■フルヴィッツ曲線(その39)

内外の包絡線を分離したいのであるが、・・・

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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、・・・

θ=β+(2k-1)π/(n-2)を代入すると

  x=(n-2)acos((n-1)β-(2k-1)π/(n-2))+nacos((n−1)β+(2k-1)π/(n-2))−2Rsin(-(2k-1)π/(n-2))+2acos((n−1)β+(2k-1)π+(2k-1)π/(n-2))

  y=-(n-2)asin((n-1)β-(2k-1)π/(n-2))+nasin((n−1)β+(2k-1)π/(n-2))−2Rcos(-(2k-1)π/(n-2))+2asin((n−1)β+(2k-1)π+(2k-1)π/(n-2))

  x=(n-2)cos((n-1)β-(2k-1)π/(n-2))+ncos((n−1)β+(2k-1)π/(n-2))−2n(n-2)sin(-(2k-1)π/(n-2))-2cos((n−1)β+(2k-1)π/(n-2))

  y=-(n-2)sin((n-1)β-(2k-1)π/(n-2))+nsin((n−1)β+(2k-1)π/(n-2))−2n(n-2)cos(-(2k-1)π/(n-2))-2sin((n−1)β+(2k-1)π/(n-2))

  x=(n-2)cos((n-1)β-(2k-1)π/(n-2))+(n-2)cos((n−1)β+(2k-1)π/(n-2))+2n(n-2)sin((2k-1)π/(n-2))

  y=-(n-2)sin((n-1)β-(2k-1)π/(n-2))+(n-2)sin((n−1)β+(2k-1)π/(n-2))−2n(n-2)cos((2k-1)π/(n-2))

  x=2(n-2)cos((n-1)β)cos((2k-1)π/(n-2))+2n(n-2)sin(-(2k-1)π/(n-2))

  y=2(n-2)cos((n-1)β)sin((2k-1)π/(n-2))−2n(n-2)cos(-(2k-1)π/(n-2))

xsin((2k-1)π/(n-2))-ycos((2k-1)π/(n-2))=2n(n-2) (直線)

βは0-2π/(n-1)

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θ=-β+(2k-1)π/(n-2)を代入すると

  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)

  x=(n-2)acos((n+1)β-(2k-1)π/(n-2))+nacos((n-3)β+(2k-1)π/(n-2))−2Rsin(2β-(2k-1)π/(n-2))+2acos(-(n−1)β+(2k-1)π+(2k-1)π/(n-2))

  y=-(n-2)asin((n+1)β-(2k-1)π/(n-2))+nasin((n-3)β+(2k-1)π/(n-2))−2Rcos(2β-(2k-1)π/(n-2))+2asin(-(n−1)nβ/(n-2)+(2k-1)π-(2k-1)π/(n-2))

  x=(n-2)acos((n+1)β-(2k-1)π/(n-2))+nacos((n-3)β+(2k-1)π/(n-2))+2Rsin(2β-(2k-1)π/(n-2))-2acos((n−1)β-(2k-1)π/(n-2))

  y=-(n-2)asin((n+1)β-(2k-1)π/(n-2))+nasin((n-3)β+(2k-1)π/(n-2))−2Rcos(2β-(2k-1)π/(n-2))+2asin((n−1)β-(2k-1)π/(n-2))

  x=(n-4)acos((n+1)β-(2k-1)π/(n-2))+nacos((n-3)β+(2k-1)π/(n-2))+2Rsin(2β-(2k-1)π/(n-2))

  y=-(n-4)asin(n(n-1)/(n-2)β-(2k-1)π/(n-2))+nasin((n-3)β+(2k-1)π/(n-2))−2Rcos(2β-(2k-1)π/(n-2))

βの定義域は、2π(1-1/(n-1)),2π)

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