91影视

Given:

$P_0$be a point on a curve C with positive curvature 魏

We have to define the radius of curvature at$P_0$.

Let $r\left(t\right)=鉄�t,伪\sin 鈦�尾t,伪\cos 鈦�尾t鉄�$and t=0

$r\left(t\right)=鉄�t,伪\sin 鈦�尾t,伪\cos 鈦�尾t鉄�$

at $t=0,r\left(0\right)=鉄�0,0,伪鉄�$

Differentiate r(t) with respect to t,

$r^{\mathrm{鈥�}}\left(t\right)=鉄�1,伪尾\cos 鈦�尾t,鈭�伪尾\sin 鈦�尾t鉄�$

and,

localid="1649998970850" $$\begin{gathered} 鈭�r^{\mathrm{鈥�}}\left(t\right)鈭�=\sqrt{1+\left(伪尾\cos 尾t^2\right)+(鈭�伪尾\sin 尾t{)}^2} \\ =\sqrt{1+{伪}^2{尾}^2} \end{gathered}$$

Now,

localid="1649999060548"

Since, $鈭�r^{\mathrm{鈥�}}\left(t\right)鈭�=\sqrt{1+{伪}^2{尾}^2}$then to have $鈭�r^{\mathrm{鈥�}}\left(0\right)鈭�=\sqrt{1+{伪}^2{尾}^2}$

Since, $鈭�T^{\mathrm{鈥�}}\left(t\right)鈭�=\frac{伪{尾}^2}{\sqrt{1+伪{尾}^2}}$which is a constant.

Then,

$鈭�T^{\mathrm{鈥�}}\left(0\right)鈭�=\frac{伪{尾}^2}{\sqrt{1+{伪}^2尾}}$

The curvature k of c at a point on the curve is given by :

$k=\frac{鈭�T^{\mathrm{鈥�}}\left(0\right)鈭�}{鈭�r^{\mathrm{鈥�}}\left(0\right)鈭�}$

localid="1649999096384" $$\begin{gathered} =\frac{伪{尾}^2}{\sqrt{1+{伪}^2{尾}^2}\sqrt{1+{伪}^2{尾}^2}} \\ =\frac{伪{尾}^2}{1+{伪}^2{尾}^2} \end{gathered}$$

Therefore, the radius of the curvature is :

localid="1649999122418" $$\begin{gathered} 蚁=\frac{1}{k} \\ 蚁=\frac{1+{伪}^2{尾}^2}{伪{尾}^2} \end{gathered}$$