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

4.3 Geometric Primitives in Linkage Simulation
to generate mX and nX is presented in step 5 and does not depend on
intermediate elements; so we can for now assume to have given points from
which we can take the lengths m and n to the origin and draw circles of
radius m around O and n around P.
One of their intersection points is chosen as a possible instance of A 1 . Joins,
i.e. defining a line (achieved by Peaucellier cells in linkages), are primitive
operations in a DGS as well as the drawing of parallels through given points,
so we can determine the point B 1 as the intersection of the parallels to OA 1
through P and A 1 P through O.
1. In Cinderella, contrary to deterministic geometry systems, we have to face the
same problems regarding the orientation of the y-axis discussed in section 3.2.
Just choosing a third free point will prevent the program to apply it‘s incidence
checking on elements depending on the axis, since the defining third point might
move later. Thus we have to take any of the two intersections of the unit circle
and the y-axis (defined as the perpendicular on OX through O) for Y.
This is just mentioned for the sake of completeness and does not cause any
trouble, because after all the same instruction sequence will always cause the
same result to appear on the screen.
2. We now generate points on the unit circle representing the angles ϕ and θ. This
can be done by dividing the distances of A 1 and B 1 to O by m and n; again, we
refer to step 5.
For clearness, in figure 4.5 (a) the points are already assumed to be on the unit
circle. Then to yield a multiple of an angle (θ, corresponding to B 1 , in the figure)
it is sufficient to iteratively draw circles around B t through B t−1 , starting with
B 1 and X. There is at most one additional intersection of these circles with the
unit circle besides B t−1 ; this point gives B t+1 with angle ∡XOB t+1 = (t + 1)θ by
congruence of △B t−1 OB t and △B t OB t+1 .
If there is no additional intersection, ∡B t−1 OB t = π, which is a degenerate case
not induced by boundary constraints other than the position of P, which is
extension of degree two of Q. The field operations +, −, · and ·· are linear operations, which can be
achieved by the ruler; additionaly, the constructible numbers are closed under √·, which corresponds
to the solving of polynomial equations in degree two, or, geometrically, intersections of circles and
lines or circles.
All operations but root extraction involved are used and presented throughout this construction; for
a more concise treatment of the topic see e.g. [Lab08] or introductory textbooks on algebraic number
theory.
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