| GE {biogeom} | R Documentation |
Calculation of the Polar Radius of the Gielis Curve
Description
GE is used to calculate polar radii of the original Gielis equation
or one of its simplified versions at given polar angles.
Usage
GE(P, phi, m = 1, simpver = NULL, nval = 1)
Arguments
P |
the parameters of the original Gielis equation or one of its simplified versions. |
phi |
the polar angle(s). |
m |
the given |
simpver |
an optional argument to use the simplified version of the original Gielis equation. |
nval |
the specified value for |
Details
When simpver = NULL, the original Gielis equation is selected:
r\left(\varphi\right) = a\left(\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}}\right)^{-\frac{1}{n_{1}}},
where r represents the polar radius at the polar angle \varphi;
m determines the number of angles within [0, 2\pi); and a, k, n_{1},
n_{2}, and n_{3} need to be provided in P.
\quad When simpver = 1, the simplified version 1 is selected:
r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}\right)^{-\frac{1}{n_{1}}},
where a, n_{1}, and n_{2} need to be provided in P.
\quad When simpver = 2, the simplified version 2 is selected:
r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}\right)^{-\frac{1}{n_{1}}},
where a and n_{1} need to be provided in P, and n_{2} should be specified in nval.
\quad When simpver = 3, the simplified version 3 is selected:
r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}+
\left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}\right)^{-\frac{1}{n_{1}}},
where a needs to be provided in P, and n_{1}
should be specified in nval.
\quad When simpver = 4, the simplified version 4 is selected:
r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}+
\left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}\right)^{-\frac{1}{n_{1}}},
where a and n_{1} need to be provided in P.
\quad When simpver = 5, the simplified version 5 is selected:
r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+
\left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{3}}\right)^{-\frac{1}{n_{1}}},
where a, n_{1}, n_{2}, and n_{3} need to be provided in P.
\quad When simpver = 6, the simplified version 6 is selected:
r\left(\varphi\right) = a\left(\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}}\right)^{-\frac{1}{n_{1}}},
where a, k, n_{1}, and n_{2} need to be provided in P.
\quad When simpver = 7, the simplified version 7 is selected:
r\left(\varphi\right) = a\left(\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}}\right)^{-\frac{1}{n_{1}}},
where a, k, and n_{1} need to be provided in P, and n_{2}
should be specified in nval.
\quad When simpver = 8, the simplified version 8 is selected:
r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}+
\left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}\right)^{-\frac{1}{n_{1}}},
where a and k are parameters that need to be provided in P, and n_{1}
should be specified in nval.
\quad When simpver = 9, the simplified version 9 is selected:
r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}+
\left|\frac{1}{k}\,\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{1}}\right)^{-\frac{1}{n_{1}}},
where a, k, and n_{1} need to be provided in P.
Value
The polar radii predicted by the original Gielis equation or one of its simplified versions.
Note
simpver here is different from that in the TGE 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
Gielis, J. (2003) A generic geometric transformation that unifies a wide range of natural
and abstract shapes. American Journal of Botany 90, 333-338. doi:10.3732/ajb.90.3.333
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. 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. doi:10.1111/nyas.14862
Shi, P., Ratkowsky, D.A., Gielis, J. (2020) The generalized Gielis geometric equation and its application. Symmetry 12, 645. doi:10.3390/sym12040645
Shi, P., Xu, Q., Sandhu, H.S., Gielis, J., Ding, Y., Li, H., Dong, X. (2015) Comparison
of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship
between spatial density of plants and total leaf area per plant. Ecology and Evolution 5,
4578-4589. doi:10.1002/ece3.1728
See Also
areaGE, curveGE, DSGE, fitGE,
SurfaceAreaSGE, TGE, VolumeSGE
Examples
GE.par <- c(2, 1, 4, 6, 3)
varphi.vec <- seq(0, 2*pi, len=2000)
r.theor <- GE(P=GE.par, phi=varphi.vec, m=5)
dev.new()
plot( varphi.vec, r.theor, cex.lab=1.5, cex.axis=1.5,
xlab=expression(italic(varphi)), ylab=expression(italic("r")),
type="l", col=4 )
graphics.off()