Calculating geometric properties of objects represented by Fourier coefficients

The author is concerned with the calculation of object features directly from the Fourier-series coefficients of boundary function r (φ) which describes the length of the radius-vector from the origin to a boundary point. The area, the coordinates of the centroid, and the second-order moments with respect to the axes passing through the origin are determined. Given these features, the orientation of the central axes, and the central moments of inertia can be easily determined. The difficulty of calculating the perimeter in terms of the Fourier coefficients of r(φ) is known. Hence, lower and upper bounds on the perimeter are established