■フルヴィッツ曲線(その64)

公転と自転の向きを逆方向にとると,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

cos(mθ+β)-cos(n-1)β=0

sin(mθ+nβ)/2sin(mθ-(n-2)β)/2=0

(mθ+nβ)/2=0,π、2π、3π、・・・

(mθ-(n-2)β)/2=0,π、2π、3π、・・・

m=n=4

  x=2cos(4β+θ)+4cos(2β-θ)−16sin(β+θ)+2cos(3θ)

  y=-2sin(4β+θ)+4sin(2β-θ)−16cos(β+θ)+2sin(3θ)

mθ+nβ=0,θ=-βを代入すると

  x=2cos(3β)+4cos(3β)+2cos(3β) =8cos(3β)

  y=-2sin(3β)+4sin(3β)−16-2sin(3β)=-16 (直線)

直線部分の長さは16

(mθ-(n-2)β)/2=0、β=2θを代入すると

  x=2cos(9θ)+4cos(3θ)−16sin(3θ)+2cos(3θ)

  y=-2sin(9θ)+4sin(3θ)−16cos(3θ)+2sin(3θ)

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