Straight Line has Zero Curvature

Theorem
A straight line has zero curvature.

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


 * $y = m x + c$

Differentiating twice $x$:

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.