α=π/n,頂点(0,1),底辺の中点(0,-cosα)

c=cosα、s=sinα、η=2c(1+c)

E=(0,1)-η(s,c)を中心として

半径r=η=2c(1+c)の弧を描けばよいことになる。

n=3のとき、η=3/2

E=(0,1)-3/2(√3/2,1/2)=(-3√3/4,1/4)

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Ex=-(1+cosα)cosψ1cosψ2/sinα

Ey=1/2(1+cosα)sin(ψ1-ψ2)/sinα+(1-cosα)/2

Er=η

頂点を通る円弧の場合、

ψ1=2α-π/2、ψ2=π/2-α

tanα=(1+cosα)/x

x=(1+cosα)/tanα

η=2xsinα=2c(1+c)

Ex=-(1+cosα)sin2α＝2c(1+c)s

Ey=-1/2(1+cosα)sin3α/sinα+(1-cosα)/2=1-2c^2(1+c)

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C(-(1+cosα)/2tanα,-cosα)=(-c(1+c)/2s,-c)

外接円の半径は1

頂点の座標は

P0(0,1)

P1(sin2α,cos2α)

P2(sin4α,cos4α)

P3(sin6α,cos6α)となる。

P0P1

y-1=tan(π-α)・x =-tanα・x

Cからの距離η

|-c(1+c)/s・s/c-c-1|/{1+(tanα)^2}^1/2

{1+(tanα)^2}^1/2=1/cより2c(c+1)

垂線の傾きはψ2=π/2-α

ψ1=α-(π/2-α)=2α-π/2

δ=(1-r/c-r+c)/2

η=2xsinα=2c(1+c)

Er=η-r(sin2α+s)/s=2c(1+c)-r(2c+1)・・・OK

Ex=-(1-r/c-r+c)sin2α＝-(1-r/c-r+c)2sc

中心(Ex,yy),半径Erの円が接点(rs,1-r/c+rc)を通る

(x-Ex)^2+(y-yy)^2=Er^2

(rs+2(1-r/c-r+c)sc)^2+(yy-1+r/c-rc)^2=(2c(1+c)-r(2c+1))^2

s^2(r+2c-2r-2cr+2c^2))^2+(yy-1+r/c-rc)^2=(2c(1+c)-r(2c+1))^2

s^2(2c(1+c)-r(1+2c))^2+(yy-1+r/c-rc)^2=(2c(1+c)-r(2c+1))^2

(yy-1+r/c-rc)^2=c^2(2c(1+c)-r(2c+1))^2

yy-1+r/c-rc=-c(2c(1+c)-r(2c+1)

y=-2c^2(1+c)+1+rc(2c+1)+rc-r/c

y=-2c^2(1+c)+1+r(2c^2+c+c-1/c}

y=-2c^2(1+c)+1+r(2c^2+2c-1/c}・・・OK

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中心(Ex,yy),半径Erの円が接点(r√3/2,1-2r+r/2)を通る

(x-Ex)^2+(y-yy)^2=Er^2

(r√3/2+(1-2r)3√3/4)^2+(yy-1+3r/2)^2=(3/2-2r)^2

(-r√3+3√3/4)^2+(y-1+3r/2)^2=(3/2-2r)^2

3r^2-9r/2+27/16+y^2+2(-1+3r/2)y+(1-3r/2)^2=(3/2-2r)^2

y^2+2(-1+3r/2)y+3r^2-9/2r+27/16+1-3r+9r^2/4=9/4-6r+4r^2

y^2+2(-1+3r/2)y+7/16-3r/2+5r^2/4=0

y^2+2(-1+3r/2)y+7/16-24r/16+25r^2/16=0

y=1-3r/2+{1-3r+9r^2/4-7/16+24r/16-25r^2/4}^1/2

y=1-3r/2+(4r-3)/4=1/4-2r/4=(1-2r)/4

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P1P2

y-cos2α=tan(π-3α)・(x-sin2α) =-tan3α・(x-sin2α)

Cからの距離η

|(-c(1+c)/s-2sc)・tan3α-c-cos2α|/{1+(tan3α)^2}^1/2

{1+(tan3α)^2}^1/2=1/cos3αより

|(-c(1+c)/2s-2sc)・sin3α-(c+cos2α)cos3α|

(-c(1+c)/s-2sc)・sin3α=(-c(1+c)/s-2sc)・(-4s^3+3s)

=(-c(1+c)-2s^2c)・(-4s^2+3)

=(-c+c^2-2c(1-c^2))・(-4(1-c^2)+3)

=(2c^3+c^2-3c)(4c^2-1)=8c^5-c^4-14c^3+c^2+3c

(c+cos2α)cos3α=(2c^2+c-1)(4c^3-3c)=8c^5+4c^4-10c^3-3c^2+3c

η=|-8c^2-4c^3+4c^2|=4c^2(2c^2+c-1)

n=5のときc=φ/2

η=φ^2(φ^2/2+φ/2-1)=(φ+1)(φ-1/2)=φ^2+φ/2-1/2=3φ/2+1/2

一方

2c(c+1)=φ(φ/2+1)=φ^2/2+φ＝=3φ/2+1/2

垂線の傾きはψ2=π/2-3α

ψ1=α-(π/2-3α)=4α-π/2

n=5のとき、

Ex=-(1+cosα)cosψ1cosψ2/sinα

Ey=1/2(1+cosα)sin(ψ1-ψ2)/sinα+(1-cosα)/2

Er=η

P0P1について

垂線の傾きはψ2=π/2-α＝3π/10

ψ1=α-(π/2-α)=2α-π/2 =-π/10

P1P2について

垂線の傾きはψ2=π/2-3α=-π/10

ψ1=α-(π/2-3α)=4α-π/2= 3π/10

Exは変わらずEyの符号が変わる・・・合致

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PkPk+1について

y-cos2kα=tan(π-(2k+1)α)・(x-sin2kα) =-tan(2k+1)α・(x-sin2kα)

Cからの距離

|(-c(1+c)/2s-sin2kα)・tan(2k+1)α-c-cos2kα|/{1+(tan(2k+)α)^2}^1/2

{1+(tan(2k+1)α)^2}^1/2=1/cos(2k+1)αより

|(-c(1+c)/2s-sin2kα)・tan(2k+1)α-c-cos2kα|cos(2k+1)α

垂線の傾きはψ2=π/2-(2k+1)α

ψ1=α-(π/2-(2k+1)α)=4kα-π/2

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