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

2 Basic Linkages
2.4 Angle Adder and Multiplicator
We finish the presentation of the basic linkworks with two constructions on angle
operations. In particular, we show that linkage designs include all necessary tools to
implement the Z-module structure of angles.
2.4.1 The Angle Adder
A common task in linkage design is the addition of a (finite) number of angles. This
can easily be achieved by inductively adding two angles by means of an angle adder
consisting of distance copiers:
Suppose we are given two angles ∡AOB and ∡CO ′ D. W.l.o.g. we can assume
|OA| = |OB| = |O ′ C| = |O ′ D|; otherwise, we use platforms to constrain the links
accordingly. We then attach two distance copiers to move CO ′ D such that the image
of O ′ C is OA, yielding ∡CO ′ D = ∡C ′ O ′′ D ′ = ∡AOD ′ , and now use another distance
copier to move |AB| to D ′ T such that AB is rotated by the angle ∡AOD ′ = ∡CO ′ D.
Since we demanded the lengths of the legs to be equal, both AB and AD ′ are chords
in a circle with center O. Therefore, ∡AOB + ∡CO ′ D = ∡AOB + ∡AOD ′ = ∡AOT.
2.4.2 The Angle Multiplicator
Now suppose we want to construct an angle multiplicator, i.e. a mechanism to multiply
an angle by an integer. Again, we can assume both legs to share the same lengths, so
we can use a distance copier in the same manner to double, triple, . . . the angle.
However, there is an easier approach to solve this special case, shown in figure 2.6.
Here OADB and OBEC are antiparallelograms; the links satisfy |OA| · |OC| = |OB| 2 ⇔
|OA|
= |OB|
|OB| |OC| .
Of course we may not allow |OA| = |OB| in this setting, which leads to the degenerate
position D = 0. This again is not a real restriction, since we can employ platforms to
get distinct lengths of the legs. In general, we will have to do so anyway, because we
have to guarantee the ratio of the bar lengths.
Since |OA| , OADB is similar to OBEC, yielding ∡AOB = ∡BOC and thus
= |OB|
|OB| |OC|
∡AOC = ∡AOB + ∡BOC = 2∡AOB. Iterating the construction, we can get a n times
angle multiplicator for arbitrary, but fixed n ∈ N.
Kempe [Kem76] uses the term angle reversor to call the multiplicator, because ∡BOA =
−∡COB; thus the mechanism also allows to change the orientation of arbitrary angles.
Together with the angle adder, we are able to subtract angles. As a side remark, which
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