ShahrOZe999
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- Sep 5, 2021
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I am currently studying Complex analysis from the book "Visual Complex Analysis" by Tristan Needham. In the chapter "Non-Euclidian Geometry" on page 282 the author says that the compostions of two Rotations is a Rotation and since a Rotation on a sphere can be decomposed into two reflections.
Rϕq∙Rθq=(RN∙RM)∙(RM∙RL)=RN∙RL=RψrRqϕ∙Rqθ=(RN∙RM)∙(RM∙RL)=RN∙RL=Rrψ
Then the author says that the angle ψψ can be found out by looking at the white spherical triangle. The area of the Triangle is AA and curvature of sphere is kk so the sum of the angles θ/2+ϕ/2+(π−ψ/2)=π+kAθ/2+ϕ/2+(π−ψ/2)=π+kA so ψ=θ+ϕ−2kAψ=θ+ϕ−2kA. But this is not the formula that is written on the book. Where am i wrong?
Rϕq∙Rθq=(RN∙RM)∙(RM∙RL)=RN∙RL=RψrRqϕ∙Rqθ=(RN∙RM)∙(RM∙RL)=RN∙RL=Rrψ
Then the author says that the angle ψψ can be found out by looking at the white spherical triangle. The area of the Triangle is AA and curvature of sphere is kk so the sum of the angles θ/2+ϕ/2+(π−ψ/2)=π+kAθ/2+ϕ/2+(π−ψ/2)=π+kA so ψ=θ+ϕ−2kAψ=θ+ϕ−2kA. But this is not the formula that is written on the book. Where am i wrong?
