PE {biogeom}

Calculation of the Abscissa, Ordinate and Distance From the Origin For an Arbitrary Point on the Preston Curve

Description

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.

Usage

PE(P, zeta, simpver = NULL)

Arguments

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.

Details

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.

Value

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.

Note

\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}.

Author(s)

Peijian Shi [e-mail], Johan Gielis [e-mail], Brady K. Quinn [e-mail].

References

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston's universal formula for avian egg shape. Ornithology 139, ukac028. 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. 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. doi:10.3390/sym15010231

Todd, P.H., Smart, I.H.M. (1984) The shape of birds' eggs. Journal of Theoretical Biology 106, 239-243. doi:10.1016/0022-5193(84)90021-3

See Also

EPE, lmPE, TSE

Examples

  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()