Automated Generation of Kempe Linkages for ... - Alexander Kobel

1 Introduction
without actually giving a hint how to build an ideal ruler in the first place. Thus,
it appears ironical that Kempe himself is best known among mathematicians for his
false proof of the four colour theorem, and he also turned out to be careless in his
linkage design. However, both these works rendered a great service to the mathematical
community and have been corrected later; both proofs involve Kempe‘s original key
ideas.
With his work, Kempe eventually settled an important question in engineering from
the theoretical point of view; although he gave a constructive description of the design
of the linkwork, the complexity of the mechanisms exceeds the practically relevant limit.
It remained an open field to the engineers to figure out easier approximate solutions;
however, simulations of the linkages turned out to be useful in modern applications,
such as CAD and robotics.
Kempe‘s conclusions have been extended later to arbitrary dimensions. With some
additional restrictions not considered by him, there still is uncharted territory left to
further research.
1.2 Dynamic Geometry in Cinderella
Everyone of us may with mixed feelings remember the geometry lessons in school.
Probably nobody dealt with the class without some torn sheets of paper, with drawings
at the wrong scale or a cluttered choice of parameters for constructions.
Dynamic geometry systems (DGS) tackle this problem, allowing to arbitrarily zoom
and shift the viewport or modify some input elements and automatically rearrange
the depending parts accordingly. The current calculating capacity of usual personal
computers, available at cheap prices and in increasing quantities found in schools,
also allow interactive animations and tracing of elements under movements of others,
capable of drastically easing the teaching of the relation of mathematics and the “real
world”. Beyond the applications in high school lessons, dynamic geometry programs
increasingly aim to include more in-depth topics, like introductory projective geometry,
which demand a great deal of imagination from the student.
Amongst a number of tools mainly targeted at the profitable market of high school
pupils, Cinderella takes a special position. Throughout the twelve-years developement
process of Cinderella up to now, the authors Ulrich Kortenkamp and Jürgen Richter-
Gebert took great care of implementing a thorough mathematical model of geometry.
Besides features such as polar views on constructions or the representation of hyperbolic
geometry, there are two major differences in design compared to standard
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