PE | R Documentation |
PE
is used to calculate the abscissa, ordinate and distance from the origin for an arbitrary point on
the Preston curve that was generated by the original Preston equation or one of its simplified versions at a given angle.
PE(P, zeta, simpver = NULL)
P |
the parameters of the original Preston equation or one of its simplified versions. |
zeta |
the angle(s) used in the Preston equation. |
simpver |
an optional argument to use the simplified version of the original Preston equation. |
When simpver = NULL
, the original Preston equation is selected:
y = a\ \mathrm{sin}\,\zeta,
x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta+c_{2}\,\mathrm{sin}^{2}\,\zeta+c_{3}\,\mathrm{sin}^{3}\,\zeta\right),
r = \sqrt{x^{2}+y^{2}},
where x
and y
represent the abscissa and ordinate of an arbitrary point on the Preston curve
corresponding to an angle \zeta
; r
represents the distance of the point from the origin; a
, b
, c_{1}
,
c_{2}
, and c_{3}
are parameters to be estimated.
\quad
When simpver = 1
, the simplified version 1 is selected:
y = a\ \mathrm{sin}\,\zeta,
x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta+c_{2}\,\mathrm{sin}^{2}\,\zeta\right),
r = \sqrt{x^{2}+y^{2}},
where x
and y
represent the abscissa and ordinate of an arbitrary point on the Preston curve
corresponding to an angle \zeta
; r
represents the distance of the point from the origin; a
, b
, c_{1}
,
and c_{2}
are parameters to be estimated.
\quad
When simpver = 2
, the simplified version 2 is selected:
y = a\ \mathrm{sin}\,\zeta,
x = b\ \mathrm{cos}\,\zeta\left(1+c_{1}\,\mathrm{sin}\,\zeta\right),
r = \sqrt{x^{2}+y^{2}},
where x
and y
represent the abscissa and ordinate of an arbitrary point on the Preston curve
corresponding to an angle \zeta
; r
represents the distance of the point from the origin; a
, b
, and c_{1}
are parameters to be estimated.
\quad
When simpver = 3
, the simplified version 3 is selected:
y = a\ \mathrm{sin}\,\zeta,
x = b\ \mathrm{cos}\,\zeta\left(1+c_{2}\,\mathrm{sin}^{2}\,\zeta\right),
r = \sqrt{x^{2}+y^{2}},
where x
and y
represent the abscissa and ordinate of an arbitrary point on the Preston curve
corresponding to an angle \zeta
; r
represents the distance of the point from the origin; a
, b
, and
c_{2}
are parameters to be estimated.
x |
the abscissa(s) of the Preston curve corresponding to the given angle(s). |
y |
the ordinate(s) of the Preston curve corresponding to the given angle(s). |
r |
the distance(s) of the Preston curve corresponding to the given angle(s) from the origin. |
\zeta
is NOT the polar angle corresponding to r
, i.e.,
y \neq r\,\mathrm{sin}\,\zeta,
x \neq r\,\mathrm{cos}\,\zeta.
Let \varphi
be the polar angle corresponding to r
. We have:
\zeta = \mathrm{arc\,sin}\frac{ r\ \mathrm{sin}\,\varphi }{a}.
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston's universal formula for avian egg shape. Ornithology 139, ukac028. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/ornithology/ukac028")}
Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying
the shape of birds' eggs. Ecology and Evolution 8, 9728-
9738. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/ece3.4412")}
Preston, F.W. (1953) The shapes of birds' eggs. The Auk 70, 160-
182.
Shi, P., Wang, L., Quinn, B.K., Gielis, J. (2023) A new program to estimate the parameters of Preston's equation, a general formula for describing the egg shape of birds. Symmetry 15, 231. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3390/sym15010231")}
Todd, P.H., Smart, I.H.M. (1984) The shape of birds' eggs. Journal of Theoretical Biology
106, 239-
243. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/0022-5193(84)90021-3")}
EPE
, lmPE
, TSE
zeta <- seq(0, 2*pi, len=2000)
Par1 <- c(10, 6, 0.325, -0.0415)
Res1 <- PE(P=Par1, zeta=zeta, simpver=1)
Par2 <- c(10, 6, -0.325, -0.0415)
Res2 <- PE(P=Par2, zeta=zeta, simpver=1)
dev.new()
plot(Res1$x, Res1$y, asp=1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
xlab=expression(italic(x)), ylab=expression(italic(y)))
lines(Res2$x, Res2$y, col=2)
dev.new()
plot(Res1$r, Res2$r, asp=1, cex.lab=1.5, cex.axis=1.5,
xlab=expression(paste(italic(r), ""[1], sep="")),
ylab=expression(paste(italic(r), ""[2], sep="")))
abline(0, 1, col=4)
graphics.off()
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.