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

3.2 Kempe‘s Straight Line Linkwork
Figure 3.2: The final translations of Kempe‘s construction
4. In the same manner, we use angle adders on the OC s,t and OY to get points D s,t
with ∡XOD s,t = ∡XOC s,t − ∡XOY = sϕ + tθ − π 2 .
5. For every coefficient a s,t and b s,t in (3.1), we attach a bar OE s,t or OF s,t of corresponding
length to scale OC s,t or OD s,t s.t. |OE s,t | = a s,t , ∡XOE s,t = sϕ + tθ and
|OF s,t | = b s,t , ∡XOF st = sϕ + tθ − π 2 .
6. Finally, we geometrically represent the summation by a chain of translators:
We translate OE 0,−d to E 0,−(d−1) to get K E 0,−(d−1) , OKE 0,−(d−1) to OE 0,−(d−2) to get
K0,−(d−2) E , . . . , OE 1,−d to K0,d E to get KE 1,−(d−1) , . . . , to get KE d,d
. In the same manner,
we add the F s,t , to ultimately obtain Kd,d F =: S.
Now, by construction, the links E s,t and F s,t have x-coordinate equal to a cos(sϕ + tθ)
and b cos ( sϕ + tθ − π 2
)
. Therefore, the x-coordinate of S is
x =
(
∑ as,t cos(sϕ + tθ) + b s,t cos ( sϕ + tϕ − π ))
2
0≤s≤d, −d≤t≤d
(s,t)̸=(0,0)
= ˜f m,n (ϕ, θ) − c
= f (x, y) − c.
When P moves along C, f (P) = f (x, y) = ˜f m,n (ϕ, θ) = 0; accordingly, for S
x = f (x, y) − c = −c
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