About Tracing Problems in Dynamic Geometry

Dynamic Geometry is the field of interactively performing geometric construction on a computer. In addition to simulating ruler-and-compass constructions we allow a drag mode. This drag mode allows to move geometric objects that have at least one degree of freedom. The remaining part of the construction should adjust automatically. Thus, during the motion, we have to trace the resulting paths of all geometric objects. This path tracking problem is known as the Tracing Problem from Dynamic Geometry. It combines the step-by-step procedure of doing geometric constructions with the continuous concept of motions. This study is based on the model for Dynamic Geometry used in the interactive geometry software Cinderella. We give a numerical solution to the Tracing Problem based on continuation methods and a reliable algorithm based on real and complex interval arithmetic. Degenerate situations like the intersection of two identical lines lead to critical points in the configuration space and are treated separately.