Confusion regarding composition of two spatial rotations of a sphere.

ShahrOZe999

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Joined
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? enter image description here
 

Dr.Peterson

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Nov 12, 2017
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There are some errors resulting from pasting (I don't suppose you used Preview to see how it would look?), and a lot that you haven't defined. Maybe an image would work better, along with a fuller explanation.
 
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