Radius of curvature

Radius of curvature
In differential geometry, the radius of curvature 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 is the absolute value of
where is the arc length from a fixed point on the curve, is the tangential angle and is the curvature.
Formula
In two dimensions
If the curve is given in Cartesian coordinates as , i.e., as the graph of a function, then the radius of curvature is (assuming the curve is differentiable up to order 2)
where and denotes the absolute value of .
If the curve is given parametrically by functions and , then the radius of curvature is
where and
Heuristically, this result can be interpreted as
where
In dimensions
If is a parametrized curve in then the radius of curvature at each point of the curve is given by
As a special case, if is a function from to , then the radius of curvature of its graph is
Derivation
Let be as above, and fix . We want to find the radius of a parametrized circle which matches in its zeroth, first, and second derivatives at . Clearly the radius will not depend on the position , only on the velocity and acceleration . There are only three independent scalars that can be obtained from two vectors and , namely and . Thus the radius of curvature must be a function of the three scalars , and .
The general equation for a parametrized circle in is
where is the center of the circle (irrelevant since it disappears in the derivatives), are perpendicular vectors of length (that is, and ), and is an arbitrary function which is twice differentiable at .
The relevant derivatives of work out to be
If we now equate these derivatives of to the corresponding derivatives of at we obtain
These three equations in three unknowns (and ) can be solved for , giving the formula for the radius of curvature:
or, omitting the parameter for readability,
Examples
Semicircles and circles
For a semi-circle of radius in the upper half-plane with
For a semi-circle of radius in the lower half-plane
The circle of radius has a radius of curvature equal to .
Ellipses
In an ellipse with major axis and minor axis , the vertices on the major axis have the smallest radius of curvature of any points, and the vertices on the minor axis have the largest radius of curvature of any points, .
The radius of curvature of an ellipse, as a function of parameter , is
where
The radius of curvature of an ellipse, as a function of , is
where the eccentricity of the ellipse is given by
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
References
Further reading
External links
The Geometry Center: Principal Curvatures
15.3 Curvature and Radius of Curvature
Differential geometry
Curvature (mathematics)
Curves
Integral calculus
Multivariable calculus
Theoretical physics