TE {biogeom}

The Troscianko Equation (TE)

Description

TE is used to calculate y values at given x values using the re-expression of Troscianko's egg-shape equation, which was proposed by Biggins et al. (2018, 2022).

Usage

TE(P, x)

Arguments

P

the parameters of the Troscianko equation, including \alpha_{0}, \alpha_{1}, and \alpha_{2}.

x

the given x values ranging from -1 to 1.

Details

The Troscianko equation is recommended as (Biggins et al., 2022):

y = \exp\left(\alpha_{0}+\alpha_{1}\,x+\alpha_{2}\,x^2\right)\sqrt{1-x^2},

where x and y represent the abscissa and ordinate of an arbitrary point on the Troscianko curve; \alpha_{0}, \alpha_{1}, and \alpha_{2} are parameters to be estimated.

Value

The y values predicted by the Troscianko equation.

Note

Here, x and y in the Troscianko equation are actually equal to y/a and x/a, respectively, in the explicit Troscianko equation, where a represents half the egg length (See ETE for details). This means that the egg length is scaled to be 2, and the maximum egg width is correspondingly adjusted to keep the same scale.

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

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

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., 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

Troscianko, J. (2014). A simple tool for calculating egg shape, volume and surface area from digital images. Ibis, 156, 874-878. doi:10.1111/ibi.12177

See Also

fitETE, lmTE

Examples

  Par <- c(-0.377, -0.29, -0.16)
  xb1 <- seq(-1, 1, len=20000)
  yb1 <- TE(P=Par, x=xb1)
  xb2 <- seq(1, -1, len=20000)
  yb2 <- -TE(P=Par, x=xb2)

  dev.new()
  plot(xb1, yb1, asp=1, type="l", col=2, ylim=c(-1, 1), cex.lab=1.5, cex.axis=1.5, 
    xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(xb2, yb2, col=4)

  graphics.off()