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##x(s)=\cos\frac{s}{\sqrt{2}}##

##y(s)=\sin\frac{s}{\sqrt{2}}##

##z(s)=\frac{s}{\sqrt{2}}##,

it is a unit-speed helix. Its curvature is ##\kappa=||\ddot{r}||=\frac{1}{2}##. Principal unit normal is ##{\mathbf n}=(\cos\frac{s}{\sqrt{2}},\sin\frac{s}{\sqrt{2}},0)##. So far so good...

But the helix in cylindrical coordinates is

##r(s)=1##

##\theta(s)=\frac{s}{\sqrt{2}}##

##z(s)=\frac{s}{\sqrt{2}}##

It is still unit-speed, ##||\dot{r}||=1##, but ##||\ddot{r}||=0##. What's wrong? How does one calculate the curvature and the principal unit normal in cylindrical coordinates?...

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# Principal normal and curvature of a helix

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