TGE: Calculation of the Polar Radius of the Twin Gielis Curve
In biogeom: Biological Geometries
TGE R Documentation
Calculation of the Polar Radius of the Twin Gielis Curve
Description
TGE is used to calculate the polar radii of the twin Gielis equation or
one of its simplified versions at given polar angles.
Usage
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.
Value
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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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()
biogeom documentation built on May 29, 2024, 8:52 a.m.
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| TGE | R Documentation |
Calculation of the Polar Radius of the Twin Gielis Curve
Description
TGE is used to calculate the polar radii of the twin Gielis equation or
one of its simplified versions at given polar angles.
Usage
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 |
simpver |
an optional argument to use the simplified version of the twin Gielis equation. |
nval |
the specified value for |
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.
Value
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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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()
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