Abstract

The radius of curvature (ROC) of a curve y = f(x) at a point (x,y = f(x)) is the radius of a circle which osculates with the curve at that point. For any point on any smooth function f, it is the circle which, when properly shifted in the x-y plane, both contains the point (x,y = f(x)) and is equal to the function in a differentially small region about that point. The most familiar example to optical scientists of an osculating circle is the circle that “matches” a parabola near its vertex. The vertex radius of curvature of the parabola is the same as the radius of curvature of that circle for which the parabola is the second order expansion (via the binomial theorem).

© 2002 Optical Society of America