Write the expression for curvature\(\left( \kappa \right)\).
\(\kappa = \left| {\frac{{dT}}{{ds}}} \right|\)-----(1)
Write the tangent vector\(\left( T \right)\)for plane curve.
\(\begin{aligned}{l}T = \left| T \right|\cos \emptyset i + \left| T \right|\sin \emptyset j\\T = \left| T \right|\left( {\cos \emptyset i + \sin \emptyset j} \right)\end{aligned}\)-----(2)
Here,
\(\emptyset \)is angle between\(T\)and\(i\).
Consider the value of\(\left| T \right| = 1\).
Substitute\(1\)for\(\left| T \right|\)in equation(2),
\(\begin{aligned}{l}T = \left( 1 \right)\left( {\cos \emptyset i + \sin \emptyset j} \right)\\ = \cos \emptyset i + \sin \emptyset j\end{aligned}\)
Re-arrange the term\(\frac{{dT}}{{ds}}\).
\(\begin{aligned}{l}\frac{{dT}}{{ds}} = \left( {\frac{{dT}}{{ds}}} \right)\left( {\frac{{d\emptyset }}{{d\emptyset }}} \right)\\\frac{{dT}}{{ds}} = \left( {\frac{{dT}}{{d\emptyset }}} \right)\left( {\frac{{d\emptyset }}{{ds}}} \right)\end{aligned}\)-----(3)
Find the value of\(\frac{{dT}}{{d\emptyset }}\).
\(\begin{aligned}{l}\frac{{dT}}{{d\emptyset }} = \frac{d}{{d\emptyset }}\left( {\cos \emptyset i + \sin \emptyset j} \right)\\ = - \sin \emptyset i + \cos \emptyset j\end{aligned}\)
Substitute\( - \sin \emptyset i + \cos \emptyset j\)for\(\frac{{dT}}{{d\emptyset }}\)in equation(3),
\(\frac{{dT}}{{ds}} = \left( { - \sin \emptyset i + \cos \emptyset j} \right)\left( {\frac{{d\emptyset }}{{ds}}} \right)\)
Apply modulus on the both sides of equation.
Substitute\(\left| {\frac{{d\emptyset }}{{ds}}} \right|\)for\(\left| {\frac{{dT}}{{ds}}} \right|\)equation(1),
\(\kappa = \left| {\frac{{d\emptyset }}{{ds}}} \right|\)
