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 surface
s, the radius of curvature is the radius of a circle that best fits a normal section
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
where is the arc length
from a fixed point on the curve, is the tangential angle
and is the curvature
If the curve is given in Cartesian coordinates
as , then the radius of curvature is (assuming the curve is differentiable up to order 2):
and denotes the absolute value of .
If the curve is given parametrically
by functions and , then the radius of curvature is
Heuristically, this result can be interpreted as
In n 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
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,
Semicircles and circles
For a semi-circle
of radius in the upper half-plane
240px|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 in the lower half-plane
of radius has a radius of curvature equal to .
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, .
*For the use in differential geometry
, see Cesàro equation
*For the radius of curvature of the earth (approximated by an oblate ellipsoid), see Radius of curvature of the earth
*Radius of curvature is also used in a three part equation for bending of beams
*Radius of curvature (optics)
*Thin films technologies
Stress in semiconductor structures
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.
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.
*Base curve radius
*Degree of curvature
*Minimum railway curve radius
*Track transition curve
The Geometry Center: Principal Curvatures