Radius of curvature (mathematics) in the context of Cesàro equation

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 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:

In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system.

The intrinsic quantities used most often are arc length ${\displaystyle s}$, tangential angle ${\displaystyle \theta }$, curvature ${\displaystyle \kappa }$ or radius of curvature, and, for 3-dimensional curves, torsion ${\displaystyle \tau }$. Specifically:

In geometry, the center of curvature of a curve is a point located at a distance from the curve equal to the radius of curvature lying on the curve normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve. The locus of centers of curvature for each point on the curve comprise the evolute of the curve. This term is generally used in physics regarding the study of lenses and mirrors (see radius of curvature (optics)).

It can also be defined as the spherical distance between the point at which all the rays falling on a lens or mirror either seems to converge to (in the case of convex lenses and concave mirrors) or diverge from (in the case of concave lenses or convex mirrors) and the lens or mirror itself.It lies on the principal axis of a mirror or lens. In case of a convex mirror it lies behind the polished, or reflecting, surface and it lies in front of the reflecting surface in case of a concave mirror.