Radius of curvature (applications)
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
Radius of curvature
Radius of curvature may refer to:
Radius of curvature (mathematics)
In geometry, the radius of curvature, R, of a curve at a point is a measure of the radius of the circular arc which best approximates the curve at that point. It is the inverse of the curvature.
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 R is the absolute value of
where s is the arc length from a fixed point on the curve, φ is the tangential angle and 
is the curvature.
If the curve is given in Cartesian coordinates as y(x), then the radius of curvature is (assuming the curve is differentiable up to order 2):
and | z | denotes the absolute value of z.
If the curve is given parametrically by functions x(t) and y(t), then the radius of curvature is
Heuristically, this result can be interpreted as
Examples
Semicircles and circles
For a semi-circle of radius a in the upper half-plane
For a semi-circle of radius a in the lower half-plane
The circle of radius a has a radius of curvature equal to a.
Radius of curvature (optics)
Radius of curvature (ROC) has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located in (x, y, z) either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis. The distance from the vertex to the center of curvature is the radius of curvature of the surface. The sign convention for the optical radius of curvature is as follows:
Thus when viewing a biconvex lens from the side, the left surface radius of curvature is positive, and the right surface has a negative radius of curvature.
Note however that in areas of optics other than design, other sign conventions are sometimes used. In particular, many undergraduate physics textbooks use an alternate sign convention in which convex surfaces of lenses are always positive. Care should be taken when using formulas taken from different sources.
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