# Straight Line has Zero Curvature

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## Theorem

A straight lines 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 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.

$\blacksquare$

## Sources

- 1952: H.T.H. Piaggio:
*An Elementary Treatise on Differential Equations and their Applications*(revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: Examples for solution: $(7)$