cinderella. We embed an algebraic
variant of this model into different fields of pure and applied mathematics,
which leads to different approaches for realizing the drag mode practically. We
develop a numerical method to solve the Tracing Problem that is based on a
generic Predictor-Corrector method. Like most numerical methods, this method
cannot guarantee the correctness of the computed solution curve, hence ambiguities
are not treated satisfactorily. To overcome this problem, we develope
a second algorithm that uses interval analysis. This algorithm is robust, and
the computed step length is small enough to break up all ambiguities. Critical
points are bypassed by detours, where the geometric objects or the corresponding
variables in the algebraic model can have complex coordinates. Here, the final
configuration depends essentially on the chosen detour, but this procedure due to
Kortenkamp and Richter-Gebert leads to a consistent treatment of degeneracies.
We investigate the connection of the used model for Dynamic Geometry to Riemann
surfaces of algebraic functions.

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