Cinderella: Computation, complexity, geometry

With Cinderella[12], you can easily create constructions, figures consisting of points, lines, circles, conics and other geometric elements, together with relations that describe the mathematical connections between the elements.[5] A standard example is a triangle, consisting of three points and three segments connecting each pair of points, and the altitudes in that triangle. The representation of the construction within the software makes it possible to move the vertices of the triangle (using the mouse) and get a new drawing for each position while the segments and the altitudes are updated accordingly. The mathematical core of Cinderella is based on projective geometry and Cayley-Klein geometries for maximal generality. All geometric constructions are designed to work in all meaningful situations and to have as few special cases as possible. At the same time it is possible to have different, simultaneous, interactive views of a geometric construction, like its 2D representation, a spherical projection, polarized versions of this or special views like the Poincaré disk for hyperbolic geometry. As soon as multi-valued geometric constructions, like the intersections of two circles or conics, or angle bisectors, are used, it is necessary to handle these with special care to avoid mathematical inconsistencies. Using a continuation technique on associated Riemann surfaces[4] Cinderella can guarantee – up to numerical errors – that continuous changes of parameters (i.e. free points) will cause continuous changes of the construction (within its realization space). As a consequence we do have a notion of “dynamic theorems”, consisting of a sequence of geometric construction steps and an element of the “fiber” over a given assignment of free values as a “germ.” The theory behind this gives rise to several mathematical questions that are interesting on their own,[6] but are not covered here. The intuitive user interface of Cinderella hides most of the mathematical details but exposes the mathematical content of geometry and makes a proper handling of geometry available not only for academic research, but also