This article deals with the intrinsic complexity of tracing and reachability questions in the context of elementary geometric constructions. We consider constructions from elementary geometry as dynamic entities: while the free points of a construction perform a continuous motion the dependent points should move consistently and continuously. We focus on constructions that are entirely built up from join, meet and angular bisector operations. In particular the last operation introduces an intrinsic ambiguity: Two intersecting lines have two different angular bisectors. Under the requirement of continuity it is a fundamental algorithmic problem to resolve this ambiguity properly during motions of the free elements. After formalizing this intuitive setup we prove the following main results of this article: It is NP-hard to trace the dependent elements in such a construction. It is NP-hard to decide whether two instances of the same construction lie in the same component of the configuration space. The last problem becomes PSPACE-hard if we allow one additional sidedness test which has to be satisfied during the entire motion. On the one hand the results have practical relevance for the implementations of Dynamic Geometry Systems. On the other hand the results can be interpreted as statements concerning the intrinsic complexity of analytic continuation.

}, keywords = {refereed}, author = {Richter-Gebert, J{\"u}rgen and Kortenkamp, Ulrich H.}, editor = {Cucker, Felipe and Rojas, J. Maurice} } @conference {Kor-MMTNVC-2002, title = {Making the move: {T}he next version of {C}inderella}, booktitle = {Proceedings of the First International Congress of Mathematical Software}, year = {2002}, note = {A slightly modified version appeared in the proceedings of CCCG 02.}, pages = {208{\textendash}216}, publisher = {World Scientific}, organization = {World Scientific}, abstract = {Cinderella is a software package for interactive or dynamic geometry. Its first version was published in 1999 and was the first geometry software to be based on the theory of complex tracing3, thus avoiding mathematical inconsistencies and unmotivated discontinuities (modulo numerical errors). At the ICMS 2002 we will introduce the next major version of Cinderella and highlight its new features and new concepts.

}, keywords = {refereed}, url = {http://www.cs.uleth.ca/~wismath/cccg/papers/ulrich.pdf}, author = {Kortenkamp, Ulrich}, editor = {Cohen, Arjeh M. and Gao, Xiao-Shan and Takayama, Nobuki} }