# Elliptic Fourier shape coefficients

Requires digitized x/y coordinates around outlines. Specimens in rows, coordinates of alternating x and y values in columns. Elliptic Fourier shape analysis is in several respects superior to simple Fourier shape analysis. One advantage is that the algorithm can handle complicated shapes which may not be expressible as a unique function in polar co-ordinates. Elliptic Fourier shapes is now a standard method of outline analysis, together with Hangle. The algorithm used in PAST is described by Ferson et al. (1985).

Cosine and sine components of x and y increments along the outline for the first 30 harmonics are given, but only the first N/2 harmonics should be used, where N is the number of digitized points. Size and positional translation are normalized away, and do not enter in the coefficients. The size (before normalization) is given in the first column. The optional standardization for rotation or starting point, following Ferson et al., sometimes flips shapes around . This should be checked with the ‘Shape view’ (see below) – it may be necessary to remove such specimens.

The coefficients can be copied to the main spreadsheet for further analysis such as discriminant analysis.

The 'Shape view' window allows graphical viewing of the elliptic Fourier shape approximation(s).

#### Reference

Ferson, S.F., F.J. Rohlf & R.K. Koehn. 1985. Measuring shape variation of two-dimensional outlines. *Systematic Zoology* 34:59-68.