■シュレーフリの公式と直角三角錐(その169)
結局、シュレーフリ記号と2面角から3行目を決めて(2,0)=2とすればよい
ユークリッド空間の基本単体では
(sinα)^2(sinγ)^2-(cosβ)^2=0
sinαsinγ-cosβ=0
(tanα)^2(tanγ)^2-(cosβ)^2/(cosα)^2(cosγ)^2=0
が導き出せればよい
4次元の場合はどれくらい複雑になるのだろうか?
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(0,2)(1,3)=(secβ)^2
(1,3)(2,4)=(secγ)^2
(1,3)=(secβ)(secγ)
(0,2)=(secβ)(cosγ)
(2,4)=(secγ)(cosβ)
(-1,1)(0,2)=(secα)^2
(2,4)(3,5)=(secδ)^2
(-1,1)=(secα)^2cosβsecγ
(3,5)=(secδ)^2secβcosγ
(-1,2)=(tanα)^2
(0,3)=(tanβ)^2
(1,4)=(tanγ)^2
(2,5)=(tanδ)^2
(tanα)^2(tanβ)^2-(secβ)(cosγ)(-1,3)=1
(-1,3)={(tanα)^2(tanβ)^2-1}/(secβ)(cosγ)
={(sinα)^2(sinβ)^2-(cosα)^2(cosβ)^2}/(cosα)^2cosβcosγ
(tanβ)^2(tanγ)^2-(secβ)(secγ)(0,4)=1
(0,4)={(tanβ)^2(tanγ)^2-1}/(secβ)(secγ)
={(sinβ)^2(sinγ)^2-(cosβ)^2(cosγ)^2}/cosβcosγ
(tanγ)^2(tanδ)^2-(secγ)(cosβ)(1,5)=1
(1,5)={(tanγ)^2(tanδ)^2-1}/(secγ)(cosβ)
={(sinγ)^2(sinδ)^2-(cosγ)^2(cosδ)^2}/(cosδ)^2cosβcosγ
(-1,3)(0,4)-(tanβ)^2(-1,4)=1
{(sinα)^2(sinβ)^2-(cosα)^2(cosβ)^2}{(sinβ)^2(sinγ)^2-(cosβ)^2(cosγ)^2}/(cosα)^2(cosβ)^2(cosγ)^2-(tanβ)^2(-1,4)=1
(0,4)(1,5)-(tanγ)^2(0,5)=1
{(sinγ)^2(sinδ)^2-(cosγ)^2(cosδ)^2}{(sinβ)^2(sinγ)^2-(cosβ)^2(cosγ)^2}/(cosδ)^2(cosβ)^2(cosγ)^2-(tanγ)^2(0,5)=1
(-1,4)(0,5)=1
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