- #1
kkz23691
- 47
- 5
Hi,
##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?...
##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?...