# Straight Line has Zero Curvature

## Theorem

A straight lines has zero curvature.

## Proof

$y = m x + c$

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.

$\blacksquare$