■ペリトロコイド曲線(その11)

  x=Rcos(β+γ−θ)+acos((n−1)β−θ)+acos((n−2)θ)

  y=Rsin(β+γ−θ)+asin((n−1)β−θ)+asin((n−2)θ)

に対して

  (∂y/∂β)(∂x/∂θ)−(∂x/∂β)(∂y/∂θ)=0

を計算すると

  θ=β−2/(n−1)arctan(Rsin((n−2)β−γ)/(Rcos((n−2)β−γ)+(n−1)a))

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∂y/∂β=Rcos(β+γ−θ)+a(n−1)cos((n−1)β−θ)

∂x/∂θ=Rsin(β+γ−θ)+asin((n−1)β−θ)-a(n−2)sin((n−2)θ)

∂x/∂β=-Rsin(β+γ−θ)-a(n−1)sin((n−1)β−θ)

∂y/∂θ=-Rcos(β+γ−θ)-acos((n−1)β−θ)+a(n−2)cos((n−2)θ)

Ra(n-2)sin(-(n-2)β+γ)+Ra(n-2)sin(β+γ−(n-1)θ)+a^2(n-1)(n-2)sin((n−1)β−(n-1)θ)=0

Rsin(β+γ−(n-1)θ)+a(n-1)sin((n−1)β−(n-1)θ)=Rsin((n-2)β-γ)

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Rsin(β+γ)cos(n-1)θ-Rcos(β+γ)sin(n-1)θ

a(n-1)sin(n−1)βcos(n-1)θ-a(n-1)cos(n−1)βsin(n-1)θ)=Rsin((n-2)β-γ)

{Rsin(β+γ)+a(n-1)sin(n−1)β}cos(n-1)θ

-{Rcos(β+γ)+a(n-1)cos(n−1)β}sin(n-1)θ=Rsin((n-2)β-γ)

A={Rsin(β+γ)+a(n-1)sin(n−1)β}

B={Rcos(β+γ)+a(n-1)cos(n−1)β}

C=Rsin((n-2)β-γ)

A^2+B^2=R^2+(a(n-1))^2+2Ra(n-1)cos((n-2)β-γ)

A^2+B^2-C^2=R^2cos^2((n-2)β-γ)+(a(n-1))^2+2Ra(n-1)cos((n-2)β-γ)

=(Rcos((n-2)β-γ)+a(n-1))^2

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  Acos((n−1)θ)-Bsin((n−1)θ)=C

の形に整理されます.

 ここで

  cosψ=A/(A^2+B^2)^(1/2),

  sinψ=B/(A^2+B^2)^(1/2),

  tanψ=B/A

とおくと,

  cos((n−1)θ+ψ)=C/(A^2+B^2)^(1/2)

より

  (n−1)θ=-arccos(A/(A^2+B^2)^(1/2))+arccos(C/(A^2+B^2)^(1/2))

 =−arctan(B/A)+arctan((A^2+B^2−C^2)^(1/2)/C)

 なかなか合致しない・・・

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