An Cesàro equation is a natural equation which expresses a curve in terms of its arc length
function
and radius of curvature (or equivalently, the curvature ). Note that while the
Cesàro equation is said to be intrinsic because it is invariant under transformations
that preserve length and angle, it is not intrinsic to a curve because it
depends on the starting point from which arc length is measured and hence on the
parametrization (see the table below for examples).

The following table summarizes the Cesàro equations for certain parametrizations of a number of curves (cf. Lawrence 1972, p. 5 and Yates 1952, p. 126).