TGE is used to calculate the polar radii of the twin Gielis equation or
one of its simplified versions at given polar angles.
TGE(P, phi, m = 1, simpver = NULL, nval = 1)The polar radii predicted by the twin Gielis equation or one of its simplified versions.
the parameters of the twin Gielis equation or one of its simplified versions.
the polar angle(s).
the given \(m\) value that determines the number of angles of the twin Gielis curve within \([0, 2\pi)\).
an optional argument to use the simplified version of the twin Gielis equation.
the specified value for \(n_{2}\) or \(n_{3}\) in the simplified versions.
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
The general form of the twin Gielis equation can be represented as follows:
$$r\left(\varphi\right) = \mathrm{exp}\left\{\frac{1}{\alpha+\beta\,\mathrm{ln}\left[r_{e}\left(\varphi\right)\right]}+\gamma\right\},$$
where \(r\) represents the polar radius of the twin Gielis curve at the polar angle \(\varphi\), and
\(r_{e}\) represents the elementary polar radius at the polar angle \(\varphi\). There is a hyperbolic
link function to link their log-transformations, i.e.,
$$\mathrm{ln}\left[r\left(\varphi\right)\right] = \frac{1}{\alpha+\beta\,\mathrm{ln}\left[r_{e}\left(\varphi\right)\right]}+\gamma.$$
The first three elements of P are \(\alpha\), \(\beta\), and \(\gamma\), and the remaining element(s) of
P are the parameters of the elementary polar function, i.e., \(r_{e}\left(\varphi\right)\).
See Shi et al. (2020) for details.
\(\quad\) When simpver = NULL, the original twin Gielis equation is selected:
$$r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{3}},$$
where \(r_{e}\) represents the elementary polar radius at the polar angle \(\varphi\);
\(m\) determines the number of angles of the twin Gielis curve within \([0, 2\pi)\);
and \(k\), \(n_{2}\), and \(n_{3}\) are the fourth to the sixth elements in P. In total, there are
six elements in P.
\(\quad\) When simpver = 1, the simplified version 1 is selected:
$$r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}},$$
where \(n_{2}\) is the fourth element in P. There are
four elements in total in P.
\(\quad\) When simpver = 2, the simplified version 2 is selected:
$$r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}, $$
where \(n_{2}\) should be specified in nval, and P only includes three elements, i.e.,
\(\alpha\), \(\beta\), and \(\gamma\).
\(\quad\) When simpver = 3, the simplified version 3 is selected:
$$r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{3}},$$
where \(n_{2}\) and \(n_{3}\) are the fourth and fifth elements in P. There are
five elements in total in P.
\(\quad\) When simpver = 4, the simplified version 4 is selected:
$$r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}},$$
where \(k\) and \(n_{2}\) are the fourth and fifth elelments in P. There are
five elements in total in P.
\(\quad\) When simpver = 5, the simplified version 5 is selected:
$$r_{e}\left(\varphi\right) = \left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}},$$
where \(k\) is the fourth elelment in P. There are
four elements in total in P. \(n_{2}\) should be specified in nval.
Li, Y., Quinn, B.K., Gielis, J., Li, Y., Shi, P. (2022) Evidence that supertriangles exist in nature from the vertical projections of Koelreuteria paniculata fruit. Symmetry 14, 23. tools:::Rd_expr_doi("10.3390/sym14010023")
Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences 1516, 123\(-\)134. tools:::Rd_expr_doi("10.1111/nyas.14862")
Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. tools:::Rd_expr_doi("10.3390/sym12040645")
areaGE, curveGE, fitGE, GE
TGE.par <- c(2.88, 0.65, 1.16, 139)
varphi.vec <- seq(0, 2*pi, len=2000)
r2.theor <- TGE(P=TGE.par, phi=varphi.vec, simpver=1, m=5)
dev.new()
plot( varphi.vec, r2.theor, cex.lab=1.5, cex.axis=1.5,
xlab=expression(italic(varphi)), ylab=expression(italic("r")),
type="l", col=4 )
starfish4 <- curveGE(TGE, P=c(0, 0, 0, TGE.par), simpver=1, m=5, fig.opt=TRUE)
graphics.off()
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