Multisection

Author

Eric W. Weisstein
March 1, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Multisection.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Multisection.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

k[α_,β_,γ_]:=2Δ/(a α+b β+c γ)
x[α_,β_,γ_]:=((k[α,β,γ]β-r+(k[α,β,γ]α-r)Cos[C])/(Sin[C]))
y[α_,β_,γ_]:=k[α,β,γ]α-r
rr[α_,β_,γ_]:={x[α,β,γ],y[α,β,γ]}

s=FullSimplify[Solve[{x==((k[α,β,γ]β-r+(k[α,β,γ]α-r)Cos[C])/(Sin[C])),y==k[α,β,γ]α-r},{α,β}]]

k-Section

Bisection

m=FullSimplify[(rr[α2,β2,γ2]+rr[α1,β1,γ1])/2]

FullSimplify[s/. Thread[{x,y}->m]]

KSection[#,2]&/@Partition[IdentityMatrix[3],2,1,1]

Trisection

m=FullSimplify[rr[α1,β1,γ1]+(rr[α2,β2,γ2]-rr[α1,β1,γ1])/3]

FullSimplify[s/. Thread[{x,y}->m]]

KSection[#,3]&/@{#,Reverse[#]}&[{{1,0,0},{0,1,0}}]

KSection[#,3]&/@{#,Reverse[#]}&[{{0,1,0},{0,0,1}}]

KSection[#,3]&/@{#,Reverse[#]}&[{{1,0,0},{0,0,1}}]

4-Section

m=FullSimplify[rr[α1,β1,γ1]+(rr[α2,β2,γ2]-rr[α1,β1,γ1])/4]

FullSimplify[s/. Thread[{x,y}->m]]

KSection[#,4]&/@Partition[IdentityMatrix[3],2,1,1]

5-Section

m=FullSimplify[rr[α1,β1,γ1]+(rr[α2,β2,γ2]-rr[α1,β1,γ1])/5]

FullSimplify[s/. Thread[{x,y}->m]]