Hangle Fourier shape analysis

Requires digitized x/y coordinates around outlines. Specimens in rows, coordinates of alternating x and y values in columns.

The “Hangle” method for analysing closed outlines, proposed by Haines & Crampton (2000) is a competitor to Elliptic Fourier Analysis. Hangle has certain advantages over EFA, the most important being that fewer coefficients are needed to capture the outline to a given precision. This is of importance for statistical testing (e.g. MANOVA) and discriminant analysis. The implementation in Past is based on the Hangle/Hmatch/Htree/Hshape package of Haines & Crampton (thanks to the authors for providing the source code).

The output consists of 46 Fourier coefficients, which are the cos and sin coefficients of the first 24 harmonics (modes), starting on harmonic number 2. Copy these numbers back to a Past spreadsheet for further multivariate shape analysis.

Starting point normalization

Usually leave at ‘Match all’, either with the ‘Hmatch’ or (perhaps preferably) the ‘Htree’ method to align all the outlines. Alternatively, select 2.-4. harmonic, which will phase shift each outline according to the selected mode (see Haines & Crampton 2000).


Increasing the smoothing parameter can reduce high-frequency noise, at the cost of dampening potentially informative high-frequency shape information.

Shape view

Use this function to inspect the shapes reconstructed from the Fourier coefficients. Check that the matching routine has not rotated any shape incorrectly. Also, use this function to select the minimum number of modes necessary for capturing the shape. In an example, the number of modes has been set to 14, which captures 99.88% of the total integrated power (amplitude squared) of the selected shape. The number of modes is shown by the red line in the power spectrum – make sure that the main features of the spectrum are to the left of this line for all the shapes.

Note: PCA visualization and regression (as for EFA) has not yet been implemented for Hangle.


Haines, A.J. & J.S. Crampton. 2000. Improvements to the method of Fourier shape analysis as applied in morphometric studies. Palaeontology 43:765-783

Published Aug. 31, 2020 8:16 PM - Last modified Aug. 31, 2020 8:17 PM