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Subsections

# 9 The Mk III auto-folder

The Mk III auto-folder is based on the working hypothesis that the folding process should only use the exterior parts of the rotation curve, so that hole-cutting becomes an active part of this process. However, there is one immediate practical difficulty: the hole-cutting algorithm requires rotation curves to be already folded. The resolution of this problem is embodied in the algorithmic definition of the Mk III auto-folder, described in Sect. 9.1, and the solution arising from it applied over the PS data is shown in Fig. 5 right.
 Figure 5: Comparison of Mk II and Mk III auto-folder solutions. The vertical lines indicate the positions of the A, B, C, D and E peaks in the PS solution

## 9.1 Algorithmic definition of the Mk III auto-folder

The Mk III auto-folder is defined by the following algorithm:
• For any given rotation curve, fold using the Mk II auto-folder;
• record the number of Fourier modes used for this fold, say , and the estimates for and ;
• apply the hole-cutting algorithm defined Roscoe 1999A to this folded rotation curve, and record the removed points;
• take the original unfolded rotation curve, and reduce it by removing these recorded points;
• fold this reduced rotation curve using the Fourier method already described, but taking care to redefine the Fourier coefficients in the manner described in the next subsection;
• for this second-stage folding process, use either the modes recorded at the first-stage folding process, or the number of modes defined by the algorithm of Sect. 6.2, whichever is the least;
• use the recorded estimates for and as the initial guess for the minimization process of this second-stage folding.

## 9.2 Partition of Fourier region

The application of the hole-cutting algorithm of Roscoe 1999A cuts a hole out of the rotation curve which is centred on the estimated dynamical centre of the galaxy; consequently, instead of representing a continuous curve, the reduced rotation curve data represents two separated continuous sections with a gap in between. This means that the Fourier cosine coefficients cannot be computed using (2), but must be computed by an integral of the form

 (3)

where X' is some positive number representing the boundaries of the cut-out hole. In practice, of course, the hole-cutting strategy will generally remove a section which is non-symmetric about the current estimate of , and so the data on one of the two sections of the reduced rotation curve must be interpolated/extrapolated to ensure that the hole which has been effectively cut out of the rotation curve is centred exactly on the current estimate of . Once the removed section is centred in this way, and the Fourier modes computed using (3), rather than (2), the process is as before.

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