# Curvature and Radius of Curvature

**Curvature** (symbol, $\kappa$) is the mathematical expression of how much a curve actually curved. It is the measure of the average *change in direction* of the curve per *unit of arc*. Imagine a particle to move along the circle from point 1 to point 2, the higher the number of $\kappa$, the more quickly the particle changes in direction. This quick change in direction is apparent in smaller circles.

$\Delta s = \rho \cdot \Delta \alpha$

$\dfrac{1}{\rho} = \dfrac{\Delta \alpha}{\Delta s}$ ← the curvature

Let 1/ρ = κ

$\kappa = \dfrac{\Delta \alpha}{\Delta s}$

It is important to note that curvature κ is reciprocal to the radius of curvature ρ according to the above equations.

As *P*_{2} approaches *P*_{1}, the ratio Δα/Δ*s* approaches a limit. This limit is the curvature of the curve at a particular point, and from the above figure that point is *P*_{1}.

$\displaystyle \kappa = \lim_{\Delta s \to 0}\dfrac{\Delta \alpha}{\Delta s}$

$\kappa = \dfrac{da}{ds}$

**Curvature in xy-Plane**

In a circle, κ is constant, however, if the curve in question is not a circle, κ represents the average curvature of the arc at a particular point.

From Analytic Geometry, the slope of the line *m* is equal to the tangent of angle of inclination, or *m* = tan α. Note that in Calculus, *m* = *dy*/*dx*.

$\tan \alpha = y'$

$\alpha = \arctan y'$

Differentiate both sides with respect to *x*:

$d\alpha = \dfrac{y'' \, dx}{1 + (y')^2}$

Note that the Differential Length of Arc in the xy-plane is given by this formula:

$ds = \sqrt{1 + (y')^2} \, dx$

Hence,

$\kappa = \dfrac{da}{ds} = \dfrac{\dfrac{y'' \, dx}{1 + (y')^2}}{\sqrt{1 + (y')^2} \, dx}$

Simplify the equation above, and we have this formula for Curvature:

And for the Radius of Curvature:

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