Radius of Curvature

Radius of Curvature

$Learning Objectives$

Understand the formal mathematical definition and geometric meaning of curvature as the rate of change of direction.
Calculate the precise radius of curvature for any twice-differentiable Cartesian function $y = f(x)$ .
Determine the exact Cartesian coordinates of the center of curvature (the center of the osculating circle).
Compute curvature algebraically for complex curves defined by parametric and polar equations without converting back to Cartesian forms.
Relate curvature to path-normal acceleration on an idealized horizontal road curve, and distinguish this kinematic relation from a roadway design check.
Relate curvature to the small-slope Euler–Bernoulli beam model and state its assumptions.

Curvature and radius quantify how sharply a smooth curve bends. In civil engineering, this geometric quantity appears in road alignment and beam theory, but it is only one part of an engineering model: road assessment also depends on design speed, superelevation, side friction, and applicable criteria, while structural response depends on loads, supports, material behavior, and section stiffness.

Curvature

$Curvature (κ)$

Curvature, denoted by $\kappa$ , measures how fast a curve is changing direction at a given point. A straight line has zero curvature. A small circle has a large curvature, while a large circle has a small curvature.

The radius of curvature is the radius of the circle that best fits the curve at a specific point. This circle is called the osculating circle (from Latin osculari, meaning "to kiss"), because it touches the curve exactly at that point and has the same tangent and curvature.

Let a curve be defined by a twice-differentiable function

y = f(x)

. The Curvature ( $\kappa$ ) is:

Curvature Formula

Curvature of a function y = f(x).

\kappa = \frac{|y''|}{[1 + (y')^2]^{3/2}}

Variables

Symbol	Description	Unit
$\kappa$	Curvature	-
$y'$	First derivative of y with respect to x	-
$y''$	Second derivative of y with respect to x	-

The Radius of Curvature ( $R$ or $\rho$ ) is the reciprocal of the curvature when

\kappa \ne 0

. Where

\kappa=0

, the radius is interpreted as infinite.

Radius of Curvature Formula

Radius of the osculating circle.

R = \frac{1}{\kappa} = \frac{[1 + (y')^2]^{3/2}}{|y''|}

Variables

Symbol	Description	Unit
$R$	Radius of curvature	-
$\kappa$	Curvature	-

Kinematics on a Horizontal Road Curve

For motion along a smooth path at speed

v

, curvature gives the magnitude of path-normal acceleration:

a_n=v^2\kappa

. On a constant-radius circular path,

\kappa=1/R

, so

a_n=v^2/R

. A change in speed adds tangential acceleration. This is a kinematic relation for a specified path.

Path-Normal Acceleration from Curvature

The acceleration needed to follow a path, with the circular-path special case.

a_n = v^2\kappa \qquad \text{and, for a circle,} \qquad a_n = \frac{v^2}{R}

Variables

Symbol	Description	Unit
$a_n$	Magnitude of acceleration normal to the path, in m/s²	-
$v$	Speed along the path, in m/s	-
$\kappa$	Path curvature, in m⁻¹	-
$R$	Radius of the circular path, in m	-

$Curvature Is Not a Safety or Design Check$

The relation above gives path-normal acceleration; it is not the roadway side-friction demand, a safe-speed criterion, or a design-compliance check. U.S. FHWA guidance describes minimum horizontal-curve radius in terms of designated design speed, superelevation, and side-friction demand. The adopted jurisdiction's criteria govern an actual assessment. The beam relationship introduced below likewise depends on the Euler–Bernoulli model and its assumptions; it does not establish structural adequacy by itself.

A circular arc shows two tangent directions separated by a small turning angle d-phi over arc length ds. For a circle, curvature equals the absolute turning rate per distance, kappa equals |d-phi/ds| equals 1/R. — Curvature as Turning per Unit Arc Length
The angle between nearby tangents is dφ and the arc length between them is ds. Their ratio approaches curvature as the points approach one another. For a circle of radius R, this gives κ = 1/R.

$Interactive Simulation$

Use the simulation below to explore the radius of curvature and its corresponding osculating circle.

Osculating Circle Simulation

Move the slider to see how the radius of curvature and the osculating circle change along the parabola $y = x^2$ .

Position on curve:

x = 0.50

Point (x, y):(0.50, 0.25)

First Derivative (y'):1.00

Radius of Curvature (R):1.41

Center of Curvature

The center of curvature

(h, k)

is the center of the osculating circle. It is located on the normal line to the curve, at a distance

R

from the point of tangency

(x, y)

, on the concave side of the curve.

The formulas for the coordinates

(h, k)

of the center of curvature are derived using the normal slope:

x-coordinate of the Center of Curvature

The h coordinate of the osculating circle's center.

h = x - \frac{y'[1 + (y')^2]}{y''}

Variables

Symbol	Description	Unit
$h$	x-coordinate of the center of curvature	-
$x$	x-coordinate of the point of tangency	-
$y', y''$	First and second derivatives	-

y-coordinate of the Center of Curvature

The k coordinate of the osculating circle's center.

k = y + \frac{1 + (y')^2}{y''}

Variables

Symbol	Description	Unit
$k$	y-coordinate of the center of curvature	-
$y$	y-coordinate of the point of tangency	-

Parametric Equations of Curvature

In many engineering and physics applications (e.g., projectile motion, orbital mechanics), a curve is given by parametric equations

x = x(t)

and

y = y(t)

rather than a direct function

y = f(x)

. Finding the curvature requires applying the chain rule to the standard formula.

If a smooth curve is given parametrically by

x = x(t)

and

y = y(t)

, the curvature

\kappa

at a regular point is computed as:

Parametric Curvature Formula

Curvature for parametrically defined curves.

\kappa = \frac{|x'y'' - y'x''|}{[(x')^2 + (y')^2]^{3/2}}

Variables

Symbol	Description	Unit
$x', y'$	First derivatives with respect to parameter t	-
$x'', y''$	Second derivatives with respect to parameter t	-

Primes denote derivatives with respect to

t

(

x' = \frac{dx}{dt}

y' = \frac{dy}{dt}

, etc.). A regular point satisfies

(x')^2+(y')^2>0

At t equals one on r(t) equals (t,t squared), velocity (1,2) is tangent and acceleration (0,2) splits into tangential and normal components. The equations give curvature as the magnitude of the velocity-acceleration determinant divided by speed cubed. — Parametric Curvature from Velocity and Acceleration
At the regular point t = 1, the acceleration splits into parts parallel and perpendicular to velocity. The normal part changes direction, with |aₙ| = |v|²κ; this yields κ = |v × a|/|v|³. Arrow lengths are schematic.

Curvature in Polar Coordinates

When curves are represented in polar coordinates (

r = f(\theta)

), such as spirals or cardioids, converting them to Cartesian or parametric form can be tedious. A direct formula for curvature in polar coordinates exists using derivatives with respect to

\theta

For a polar curve

r = f(\theta)

at a regular point, the curvature

\kappa

is given by:

Polar Curvature Formula

Curvature for curves in polar coordinates.

\kappa = \frac{|r^2 + 2(r')^2 - r r''|}{[r^2 + (r')^2]^{3/2}}

Variables

Symbol	Description	Unit
$r$	Radial distance as a function of \theta	-
$r', r''$	Derivatives with respect to \theta	-

Where

r'

and

r''

are the first and second derivatives of

r

with respect to

\theta

A three-step derivation converts x equals r cosine theta and y equals r sine theta into the polar curvature formula. The parametric speed squared becomes r squared plus r-prime squared, and the determinant becomes r squared plus two r-prime squared minus r r-double-prime. — Deriving Curvature in Polar Coordinates
Substitute x = r cos θ and y = r sin θ into the parametric formula. The speed denominator simplifies to [r² + (r′)²]^(3/2), while the determinant becomes r² + 2(r′)² − rr″. The algebra is exact at regular points.

Curvature in a Bernoulli–Euler Beam Model

For a linearly elastic Bernoulli–Euler beam, transverse shear deformation is neglected. When the deflection slope is small, signed curvature is approximately the second derivative of transverse deflection,

\kappa \approx v''(x)

. With the sign convention shown below, the internal bending moment is

M(x)=E I(x)\kappa(x)

, where

E I

is the flexural rigidity. Reversing the moment sign convention reverses the corresponding curvature sign as well.

A gently deflected beam and its straight reference axis are shown with deflection v, tangent angle theta, internal moment M at a section, and flexural rigidity EI. For small slope, curvature is approximately v double-prime and the chosen sign convention gives M equals EI times curvature. — Beam Moment, Flexural Rigidity, and Curvature
The Euler–Bernoulli relation M = EIκ uses a linear elastic constitutive law and neglects shear deformation. The small-slope approximation gives κ ≈ v″. The diagram exaggerates deflection and does not establish structural adequacy.

Key Takeaways

Curvature ( $\kappa$ ) measures how sharply a curve bends. It relies heavily on the second derivative.
Radius of Curvature ( $R$ ) is the reciprocal of curvature. It is the radius of the osculating circle that best approximates the curve at a point.
The standard formula for a function $y=f(x)$ is $\kappa = \frac{|y''|}{[1 + (y')^2]^{3/2}}$ , with $R=1/\kappa$ when curvature is nonzero.
The Center of Curvature $(h, k)$ locates the exact center of this osculating circle using the normal vector direction.
Parametric and Polar Forms provide specialized formulas for computing curvature directly from parameters $t$ or angles $\theta$ without needing to eliminate them.
In a linear elastic Euler–Bernoulli beam with small slope, curvature is approximately $v''$ and the signed moment-curvature relation is $M=EI\kappa$ under a stated sign convention.
For motion on a smooth path, path-normal acceleration is $a_n=v^2\kappa$ ; on a circular road curve it is $a_n=v^2/R$ . This kinematic relation does not establish roadway safety or design compliance.

$References$

Previous TopicPartial Differentiation - Examples & Applications

Quiz Me

Next TopicRadius of Curvature - Examples & Applications

Prev Next

Quiz Me