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 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: $(7)$