TGE function

Calculation of the Polar Radius of the Twin Gielis Curve

TGE is used to calculate the polar radii of the twin Gielis equation or one of its simplified versions at given polar angles. UTF-8

TGE(P, phi, m = 1, simpver = NULL, nval = 1)

Arguments

• P: the parameters of the twin Gielis equation or one of its simplified versions.
• phi: the polar angle(s).
• m: the given $m$ value that determines the number of angles of the twin Gielis curve within $[0, 2\pi)$.
• simpver: an optional argument to use the simplified version of the twin Gielis equation.
• nval: the specified value for $n_{2}$ or $n_{3}$ in the simplified versions.

Details

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.

Returns

The polar radii predicted by the twin Gielis equation or one of its simplified versions.

Note

simpver here is different from that in the GE function.

Author(s)

Peijian Shi pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca .

References

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")

See Also

areaGE, curveGE, fitGE, GE

Examples

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()