When simpver = NULL, the original Preston equation is selected:
$$y = a\ \mathrm{sin}\,\zeta, $$
$$x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta+c_{2}\,\mathrm{sin}^{2}\,\zeta+c_{3}\,\mathrm{sin}^{3}\,\zeta\right), $$
$$r = \sqrt{x^{2}+y^{2}}, $$
where \(x\) and \(y\) represent the abscissa and ordinate of an arbitrary point on the Preston curve
corresponding to an angle \(\zeta\); \(r\) represents the distance of the point from the origin; \(a\), \(b\), \(c_{1}\),
\(c_{2}\), and \(c_{3}\) are parameters to be estimated.
\(\quad\) When simpver = 1, the simplified version 1 is selected:
$$y = a\ \mathrm{sin}\,\zeta, $$
$$x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta+c_{2}\,\mathrm{sin}^{2}\,\zeta\right), $$
$$r = \sqrt{x^{2}+y^{2}}, $$
where \(x\) and \(y\) represent the abscissa and ordinate of an arbitrary point on the Preston curve
corresponding to an angle \(\zeta\); \(r\) represents the distance of the point from the origin; \(a\), \(b\), \(c_{1}\),
and \(c_{2}\) are parameters to be estimated.
\(\quad\) When simpver = 2, the simplified version 2 is selected:
$$y = a\ \mathrm{sin}\,\zeta, $$
$$x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta\right), $$
$$r = \sqrt{x^{2}+y^{2}}, $$
where \(x\) and \(y\) represent the abscissa and ordinate of an arbitrary point on the Preston curve
corresponding to an angle \(\zeta\); \(r\) represents the distance of the point from the origin; \(a\), \(b\), and \(c_{1}\)
are parameters to be estimated.
\(\quad\) When simpver = 3, the simplified version 3 is selected:
$$y = a\ \mathrm{sin}\,\zeta, $$
$$x = b\ \mathrm{cos}\,\zeta\left(1+c_{2}\,\mathrm{sin}^{2}\,\zeta\right), $$
$$r = \sqrt{x^{2}+y^{2}}, $$
where \(x\) and \(y\) represent the abscissa and ordinate of an arbitrary point on the Preston curve
corresponding to an angle \(\zeta\); \(r\) represents the distance of the point from the origin; \(a\), \(b\), and
\(c_{2}\) are parameters to be estimated.