Straight Line has Zero Curvature

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A straight lines has zero curvature.


From Equation of Straight Line in Plane: Slope-Intercept Form, a straight line has the equation:

$y = m x + c$

Differentiating twice with respect to $x$:

\(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds m\) Power Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds \dfrac {\d^2 y} {\d x^2}\) \(=\) \(\ds 0\)

By definition, the curvature of a curve is defined as:

$\kappa = \dfrac {y''} {\paren {1 + y'^2}^{3/2} }$

But we have that:

$y'' := \dfrac {\d^2 y} {\d x^2} = 0$

and so, as in general $y' := \dfrac {\d y} {\d x} = m \ne 0$:

$\kappa = \dfrac 0 {\paren {1 + m^2}^{3/2} }$

the curvature is zero.