■フルヴィッツ曲線(その66)

公転と自転の向きを同じ方向にとると,フルヴィッツ曲線の運動族は

  x=(n-2)acos(nβ-θ)+nacos((n−2)β+θ)−2Rsin(β-θ)+2acos((n−1)θ)

  y=-(n-2)asin(nβ-θ)+nasin((n−2)β+θ)−2Rcos(β-θ)+2asin((n−1)θ)

m=n-2

cos(mθ+β)+cos(n-1)β=0

cos(mθ+nβ)/2cos(mθ-(n-2)β)/2=0

(mθ+nβ)/2=π/2、3π/2、5π/2、・・・

(mθ-(n-2)β)/2=π/2、3π/2、5π/2、・・・

m=1、n=3

  x=cos(3β-θ)+3cos(β+θ)−6sin(β-θ)+2cos(2θ)

  y=-sin(3β-θ)+3sin(β+θ)−6cos(β-θ)+2sin(2θ)

(mθ+nβ)/2=π/2。θ=-3β+πを代入

  x=cos(6β-π)+3cos(-2β+π)−6sin(4β-π)+2cos(-6β+2π)

  y=-sin(6β-π)+3sin(-2β+π)−6cos(4β-π)+2sin(-6β+2π)

  x=-cos(6β)-cos(2β)+6sin(4β)+2cos(6β)

  y=sin(6β)+3sin(2β)+6cos(4β)-2sin(6β)

(mθ-(n-2)β)/2=π/2,θ=β+πを代入

  x=cos(2β-π)+3cos(2β+π)+2cos(2β)

  y=-sin(2β-π)+3sin(2β+π)+6+2asin(2β)

  x=-cos(2β)-3cos(2β)+2cos(2β)=−2cos(2β)

  y=+sin(2β)-3sin(2β)+6+2asin(2β)=6  (直線)

直線部分の長さは4と思われる

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