The distance from the center of a circle or sphere to its surface is its radius.
For other curved lines or surfaces, the radius of curvature at a given point is
the radius of a circle that mathematically best fits the curve at
that point. In the case of a surface, the radius of curvature is the radius of
a circle that best fits a normal section.
Explanation Imagine driving a car on a curvy road on
a completely flat surface. At any one point along the way, lock the steering
wheel in its position, so that the car thereafter follows a perfect circle. The
car will, of course, deviate from the road, unless the road is also a perfect
circle. The radius of that circle the car makes is the radius of curvature of
the curvy road at the point at which the steering wheel was locked. The more
sharply curved the road is at the point you locked the steering wheel, the
smaller the radius of curvature. Formula
If is a parameterized curve in then the radius of curvature at each point of
the curve, , is given by As a special case, if f(t) is a function
from to , then the curvature of its graph, , is
= Derivation = Let be as above, and fix . We want to
find the radius of a parameterized 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 v and w, namely v·v,
v·w, and w·w. Thus the radius of curvature must be a function of the
three scalars , and . The general equation for a parameterized
circle in is where is the center of the circle, are
perpendicular vectors of length , and is an arbitrary function which is twice
differentiable at t. The relevant derivatives of g work out
to be If we now equate these derivatives of g
to the corresponding derivatives of at t we obtain
These three equations in three unknowns can be solved for , giving the formula
for the radius of curvature: or, omitting the parameter for
readability, Applications and examples
For the use in differential geometry, see Cesàro equation.
For the radius of curvature of the earth, 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 applied to measurements of the stress in the
semiconductor structures Stress in the semiconductor structure
involving evaporated thin films usually results from the thermal expansion
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 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 m and more. References:
^ http:flipchips.comprocess and references within
ww.qucosa.dedatadocumentsdata/Stoney.pdf ^ http:www.zebraoptical.com/ModelX.html
See also External links
The Geometry Center: Principal Curvatures
15.3 Curvature and Radius of Curvature Weisstein, Eric W., "Principal
Curvatures", MathWorld. Weisstein, Eric W., "Principal Radius of
Curvature", MathWorld.