@article {RicKor-PSMP-2010, title = {The power of scripting: {DGS} meets programming}, journal = {Acta Didactica Napocensia}, volume = {3}, number = {2}, year = {2010}, pages = {67-78}, abstract = {

In this article we demonstrate how the combination of a system for dynamic geometry with a freely programmable scripting environment can be advantageously used in teaching and research. We explain the reasons behind various design decisions that were made by us when designing the language CindyScript and give examples that proof how they lead to easy and understandable code that can be used in education. We give several concrete application scenarios of the language that was developed by the authors and seamlessly interacts with the dynamic geometry system Cinderella.

}, keywords = {refereed}, url = {http://dppd.ubbcluj.ro/adn/article_3_2_7.pdf}, author = {Richter-Gebert, J{\"u}rgen and Kortenkamp, Ulrich} } @conference {KorRic-BEW-2009, title = {Blended Experimentation with DGS}, booktitle = {Proceedings of CADGME 2009}, year = {2009}, abstract = {

Experimentation is an important element of science teaching. Students should experience real-world phenomena, and support or disprove their hypotheses about the world by setting up appropriate situations, conducting the experiments, gathering data and drawing conclusions. Unfortunately, many experiments are unsuitable for school teaching out of various reasons. They might be too dangerous, too expensive, too complex, too unreliable, or they take too much time to complete. Using simulation software, many experiments can be replaced by safer, cheaper, easier, more reliable or sped-up virtual counterparts. Such simulations range from showing videos of repeated runs of the real experiment, together with interaction facilities, to mathematical simulations using numerical solutions of partial differential equations. However, there is a pedagogical drawback of the simulation approach: The connection to the real world situation is (at least partially) lost. This might harm both the motivation of the students and their belief in the conclusions. In this paper we describe the approach of blended experimentation, where simulations and the real world are connected via sensors and actors. The sensors are able to influence the simulation, while the actors can change the real world. We show first implementations and examples using Dynamic Geometry Software and other Mathematical Software as simulation environments.

}, keywords = {refereed}, url = {http://www.risc.jku.at/conferences/cadgme2009/?content=proceedings}, author = {Kortenkamp, Ulrich and Richter-Gebert, J{\"u}rgen} } @conference {KorRic-UATPIUGS-2004, title = {Using Automatic Theorem Proving to Improve the Usability of Geometry Software}, booktitle = {Proceedings of MathUI 2004}, year = {2004}, abstract = {

Dynamic or interactive Geometry software (DGS) is the mathematical version of vector based drawing software: the objects (points, lines, circles, conics, polygons, etc.) are both graphical and mathematical entities. This allows adding relations between the objects that govern their behavior. Thus, DGS is used as an input tool for constructions, as opposed to simple drawings. The additional information that is present in a construction can be used to greatly enhance the usability of DGS. We show how automatic theorem proving can be used to remove redundant elements in a construction that obstruct a smooth work flow clarify the semantics of user actions, and improve the graphical rendering of elements. Finally we discuss the various possibilities of transforming the mathematical tool DGS into an educational tool. Here, automatic theorem proving is used to analyze user actions and to react properly.

}, keywords = {refereed}, url = {http://kortenkamps.net/papers/2004/ATP-UI-article.pdf}, author = {Kortenkamp, Ulrich and Richter-Gebert, J{\"u}rgen}, editor = {Libbrecht, Paul} } @conference {RicKor-CIDG-2002, title = {Complexity issues in Dynamic Geometry}, booktitle = {Foundations of Computational Mathematics (Proceedings of the Smale Fest 2000)}, year = {2002}, note = {Also available as technical report TRB-2000/22, Freie Universit{\"a}t Berlin}, publisher = {World Scientific}, organization = {World Scientific}, abstract = {

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 {KorRic-DCDG-2001, title = {Decision complexity in {D}ynamic {G}eometry}, booktitle = {Proceedings of ADG 2000}, series = {Lecture Notes in Artificial Intelligence}, number = {2061}, year = {2001}, pages = {167{\textendash}172}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, address = {Heidelberg}, abstract = {

Geometric straight-line programs [5,8] can be used to model geometric constructions and their implicit ambiguities. In this paper we discuss the complexity of deciding whether two instances of the same geometric straight-line program are connected by a continuous path, the Complex Reachability Problem.

}, keywords = {refereed}, url = {http://kortenkamps.net/papers/2001/36_DecisionComplexity.pdf}, author = {Kortenkamp, Ulrich and Richter-Gebert, J{\"u}rgen}, editor = {Wang, Dongming} } @conference {RicKor-DSEG-2001, title = {A dynamic setup for elementary geometry}, booktitle = {Proceedings of MTCM 2000}, year = {2001}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, abstract = {

In this article we survey the theoretical background that is required to build a consistent and continuous setup of dynamic elementary geometry. Unlike in static elementary geometry in dynamic elementary geometry the elements of a construction are allowed to move around as long as the geometric constraints intended by the construction are not violated. A typical problem in such a scenario is to resolve ambiguous situations that arise from geometric operations like intersecting a circle and a line. After introducing a formal framework for dealing with dynamic geometric constructions, we will demonstrate that a suitable resolution of these ambiguities requires the consideration of complex projective spaces. We will discuss several aspects where one can benefit from such a rather general approach. Finally, we will sketch some proofs that show that several fundamental algorithmic problems arising in such a context are NP-hard or even harder.

}, keywords = {refereed}, url = {http://kortenkamps.net/papers/2001/35\_DynamicSetup.pdf}, author = {Richter-Gebert, J{\"u}rgen and Kortenkamp, Ulrich} } @inbook {RicKor-GDG-2001, title = {Grundlagen Dynamischer Geometrie}, booktitle = {Zeichnung {\textendash} Figur {\textendash} Zugfigur}, year = {2001}, pages = {123{\textendash}144}, publisher = {Franzbecker}, organization = {Franzbecker}, address = {Hildesheim, Berlin}, abstract = {

In this article we present fundamental definitions that can be used to introduce a mathematical model for dynamic geometry. Starting from reasonable expectations that such a model should meet we will formalize the terms (dynamic) construction, instance of a construction and Dynamic-Geometry-System (DGS). The behavior of a DGS will be described by the terms conservatism and continuity. One of the main results of this article is that we can find a continuous DGS for any construction Z that is built up using algebraic basic construction steps only.

}, keywords = {refereed}, url = {http://kortenkamps.net/papers/2001/DG_OW1.pdf}, author = {Richter-Gebert, J{\"u}rgen and Kortenkamp, Ulrich}, editor = {Henn, Hans-Wolfgang and Elschenbroich, Hans-J{\"u}rgen and Gawlick, Thomas} } @conference {KorRic-DCDG-2000, title = {Decision complexity in {D}ynamic {G}eometry}, booktitle = {Proceedings of ADG 2000}, series = {International Workshop on Automatic Deduction in Geometry}, volume = {3}, year = {2000}, note = {Extended Abstract}, pages = {216{\textendash}220}, abstract = {

Geometric straight-line programs [6, 10] can be used to model geo- metric constructions and their implicit ambiguities. In this paper we discuss the complexity of deciding whether two instances of the same geometric straight-line program are connected by a continuous path, the Complex Reachability Problem.

}, keywords = {refereed}, url = {http://kortenkamps.net/papers/2000/ADG2000_UK.pdf}, author = {Kortenkamp, Ulrich and Richter-Gebert, J{\"u}rgen}, editor = {Wang, Dongming} } @conference {RicKor-DACG-2000, title = {Dynamic aspects in computational geometry}, booktitle = {Proceedings of the EACA 2000}, year = {2000}, pages = {51{\textendash}61}, address = {Barcelona}, abstract = {

Computational Geometry very often focuses on static problems, like computing the convex hull or Voronoi complex of a given set of points. Fundamentally new questions arise when the objects under consideration are no longer static, but may move around with respect to certain constraints. This scenario is not unusual, for instance every mechanism can be considered as a dynamic geometry entity. Here we focus on the new areas of problems that arise from genuinely dynamic effects. Constructions from elementary geometry play a crucial role in this context, since they form first natural instances of non-trivial examples where it is reasonable to study the dynamic behaviour. One of the most fundamental problems arises when one considers one point of an intersection of a line and a circle and allows the line to move around. Since the point of intersection is not unique, a computer program has to decide for every "discrete snapshot" which of the two intersection points is meant. If this decision is not made correctly a "path-jumping" of the point may occur. A careful analysis of the situation shows that for a satisfactory resolution of the problem one has to embed the configuration in an ambient complex projective space. One even has to take monodromy effects and underlying Riemann surfaces into account. We will develop a theory for the study of these phenomena. After this, we will investigate the algorithmic complexity of "making the right decision". It will turn out that even in very weak versions this problem is NP-hard. In some stronger versions it is PSPACE hard or even undecidable.

}, keywords = {refereed}, author = {Richter-Gebert, J{\"u}rgen and Kortenkamp, Ulrich}, editor = {Montes, Antonio} } @article {RicKor-ENGC-2000, title = {Euklidische und nicht-{E}uklidische {G}eometrie in {C}inderella}, journal = {Journal f{\"u}r Mathematikdidaktik}, volume = {22}, year = {2000}, pages = {303{\textendash}324}, abstract = {

Zusammenfassung: Wir erinnern an die Konzeption euklidischer und nicht-euklidischer Geometrie durch Klein und Cayley und weisen auf ihre Bedeutung fu ̈r moderne Software zur interaktiven (oder dynamischen) Geometrie hin. Insbesondere zeigt es sich, dass sich komplexe Zahlen vorzu ̈glich zu einer allgemein Behandlung hyperbolischer, elliptischer, euklidischer und anderer Geometrie eignen. Summary: We recall the concepts of Euclidean and non-Euclidean Geometry in the sense of Klein and Cayley and illustrate their importance for modern interactive (dynamic) geometry software. In particular, the use of complex numbers turns out to be the key to the most general approach to measurements in hyperbolic, elliptic, Euclidean and other geometries.

}, keywords = {refereed}, url = {http://kortenkamps.net/papers/2000/Euklidisch-NichtEuklidisch.pdf}, author = {Richter-Gebert, J{\"u}rgen and Kortenkamp, Ulrich} } @inbook {KorRic-C-1999, title = {Das {C}inderella-{P}rojekt}, booktitle = {Erfahrungen mit Java}, year = {1999}, pages = {381{\textendash}401}, publisher = {dpunkt.verlag}, organization = {dpunkt.verlag}, chapter = {17}, address = {Heidelberg}, keywords = {refereed}, url = {http://www.amazon.de/exec/obidos/ASIN/3932588339/theinteractive07}, author = {Kortenkamp, Ulrich H. and Richter-Gebert, J{\"u}rgen}, editor = {Maffeis, Silvano and Toenniessen, Fridtjof and Zeidler, Christian} } @conference {RicKor-DGPC-1999, title = {Dynamic Geometry {I}: The Problem of Continuity}, booktitle = {Abstracts 15th European Workshop Comput. Geom.}, year = {1999}, pages = {51{\textendash}53}, publisher = {INRIA Sophia-Antipolis}, organization = {INRIA Sophia-Antipolis}, keywords = {refereed}, url = {http://www.cinderella.de/papers/antibes-1.pdf}, author = {Richter-Gebert, J{\"u}rgen and Kortenkamp, Ulrich H.} } @conference {KorRic-DGA-1999, title = {Dynamic Geometry {II}: Applications}, booktitle = {Abstracts 15th European Workshop Comput. Geom.}, year = {1999}, pages = {109{\textendash}111}, publisher = {INRIA Sophia-Antipolis}, organization = {INRIA Sophia-Antipolis}, keywords = {refereed}, url = {http://cinderella.de/papers/antibes-2.pdf}, author = {Kortenkamp, Ulrich H. and Richter-Gebert, J{\"u}rgen} } @conference {KorRic-GEI-1998, title = {Geometry and education in the {I}nternet age}, booktitle = {Ed-Media \& Ed-Telecom 98. Proceedings of the Tenth World Conference on Educational Multimedia and Hypermedia \& World Conference on Educational Telecommunications, Freiburg, Germany, June 20-25, 1998}, year = {1998}, publisher = {AACE}, organization = {AACE}, address = {Charlottesville}, abstract = {

Interactive Geometry is a major tool in modern geometry education and various software tools are available. We discuss the requirements of such tools and how they can be fulfilled. We also explain how a geometry tool can benefit from the Internet and present Cinderella{\textquoteright}s Caf{\'e}, which is an internet-aware geometry tool with a high mathematical background.

}, keywords = {refereed}, url = {http://www.cinderella.de/papers/geo-i.pdf.gz}, author = {Kortenkamp, Ulrich H. and Richter-Gebert, J{\"u}rgen}, editor = {Ottmann, Thomas and Tomek, Ivan} } @article {KorRicSarZie-EP0-1997, title = {Extremal properties of 0/1-polytopes}, journal = {Discrete \& Computational Geometry}, volume = {17}, year = {1997}, pages = {439{\textendash}448}, abstract = {

We provide lower and upper bounds for the maximal number of facets of a d-dimensional 0/1-polytope, and for the maximal number of vertices that can appear in a two-dimensional projection ("shadow") of such a polytope.

}, keywords = {refereed}, url = {http://link.springer.de/link/service/journals/00454/htabst/17n4p439.html}, author = {Kortenkamp, Ulrich and Richter-Gebert, J{\"u}rgen and Sarangarajan, A. and Ziegler, G{\"u}nter M.} }