AbstractAn arbitrary closed curve on a plane can be represented as a Fourier series. Curves may be useful particularly for the experimental study on description and recognition of closed curves, provided that the arc length is used as the parameter for their parametric representation. This paper introduces by formal discussion a method to generate a closed curve with Fourier representation which passes through a sequence of points or their neighborhood, in the sequence order, given such a point sequence. In particular, a simple method is introduced which uses the Fourier transform and interpolating filters for a case in which the sequence comprises a relatively small number of points and in which the distances between successive points are not uniform. In addition, a conversion is described by which the arc length becomes the parameter. Fourier representation of a closed curve has often been applied to pattern recognition of images and figures. As a result, Fourier coefficients have been calculated based on the discrete Fourier transform of the coordinate sequence comprising the grid points or of their interpolated points. However, the mathematical meaning of the Fourier transform method has not been clarified. The formal discussion in this paper also clarifies this meaning.