biogeom (version 1.4.3)

TE: 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)

Value

The \(y\) values predicted by the Troscianko equation.

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.

Author

Peijian Shi , Johan Gielis , Brady K. Quinn .

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.

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. 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. tools:::Rd_expr_doi("10.1002/ece3.4412")

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308\(-\)313. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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()