Radius of curvature

Radius of curvature and center of curvature

In differential geometry, the radius of curvature, Template:Mvar, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.

Definition

In the case of a space curve, the radius of curvature is the length of the curvature vector.

In the case of a plane curve, then Template:Mvar is the absolute value of

$R\equiv \left|\frac{ds}{d\phi }\right|=\frac{1}{\kappa },$

where Template:Mvar is the arc length from a fixed point on the curve, Template:Mvar is the tangential angle and Template:Mvar is the curvature.

Formula

In two dimensions

If the curve is given in Cartesian coordinates as Template:Math, i.e., as the graph of a function, then the radius of curvature is (assuming the curve is differentiable up to order 2)

$$R=\left|\frac{{(1+y{\text{'}}^2)}^{\frac{3}{2}}}{y^{″}}\right|,$$

where $y^{\prime }=\frac{dy}{dx},$ $y^{″}=\frac{d^{2}y}{d{x}^2},$ and Template:Math denotes the absolute value of Template:Mvar.

If the curve is given parametrically by functions Template:Math and Template:Math, then the radius of curvature is

$$R=\left|\frac{ds}{d\phi }\right|=\left|\frac{{\left({\dot{x}}^2+{\dot{y}}^2\right)}^{\frac{3}{2}}}{\dot{x}\ddot{y}-\dot{y}\ddot{x}}\right|$$

where $\dot{x}=\frac{dx}{dt},$ $\ddot{x}=\frac{d^{2}x}{d{t}^2},$ $\dot{y}=\frac{dy}{dt},$ and $\ddot{y}=\frac{d^{2}y}{d{t}^2}.$

Heuristically, this result can be interpreted as

$$R=\frac{{\left|𝐯\right|}^3}{|𝐯\times \dot{𝐯}|},$$

where

$$\left|𝐯\right|=|(\dot{x},\dot{y})|=R\frac{d\phi }{dt}.$$

In Template:Mvar dimensions

If Template:Math is a parametrized curve in Template:Math then the radius of curvature at each point of the curve, Template:Math, is given by

$$\rho =\frac{{\left|{\mathit{\gamma }}^{\prime }\right|}^3}{\sqrt{{\left|{\mathit{\gamma }}^{\prime }\right|}^2{\left|{\mathit{\gamma }}^{″}\right|}^2-{({\mathit{\gamma }}^{\prime }\cdot {\mathit{\gamma }}^{″})}^2}}.$$

As a special case, if Template:Math is a function from Template:Math to Template:Math, then the radius of curvature of its graph, Template:Math, is

$$\rho (t)=\frac{{|1+f{\text{'}}^2(t)|}^{\frac{3}{2}}}{|f^{″}(t)|}.$$

Derivation

Let Template:Math be as above, and fix Template:Mvar. We want to find the radius Template:Mvar of a parametrized circle which matches Template:Math in its zeroth, first, and second derivatives at Template:Mvar. Clearly the radius will not depend on the position Template:Math, only on the velocity Template:Math and acceleration Template:Math. There are only three independent scalars that can be obtained from two vectors Template:Math and Template:Math, namely Template:Math, Template:Math, and Template:Math. Thus the radius of curvature must be a function of the three scalars Template:Math, Template:Math and Template:Math.

The general equation for a parametrized circle in Template:Math is

$$𝐠(u)=𝐚\cos (h(u))+𝐛\sin (h(u))+𝐜$$

where Template:Math is the center of the circle (irrelevant since it disappears in the derivatives), Template:Math are perpendicular vectors of length Template:Mvar (that is, Template:Math and Template:Math), and Template:Math is an arbitrary function which is twice differentiable at Template:Mvar.

The relevant derivatives of Template:Math work out to be

$$\begin{array}{cc}|{𝐠}^{\prime }{|}^2& ={\rho }^2(h^{\prime }{)}^2\\ {𝐠}^{\prime }\cdot {𝐠}^{″}& ={\rho }^2{h}^{\prime }{h}^{″}\\ |{𝐠}^{″}{|}^2& ={\rho }^2((h^{\prime }{)}^4+(h^{″}{)}^2)\end{array}$$

If we now equate these derivatives of Template:Math to the corresponding derivatives of Template:Math at Template:Mvar we obtain

$$\begin{array}{cc}|{\mathit{\gamma }}^{\prime }(t){|}^2& ={\rho }^{2}h{\text{'}}^2(t)\\ {\mathit{\gamma }}^{\prime }(t)\cdot {\mathit{\gamma }}^{″}(t)& ={\rho }^2{h}^{\prime }(t)h^{″}(t)\\ |{\mathit{\gamma }}^{″}(t){|}^2& ={\rho }^2(h{\text{'}}^4(t)+h^{\prime }{\text{'}}^2(t))\end{array}$$

These three equations in three unknowns (Template:Mvar, Template:Math and Template:Math) can be solved for Template:Mvar, giving the formula for the radius of curvature:

$$\rho (t)=\frac{{|{\mathit{\gamma }}^{\prime }(t)|}^3}{\sqrt{{|{\mathit{\gamma }}^{\prime }(t)|}^2{|{\mathit{\gamma }}^{″}(t)|}^2-({\mathit{\gamma }}^{\prime }(t)\cdot {\mathit{\gamma }}^{″}(t){)}^2}},$$

or, omitting the parameter Template:Mvar for readability,

$$\rho =\frac{{\left|{\mathit{\gamma }}^{\prime }\right|}^3}{\sqrt{{\left|{\mathit{\gamma }}^{\prime }\right|}^2{\left|{\mathit{\gamma }}^{″}\right|}^2-{({\mathit{\gamma }}^{\prime }\cdot {\mathit{\gamma }}^{″})}^2}}.$$

Examples

Semicircles and circles

For a semi-circle of radius Template:Mvar in the upper half-plane with $R=|-a|=a,$

$$\begin{array}{cc}y& =\sqrt{a^2-x^2}\\ y^{\prime }& =\frac{-x}{\sqrt{a^2-x^2}}\\ y^{″}& =\frac{-a^2}{{(a^2-x^2)}^{\frac{3}{2}}}.\end{array}$$

An ellipse (red) and its evolute (blue). The dots are the vertices of the ellipse, at the points of greatest and least curvature.

For a semi-circle of radius Template:Mvar in the lower half-plane $$y=-\sqrt{a^2-x^2}.$$

The circle of radius Template:Mvar has a radius of curvature equal to Template:Mvar.

Ellipses

In an ellipse with major axis Template:Math and minor axis Template:Math, the vertices on the major axis have the smallest radius of curvature of any points, Template:Nowrap and the vertices on the minor axis have the largest radius of curvature of any points, Template:Math.

The radius of curvature of an ellipse as a function of the geocentric coordinate $t$ with $\tan t=\frac{y}{x}$ is$$R(t)=\frac{(b^2{\cos }^{2}t+a^2{\sin }^{2}t{)}^{3/2}}{ab}.$$It has its minima at $t=0$ and $t=180^{\circ }$ and its maxima at $t=\pm 90^{\circ }$.

Applications

• For the use in differential geometry, see Cesàro equation.
• For the radius of curvature of the Earth (approximated by an oblate ellipsoid); see also: arc measurement
• Radius of curvature is also used in a three part equation for bending of beams.
• Radius of curvature (optics)
• Thin films technologies
• Printed electronics
• Minimum railway curve radius
• AFM probe

Stress in semiconductor structures

Stress in the semiconductor structure involving evaporated thin films usually results from the thermal expansion (thermal stress) during the manufacturing process. Thermal stress occurs because film depositions are usually made above room temperature. Upon cooling from the deposition temperature to room temperature, the difference in the thermal expansion coefficients of the substrate and the film cause thermal stress.

Intrinsic stress results from the microstructure created in the film as atoms are deposited on the substrate. Tensile stress results from microvoids (small holes, considered to be defects) in the thin film, because of the attractive interaction of atoms across the voids.

The stress in thin film semiconductor structures results in the buckling of the wafers. The radius of the curvature of the stressed structure is related to stress tensor in the structure, and can be described by modified Stoney formula. The topography of the stressed structure including radii of curvature can be measured using optical scanner methods. The modern scanner tools have capability to measure full topography of the substrate and to measure both principal radii of curvature, while providing the accuracy of the order of 0.1% for radii of curvature of 90 meters and more.

See also

• Base curve radius
• Bend radius
• Degree of curvature (civil engineering)
• Osculating circle
• Track transition curve

Further reading

• Template:Cite book

External links

• The Geometry Center: Principal Curvatures
• 15.3 Curvature and Radius of Curvature Template:Webarchive
• Template:MathWorld
• Template:MathWorld