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

3.1 Trigonometric Algebra
identities
sin α = cos ( α − π )
2 , (3.3)
cos n α = 1 n
( n
2 ∑ n cos ((n − 2k)α) and (3.4)
k)
k=0
cos α cos β = 1 (cos(α + β) + cos(α − β)) (3.5)
2
yields
f (x, y) (3.2)
= ∑ a i,j (m cos ϕ + n cos θ) i (m sin ϕ + n sin θ) j
0≤i+j≤d
(3.3)
= ∑
0≤i+j≤d
a i,j (m cos ϕ + n cos θ) i ( m cos ( ϕ − π ) ( ))
2 + n cos θ −
π j
2
(
i
= ∑ a i,j ∑ c i,k m k n i−k cos k ϕ cos i−k θ
0≤i+j≤d k=0
= ∑ a i,j
0≤i+j≤d
(3.4)
=
(3.5)
∑
∑
i
∑
k=0
0≤s≤d −d≤t≤d
( j
· ∑ c j,l m l n j−l cos l ( ϕ − π )
2 cos
j−l ( θ − π ) )
2
l=0
j
∑
l=0
)
·
c i,k c j,l m k+l n i+j−k−l cos k ϕ cos i−k θ cos l ( ϕ − π 2
)
cos
j−l ( θ − π 2
(
as,t cos(sϕ + tθ) + b s,t cos ( sϕ + tϕ − π ))
2 .
For s = t = 0 we can isolate the constant terms to finally get
)
f (x, y) = c +
∑
0≤s≤d, −d≤t≤d
(s,t)̸=(0,0)
(
as,t cos(sϕ + tθ) + b s,t cos ( sϕ + tϕ − π 2
))
(3.6)
where a s,t , b s,t , c ∈ R.
Gao et al. [GZCG02] point out that we can simplify this even further to get
f (x, y) = c +
∑
0≤s≤d, −d≤t≤d
(s,t)̸=(0,0)
(d s,t cos(sϕ + tθ + ψ s,t ))
where c, d s,t , ψ s,t ∈ R. This simplification however is, despite it‘s minor impact on the
theoretical complexity of the construction, objectionable for our needs, since ψ s,t in
general is not constructible by ruler-compass constructions even if f (x, y) ∈ Q[x, y].
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