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

2 Basic Linkages
(a)
(b)
Figure 2.4: Two translators.
Note that the length of the bars is both a necessary and sufficient condition for the
shape of the linkwork, as long as we guarantee that no crossings of the bars occur,
leading to an antiparallelogram.
While this simple mechanism works flawlessly in theory, it is not convenient, since
the distances |AB| and |AS| are fixed. We can do better with a combination of four
such translators (see figure 2.4 b). The correctness of this linkage directly results from
the fact that parallelity is an equivalence relation of lines in the plane and, as such, in
particular transitive.
For the sake of simplicity, we can assume that any parallel bars have the same length.
This assumption is without loss of generality; the only real restriction is given by the
sum of the lengths of the bars compared to |AB| and |AS|.
When in the following we say that a segment or distance is translated to another
point, we refer to the use of this refined linkage to achieve this.
2.3 The Rotator
To complete the movements of segments in the plane, in addition to translation we
have to be able to perform rotations. Figure 2.5 shows a linkwork to achieve this.
Here, |OA| = |OB|, |AC| = |BC|, |DS| = |DT|, and |DU| = |DV|. |OA| = |OB| is
not a real restriction; if source and target direction (along OA or OB) are given with
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