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for a geodesic right cylinder with radius R. Find the curve taht minimizes the distance between two points r1 and r2. where r = (R,phi, z) in cylindrical polar coordinates. Express your answer as z = z(phi)

pardon the sloppy math

not the most fun to get this type of question on a 50 minute test! Anyway,

x = r cos t

y = r sin t

z = z

[tex] \mbox{distance} = \int_{1}^{2} \sqrt{r^2 + \left( \frac{\partial z}{\partial \theta}\right)^2} [/tex]

ist hat the distance between two points on a cylinder?

Or would this distane be represented by something of a helix? I mean a cylinder could be considered as helix... right?

pardon the sloppy math

not the most fun to get this type of question on a 50 minute test! Anyway,

x = r cos t

y = r sin t

z = z

[tex] \mbox{distance} = \int_{1}^{2} \sqrt{r^2 + \left( \frac{\partial z}{\partial \theta}\right)^2} [/tex]

ist hat the distance between two points on a cylinder?

Or would this distane be represented by something of a helix? I mean a cylinder could be considered as helix... right?

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