Given the hyperbolic metric d s 2 </msup> = d

desertiev5 2022-07-10 Answered
Given the hyperbolic metric d s 2 = d x 2 + d y 2 x 2 on the half plane x > 0, find the length of the arc of the circle x 2 + y 2 = 1 from ( cos α , sin α ) to ( cos β , sin β )

I found that d s 2 = d θ 2 cos 2 θ but when I try to plug in π / 3 , π / 3, which should give me the arc length of 2 π / 3,
I get 4 π / 3 = ( π / 3 ( π / 3 ) ) 2 c o s 2 ( π / 3 )
I feel like I'm making a simple mistake but I cant place it
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Answers (1)

thatuglygirlyu
Answered 2022-07-11 Author has 14 answers
The circle x 2 + y 2 = 1 can be parametrised by ( cos θ , sin θ ). If x ( θ ) = cos θ and y ( θ ) = sin θ then
d s 2 = d x 2 + d y 2 x 2 = ( sin 2 θ + cos 2 θ ) d θ 2 cos 2 θ = sec 2 θ d θ 2 .
The arc-length that you are interested in is given by:
s = d s 2 = α β | sec θ | d θ .
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