■フルヴィッツ曲線(その29)

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、・・・

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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、・・・

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辺縁は円だろうか?

m=2,n=4

(mθ+nβ)/2=π/2

(2θ+4β)=π

θ=π/2-2β

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

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

  x=2cos(6β-π/2)+4cos(π/2)−16sin(3β-π/2)+2cos(3π/2-6β)

  y=-2sin(6β-π/2)+4sin(π/2)−16cos(3β-π/2)+2sin(3π/2-6β)

  x=2sin(6β)+16cos(3β)-2sin(6β)

  y=2cos(6β)+4−16sin(3β)-2cos(6β)

  x=+16cos(3β)

  y=4−16sin(3β)

x^2+(y-4)^2=16^2

これは円である。

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辺縁は直線だろうか?

m=2,n=4

(mθ-(n-2)β)/2=π/2

(2θ-2β)=π

θ=π/2+β

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

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

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

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

  x=2sin(3β)-4sin(3β)+16+2sin(3β)

  y=2cos(3β)+4cos(3β)-2cos(3β)

  x=16

  y=4cos(3β)

これは直線である。

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