How Much Shorter is the Straight Line?
Consider a sphere of radius R = 9 centered on the origin ( x , y , z ) = ( 0 , 0 , 0 ) . Find the minimum path length from ( 8 , 4 , 1 ) to ( − 3 , − 6 , − 6 ) , subject to the constraint that the path must not leave the sphere's surface.
Enter your answer as the ratio of this distance to the straight-line distance between the two points.
The answer is 1.26.
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3 solutions
S t r a i g h t − l i n e d i s t a n c e = [ 8 − ( − 3 ) ] 2 + [ 4 − ( − 6 ) ] 2 + [ 1 − ( − 6 ) ] 2 = 2 7 0 This line forms an isosceles triangle at the center with the two radius, and say vertex angle X. X = ( 2 ∗ S i n − 1 2 ∗ 9 2 7 0 ) c . S o t h e a r c d i s t a n c e = ( 2 ∗ S i n − 1 2 ∗ 9 2 7 0 ) c ∗ 9 . ∴ r e q u i r e d r a t i o = 2 7 0 2 ∗ S i n − 1 2 ∗ 9 2 7 0 c ∗ 9 . = 1 . 2 6 0 0
Take a circle of radius 9 . Now let a given chord of thus circle subtend angle θ at centre.
The minor arc represnts the smallest path on sphere and path along chord is straigh line distance.
Thus 1 8 s i n ( θ / 2 ) = 2 7 0 .
And arc length = 9 × θ
Key to answer : Visualisation in 3-D + 2-D .