GE: Calculation of the Polar Radius of the Gielis Curve
In biogeom: Biological Geometries
GE 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 m value that determines the number of angles of
the Gielis curve within [0, 2\pi).
simpver
an optional argument to use the simplified version of the original Gielis equation.
nval
the specified value for n_{1} or n_{2} or n_{3} in the simplified versions.
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. \Sexpr[results=rd]{tools:::Rd_expr_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. \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")}
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. \Sexpr[results=rd]{tools:::Rd_expr_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()
biogeom documentation built on May 29, 2024, 8:52 a.m.
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| GE | 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. \Sexpr[results=rd]{tools:::Rd_expr_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. \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")}
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. \Sexpr[results=rd]{tools:::Rd_expr_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()
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