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: $(7)$